Cylindrical model structures
Richard Williamson
We build a model structure from the simple point of departure of a structured interval in a monoidal category --- more generally, a structured cylinder and a structured co-cylinder in a category.
cofibrations_and_fibrations.tex
\begin{section}{Cofibrations and fibrations} \label{CofibrationsAndFibrationsSection}

We introduce a notion of a cofibration and of a trivial cofibration with respect to a cylinder $\cylinder$ or a co-cylinder $\cocylinder$ in a formal category $\cl{A}$. We introduce a dual notion of a fibration and of a trivial fibration with respect to $\cocylinder$ or $\cylinder$. 

If $\cl{A}$ admits both a cylinder $\cylinder$ and a co-cylinder $\cocylinder$ and if $\cylinder$ is left adjoint to $\cocylinder$ we characterise fibrations with respect to $\cocylinder$ as exactly fibrations with respect to $\cylinder$, and characterise cofibrations with respect to $\cylinder$ as exactly cofibrations with respect to $\cocylinder$. 

Thus far everything is directly analogous to topology. In an abstract setting cofibrations, fibrations, and the afore-mentioned characterisations are discussed for example in \cite{KampsPorterAbstractHomotopyAndSimpleHomotopyTheory}. There are considerably older references --- see for example \cite{KampsKanBedingungenUndAbstrakteHomotopietheorie}. 

To equip a category $\cl{A}$ admitting both a cylinder $\cylinder$ and a co-cylinder $\cocylinder$ with a model structure we must go a little further. We introduce the notion of a normally cloven fibration with respect to $\cocylinder$ or $\cylinder$. This is a strengthening of the notion of a fibration. Roughly speaking, we impose two requirements upon the lifts of homotopies that define a fibration, that these lifts are compatible and that identity homotopies lift to identity homotopies. If $\cylinder$ is left adjoint to $\cocylinder$ we characterise normally cloven fibrations with respect to $\cocylinder$ as exactly normally cloven fibrations with respect to $\cylinder$. 

In addition we introduce the dual notion of a normally cloven cofibration with respect to $\cylinder$ or $\cocylinder$. If $\cylinder$ is left adjoint to $\cocylinder$ we characterise normally cloven cofibrations with respect to $\cylinder$ as exactly normally cloven cofibrations with respect to $\cocylinder$.

A composition of cofibrations is a cofibration and that a composition of fibrations is a fibration. Suppose that we have commutative diagrams \twosq{a_{2},a_{0},a_{3},a_{1},g_{0},j,j',g_{1},a_{0},a_{2},a_{1},a_{3},r_{0},j',j,r_{1}} in $\cl{A}$ such that $r_{0}$ is a retraction of $g_{0}$, and such that $r_{1}$ is a retraction of $g_{1}$. If $j$ is a cofibration with respect to $\cylinder$, then $j'$ is also a cofibration with respect to $\cylinder$. We also prove this for normally cloven cofibrations with respect to $\cylinder$. We obtain dual results for fibrations and normally cloven fibrations with respect to $\cocylinder$. 

In our abstract setting Garner and van den Berg recently arrived at the notion of a normally cloven fibration independently of the author --- see around Proposition 6.1.5 in \cite{VanDenBergGarnerTopologicalAndSimplicialModelsOfIdentityTypes}. In \ref{MappingCylindersAndMappingCoCylindersSection} we will present further ideas from this paper. 

In \ref{ExampleSection} we will construct a homotopy theory of categories by means of our abstract theory. A normally cloven fibration in our sense with respect to this homotopy theory is a normally cloven iso-fibration. This motivates our choice of terminology. 

\begin{assum} Let $\cl{C}$ be a 2-category with a final object such that pushouts and pullbacks of 2-arrows of $\cl{C}$ give rise to pushouts and pullbacks in formal categories in the sense of Definition \ref{PushoutsPullbacks2ArrowsArePushoutsPullbacksInFormalCategoriesTerminology}. Let $\cl{A}$ be an object of $\cl{C}$. As before we view $\cl{A}$ as a formal category, writing of objects and arrows of $\cl{A}$. \end{assum}

\begin{defn} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$. An arrow \ar{a_{0},a_{1},j} of $\cl{A}$ is a {\em cofibration} with respect to $\cylinder$ if for any commutative diagram \sq{a_{0},\cyl(a_{0}),a_{1},a_{2},{i_{0}(a_{0})},h,j,f} in $\cl{A}$ there is an arrow \ar{\cyl(a_{1}),a_{2},k} with respect to $\cylinder$ such that the following diagram in $\cl{A}$ commutes. \pushout[{4,3,-1}]{a_{0},\cyl(a_{0}),a_{1},\cyl(a_{1}),a_{2},{i_{0}(a_{0})},\cyl(j),j,{i_{0}(a_{1})},h,f,k} \end{defn}

\begin{defn} Let $\cocylinder = \big( \cocyl, e_0, e_1 \big)$ be a co-cylinder in $\cl{A}$. An arrow \ar{a_{0},a_{1},f} of $\cl{A}$ is a {\em fibration} with respect to $\cocylinder$ if $f^{op}$ is a cofibration with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$. \end{defn}

\begin{defn} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$. An arrow \ar{a_{0},a_{1},j} of $\cl{A}$ is a {\em trivial cofibration} with respect to $\cylinder$ if it is both a cofibration and a homotopy equivalence with respect to $\cylinder$. \end{defn}

\begin{defn} Let $\cocylinder = \big( \cocyl, e_0, e_1 \big)$ be a co-cylinder in $\cl{A}$. An arrow \ar{a_{0},a_{1},j} of $\cl{A}$ is a {\em trivial fibration} with respect to $\cocylinder$ if it is both a fibration and a homotopy equivalence with respect to $\cocylinder$. \end{defn}

\begin{defn} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$. An arrow \ar{a_{1},a_{2},f} of $\cl{A}$ is a {\em fibration} with respect to $\cylinder$ if for any commutative diagram \sq{a_{0},a_{1},\cyl(a_{0}),a_{2},g,f,i_{0}(a_{0}),h} in $\cl{A}$ there is a homotopy \ar{\cyl(a_{0}),a_{1},l} with respect to $\cylinder$ such that the following diagram in $\cl{A}$ commutes. \liftingsquare{a_{0},a_{1},\cyl(a_{0}),a_{2},g,f,i_{0}(a_{0}),h,l} \end{defn}

\begin{prpn} \label{FibrationCoCylinderIffFibrationCylinderProposition} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$, and let $\cocylinder = \big( \cocyl, e_0, e_1 \big)$ be a co-cylinder in $\cl{A}$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$. An arrow \ar{a_{1},a_{2},f} is a fibration with respect to $\cocylinder$ if and only if it is a fibration with respect to $\cylinder$. \end{prpn}

\begin{proof} We first prove that if $f$ is a fibration with respect to $\cocylinder$ then it is a fibration with respect to $\cylinder$. Suppose that we have a commutative diagram in $\cl{A}$ as follows. \sq{a_{0},a_{1},\cyl(a_{0}),a_{2},g,f,i_{0}(a_{0}),h} %
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Adopting the shorthand of Recollection \ref{AdjunctionRecollection} let \ar{{\mathsf{Hom}_{\cl{A}}\big(\cyl(-),- \big)},{\mathsf{Hom}_{\cl{A}}\big(-,\cocyl(-) \big)},\mathsf{adj}} denote the natural isomorphism which the adjunction between $\cyl$ and $\cocyl$ gives rise to. 

Since $\cylinder$ is left adjoint to $\cocylinder$ the following diagram in $\cl{A}$ commutes. \sq{a_{0},\cyl(a_{0}),\cocyl(a_{2}),a_{2},i_{0}(a_{0}),h,\mathsf{adj}(h),e_{0}(a_{2})} %
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By the commutativity of the last two diagrams the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{a_{0},a_{1},a_{2},\cocyl(a_{2}),a_{2},g,f,\mathsf{adj}(h),e_{0}(a_{2})} Thus there is a homotopy \ar{a_{0},\cocyl(a_{1}),k} with respect to $\cocylinder$ such that the following diagram in $\cl{A}$ commutes, since $f$ is a fibration with respect to $\cocylinder$. \pullback[{4,4,-1}]{\cocyl(a_{1}),a_{1},\cocyl(a_{2}),a_{2},a_{0},e_{0}(a_{1}),f,\cocyl(f),e_{0}(a_{2}),g,{\mathsf{adj}(h)},k} %
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By the naturality of $\mathsf{adj}$ we also have that the following diagram in $\cl{A}$ commutes. \tri[{4,3}]{a_{0},\cocyl(a_{1}),\cocyl(a_{2}),k,\cocyl(f),{\mathsf{adj}\Big(f \circ \mathsf{adj}^{-1}(k) \Big)}} We deduce that $\mathsf{adj}\big( f \circ \mathsf{adj}^{-1}(k) \big) = \mathsf{adj}(h)$ and hence that the following diagram in $\cl{A}$ commutes. \tri[{5,3}]{\cyl(a_{0}),a_{1},a_{2},{\mathsf{adj}^{-1}(k)},f,h} %
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Moreover since $\cylinder$ is left adjoint to $\cocylinder$ the following diagram in $\cl{A}$ commutes. \sq{a_{0},\cocyl(a_{1}),\cyl(a_{0}),a_{1},k,e_{0}(a_{1}),i_{0}(a_{0}),\mathsf{adj}^{-1}(k)} Hence by the commutativity of the diagram which defines $k$ the following diagram in $\cl{A}$ commutes. \triother[{5,3}]{a_{0},\cyl(a_{0}),a_{1},i_{0}(a_{0}),\mathsf{adj}^{-1}(k),g} %
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Putting this all together we have shown that the following diagram in $\cl{A}$ commutes, concluding this direction of the proof. \liftingsquare[{9,4}]{a_{0},a_{1},\cyl(a_{0}),a_{2},g,f,i_{0}(a_{0}),h,\mathsf{adj}^{-1}(k)} %#
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We now prove that if $f$ is a fibration with respect to $\cylinder$ then $f$ is a fibration with respect to $\cocylinder$. To this end suppose now that we have a commutative diagram in $\cl{A}$ as follows. \sq[{4,3}]{a_{0},a_{1},\cocyl(a_{2}),a_{2},g,f,h,e_{0}(a_{2})} %
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Adopting again the shorthand of Recollection \ref{AdjunctionRecollection} let \ar{{\mathsf{Hom}_{\cl{A}}\big(\cyl(-),- \big)},{\mathsf{Hom}_{\cl{A}}\big(-,\cocyl(-) \big)},\mathsf{adj}} denote the natural isomorphism which the adjunction between $\cyl$ and $\cocyl$ gives rise to. Since $\cylinder$ is left adjoint to $\cocylinder$ the following diagram in $\cl{A}$ commutes. \sq{a_{0},\cocyl(a_{2}),\cyl(a_{0}),a_{2},h,e_{0}(a_{2}),i_{0}(a_{0}),\mathsf{adj}^{-1}(h)} %
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By the commutativity of the last two diagrams the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{a_{0},a_{1},\cyl(a_{0}),a_{2},g,f,i_{0}(a_{0}),\mathsf{adj}^{-1}(h)} Thus there is a homotopy \ar{\cyl(a_{0}),a_{1},l} with respect to $\cylinder$ such that the following diagram in $\cl{A}$ commutes, since $f$ is a fibration with respect to $\cylinder$. \liftingsquare[{6,4}]{a_{0},a_{1},\cyl(a_{0}),a_{2},g,f,i_{0}(a_{0}),{\mathsf{adj}^{-1}(h)},l} %
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By the naturality of $\mathsf{adj}$ we also have that the following diagram in $\cl{A}$ commutes. \tri[{4,3}]{a_{0},\cocyl(a_{1}),\cocyl(a_{2}),{\mathsf{adj}(l)},\cocyl(f),{\mathsf{adj}\big(f \circ l \big)}} We deduce that the following diagram in $\cl{A}$ commutes. \tri[{5,3}]{a_{0},\cocyl(a_{1}),\cocyl(a_{2}),{\mathsf{adj}(l)},\cocyl(f),h} %
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Moreover since $\cylinder$ is left adjoint to $\cocylinder$ the following diagram in $\cl{A}$ commutes. \sq{a_{0},\cyl(a_{0}),\cocyl(a_{1}),a_{1},i_{0}(a_{0}),k,\mathsf{adj}(l),e_{0}(a_{1})} Hence by the commutativity of the diagram which defines $l$ the following diagram in $\cl{A}$ commutes. \triother[{4,3}]{a_{0},\cocyl(a_{1}),a_{1},\mathsf{adj}(l),e_{0}(a_{1}),g} %
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Putting this all together we have shown that the following diagram in $\cl{A}$ commutes concluding this direction of the proof. \pullback[{4,4,-1}]{\cocyl(a_{1}),a_{1},\cocyl(a_{2}),a_{2},a_{0},e_{0}(a_{1}),f,\cocyl(f),e_{0}(a_{2}),g,h,{\mathsf{adj}(l)}} \end{proof} 

\begin{defn} Let $\cocylinder = \big( \cocyl, i_0, i_1 \big)$ be a co-cylinder in $\cl{A}$. An arrow \ar{a_{0},a_{1},j} of $\cl{A}$ is a {\em cofibration} with respect to $\cocylinder$ if for any commutative diagram \sq{a_{0},\cocyl(a_{2}),a_{1},a_{3},h,e_{0}(a_{2}),j,g} in $\cl{A}$ there is a homotopy \ar{a_{1},\cocyl(a_{2}),l} with respect to $\cocylinder$ such that the following diagram in $\cl{A}$ commutes. \liftingsquare{a_{0},\cocyl(a_{2}),a_{1},a_{2},h,e_{0}(a_{2}),j,g,l} \end{defn}

\begin{rmk}  Let $\cocylinder = \big( \cocyl, i_0, i_1 \big)$ be a co-cylinder in $\cl{A}$. An arrow \ar{a_{0},a_{1},j} of $\cl{A}$ is a cofibration with respect to $\cocylinder$ if and only if $j^{op}$ is a fibration with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$. \end{rmk} 

\begin{cor} \label{CofibrationCoCylinderIffCofibrationCylinderCorollary} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$, and let $\cocylinder = \big( \cocyl, e_0, e_1 \big)$ be a co-cylinder in $\cl{A}$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$. 

An arrow \ar{a_{0},a_{1},j} is a cofibration with respect to $\cylinder$ if and only if it is a cofibration with respect to $\cocylinder$. \end{cor}

\begin{proof} Follows immediately from Proposition \ref{FibrationCoCylinderIffFibrationCylinderProposition} by duality. \end{proof}

\begin{prpn} \label{IdentityIsCofibrationProposition} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$. Then for any object $a$ of $\cl{A}$ the arrow \ar{a,a,id} of $\cl{A}$ is a cofibration with respect to $\cylinder$. \end{prpn} 

\begin{proof} Suppose that we have a commutative diagram in $\cl{A}$ as follows. \sq{a,\cyl(a),a,a',i_{0}(a),h,id,g} Then the following diagram in $\cl{A}$ commutes. \pushout[{4,3,-1}]{a,\cyl(a),a,\cyl(a),a',i_{0}(a),id,id,i_{0}(a),h,g,h} \end{proof} 

\begin{prpn} \label{CompositionOfCofibrationsIsACofibrationProposition} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$. Let \ar{a_{0},a_{1},j_{0}} and \ar{a_{1},a_{2},j_{1}} be arrows of $\cl{A}$ which are cofibrations with respect to $\cylinder$. Then $j_{1} \circ j_{0}$ is a cofibration with respect to $\cylinder$. 
\end{prpn} 

\begin{proof} Suppose that we have a commutative diagram in $\cl{A}$ as follows. \sq{a_{0},\cyl(a_{0}),a_{2},a_{3},i_{0}(a_{0}),h,j_{1} \circ j_{0},g} %
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Since $j_{0}$ is a cofibration with respect to $\cylinder$ there is an arrow \ar{\cyl(a_{1}),a_{3},k_{0}} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),a_{2},\cyl(a_{2}),a_{3},i_{0}(a_{0}),\cyl(j_{0}),j_{0},i_{0}(a_{1}),h,g \circ j_{1},k_{0}} %
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Since $j_{1}$ is a cofibration with respect to $\cylinder$ there is an arrow \ar{\cyl(a_{2}),a_{3},k_{1}} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout{a_{1},\cyl(a_{1}),a_{2},\cyl(a_{2}),a_{3},i_{0}(a_{1}),\cyl(j_{1}),j_{1},i_{0}(a_{2}),k_{0},g,k_{1}} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \pushout[{3.5,3,2}]{a_{0},\cyl(a_{0}),a_{2},\cyl(a_{2}),a_{3},i_{0}(a_{0}),\cyl(j_{1} \circ j_{0}),j_{1} \circ j_{0},i_{0}(a_{2}),h,g,k_{1}}  
\end{proof}

\begin{cor} \label{CompositionOfFibrationsIsAFibrationCorollary} Let $\cocylinder = \big( \cocyl, e_0, e_1 \big)$ be a co-cylinder in $\cl{A}$. Let \ar{a_{0},a_{1},f_{0}} and \ar{a_{1},a_{2},f_{1}} be arrows of $\cl{A}$ which are fibrations with respect to $\cocylinder$. Then $f_{1} \circ f_{0}$ is a fibration with respect to $\cocylinder$. \end{cor} 

\begin{proof} Follows immediately from Proposition \ref{CompositionOfCofibrationsIsACofibrationProposition} by duality. \end{proof}

\begin{prpn} \label{RetractionCofibrationIsCofibrationProposition} Let $\cylinder = \big( \cyl, i_{0}, i_{1} \big)$ be a cylinder in $\cl{A}$. Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$ which is a cofibration with respect to $\cylinder$. 

Suppose that we have commutative diagrams \twosq{a_{2},a_{0},a_{3},a_{1},g_{0},j,j',g_{1},a_{0},a_{2},a_{1},a_{3},r_{0},j',j,r_{1}} in $\cl{A}$ such that $r_{0}$ is a retraction of $g_{0}$ and such that $r_{1}$ is a retraction of $g_{1}$. Then $j'$ is a cofibration with respect to $\cylinder$. \end{prpn} 

\begin{proof} Suppose that we have a commutative diagram in $\cl{A}$ as follows. \sq{a_{2},\cyl(a_{2}),a_{3},a_{4},i_{0}(a_{2}),h,j',f} %
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Then the following diagram in $\cl{A}$ commutes. \trapeziumstwo[{3,3.5,0,0}]{a_{0},\cyl(a_{0}),a_{1},a_{2},\cyl(a_{2}),a_{3},a_{4},i_{0}(a_{0}),\cyl(r_{0}),h,j,r_{1},f,r_{0},i_{0}(a_{2}),j'} %
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Thus since $j$ is a cofibration with respect to $\cylinder$ there is an arrow \ar{\cyl(a_{1}),a_{4},k} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),a_{1},\cyl(a_{1}),a_{4},i_{0}(a_{0}),\cyl(j),j,i_{0}(a_{1}),h \circ \cyl(r_{0}),f \circ r_{1}, k} %
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Let \ar{\cyl(a_{3}),a_{4},l} denote the arrow $k \circ \cyl(g_{1})$ of $\cl{A}$. We claim that the following diagram in $\cl{A}$ commutes. \pushout{a_{2},\cyl(a_{2}),a_{3},\cyl(a_{3}),a_{4},i_{0}(a_{2}),\cyl(j'),j',i_{0}(a_{3}),h,f,l} % 
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Firstly we have that the following diagram in $\cl{A}$ commutes. \trapeziumsfive{a_{3},\cyl(a_{3}),a_{1},\cyl(a_{1}),a_{3},a_{4},i_{0}(a_{1}),l,id,f,g_{1},\cyl(g_{1}),i_{0}(a_{2}),r_{1},k} %
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Secondly we have that the following diagram in $\cl{A}$ commutes. \trapeziumsfive{\cyl(a_{2}),\cyl(a_{3}),\cyl(a_{0}),\cyl(a_{1}),\cyl(a_{3}),a_{4},\cyl(j'),l,id,h,\cyl(g_{0}),\cyl(g_{1}),\cyl(j),\cyl(r_{0}),k}  
    
\end{proof}

\begin{cor} \label{RetractionTrivialCofibrationIsTrivialCofibrationCorollary} Let $\cylinder = \big( \cyl, i_{0}, i_{1} \big)$ be a cylinder in $\cl{A}$. Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$ which is a trivial cofibration with respect to $\cylinder$. 

Suppose that we have commutative diagrams \twosq{a_{2},a_{0},a_{3},a_{1},g_{0},j,j',g_{1},a_{0},a_{2},a_{1},a_{3},r_{0},j',j,r_{1}} in $\cl{A}$ such that $r_{0}$ is a retraction of $g_{0}$, and such that $r_{1}$ is a retraction of $g_{1}$. Then $j'$ is a trivial cofibration with respect to $\cylinder$. \end{cor}

\begin{proof} Follows immediately from Proposition \ref{RetractionCofibrationIsCofibrationProposition} and Proposition \ref{RetractionHomotopyEquivalenceIsHomotopyEquivalenceProposition}. \end{proof}

\begin{cor} \label{RetractionFibrationIsFibrationCorollary} Let $\cocylinder = \big( \cocyl, e_{0}, e_{1} \big)$ be a co-cylinder in $\cl{A}$. Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$ which is a fibration with respect to $\cocylinder$. Suppose that we have commutative diagrams \twosq{a_{2},a_{0},a_{3},a_{1},g_{0},f,f',g_{1},a_{0},a_{2},a_{1},a_{3},r_{0},f',f,r_{1}} in $\cl{A}$ such that $r_{0}$ is a retraction of $g_{0}$ and such that $r_{1}$ is a retraction of $g_{1}$. Then $f'$ is a fibration with respect to $\cocylinder$. \end{cor} 

\begin{proof} Follows immediately from Proposition \ref{RetractionCofibrationIsCofibrationProposition} by duality. \end{proof} 

\begin{cor} \label{RetractionTrivialFibrationIsTrivialFibrationCorollary} Let $\cocylinder = \big( \cocyl, e_{0}, e_{1} \big)$ be a co-cylinder in $\cl{A}$. Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$ which is a trivial fibration with respect to $\cocylinder$. 

Suppose that we have commutative diagrams \twosq{a_{2},a_{0},a_{3},a_{1},g_{0},f,f',g_{1},a_{0},a_{2},a_{1},a_{3},r_{0},f',f,r_{1}} in $\cl{A}$ such that $r_{0}$ is a retraction of $g_{0}$ and such that $r_{1}$ is a retraction of $g_{1}$. Then $f'$ is a trivial fibration with respect to $\cocylinder$. \end{cor} 

\begin{proof} Follows immediately from Corollary \ref{RetractionFibrationIsFibrationCorollary} and Proposition \ref{RetractionHomotopyEquivalenceIsHomotopyEquivalenceProposition}. \end{proof} 

\begin{notn} Let $\cylinder = \big( \cyl,i_0,i_1 \big)$ be a cylinder in $\cl{A}$. Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$, and let $a_{2}$ be an object of $\cl{A}$. 

Let $\Sigma_{j,a_{2}}^{\cylinder}$ denote the set of pairs $(g,h)$ consisting of an arrow \ar{a_{1},a_{2},g} of $\cl{A}$, and a homotopy \ar{\cyl(a_{0}),a_{2},h} with respect to $\cylinder$, such that the following diagram in $\cl{A}$ commutes. \sq{a_{0},\cyl(a_{0}),a_{1},a_{2},i_{0}(a_{0}),h,j,g} 

Let $\Upsilon_{j,a_{2}}^{\cylinder}$ denote the set of homotopies \ar{\cyl(a_{1}),a_{2},k} with respect to $\cylinder$ such that the following diagram in $\cl{A}$ commutes. \pushout[{4,3,-1}]{a_{0},\cyl(a_{0}),a_{1},\cyl(a_{1}),a_{2},i_{0}(a_{0}),\cyl(j),j,i_{0}(a_{1}),h,g,k} \end{notn}

\begin{defn} Let $\cylinder = \big( \cyl,i_0,i_1 \big)$ be a cylinder in $\cl{A}$. A {\em cofibration equipped with a cleavage} with respect to $\cylinder$ is an arrow \ar{a_{0},a_{1},j} of $\cl{A}$ together with a map \ar{\Sigma_{j,a_{2}}^{\cylinder},\Upsilon_{j,a_{2}}^{\cylinder},k_{a_{2}}} for every object $a_{2}$ of $\cl{A}$. \end{defn}   

\begin{defn} \label{NormallyClovenCofibrationCylinderDefinition} Let $\cylinder = \big( \cyl,i_0,i_1,p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. A {\em normally cloven cofibration} with respect to $\cylinder$ is an arrow \ar{a_{0},a_{1},j} of $\cl{A}$ which is a cofibration equipped with a cleavage with respect to $\cylinder$ satisfying the following conditions, for which we denote by \ar{\Sigma_{j,a_{2}}^{\cylinder},\Upsilon_{j,a_{2}}^{\cylinder},k_{a_{2}}} the map of the cleavage corresponding to an object $a_{2}$ of $\cl{A}$. 

\begin{itemize}[itemsep=1em,topsep=1em] 

\item[(i)] Suppose that we have a commutative diagram in $\cl{A}$ as follows. \triother{a_{0},a_{1},a_{2},j,g_{1},g_{0}} %
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Then the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{1}),a_{1},a_{2},p(a_{1}),g_{1},{k_{a_{2}}\big(g_{1},g_{0} \circ p(a_{0})\big)}} 

\item[(ii)] Suppose that we have a commutative diagram in $\cl{A}$ as follows. \sq{a_{0},\cyl(a_{0}),a_{1},a_{2},i_{0}(a_{0}),h,j,g_{1}} Then for any arrow \ar{a_{2},a_{3},g_{2}} of $\cl{A}$, the following diagram in $\cl{A}$ commutes. \tri[{5,3}]{\cyl(a_{1}),a_{2},a_{3},{k_{a_{2}}(g_{1},h)},g_{2},{k_{a_{3}}(g_{2} \circ g_{1}, g_{2} \circ h)}} \end{itemize} \end{defn}

\begin{rmk} Let $\cylinder = \big( \cyl,i_0,i_1,p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. Let $j$ be an arrow of $\cl{A}$ which is a cofibration equipped with a cleavage with respect to $\cylinder$.

We will refer to condition (i) of Definition \ref{NormallyClovenCofibrationCylinderDefinition} as {\em lifting of identities}, and to condition (ii) of Definition \ref{NormallyClovenCofibrationCylinderDefinition} as {\em compatibility of liftings}. \end{rmk}

\begin{rmk} Let $\cylinder = \big( \cyl,i_0,i_1,p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. If an arrow \ar{a_{0},a_{1},j} of $\cl{A}$ is a cofibration with respect to $\cylinder$ we can think of $\cyl(a_{1})$ as a `weak pushout' of $j$ along the arrow \ar{a_{0},\cyl(a_{0}),i_{0}(a_{0})} of $\cl{A}$. 

The lifting of identities and compatibility of liftings conditions bring $\cyl(a_{1})$ closer to an actual pushout of $j$ along $i_{0}(a_{0})$. \end{rmk}

\begin{terminology} \label{NormallyClovenCofibrationTerminology} Let $\cylinder = \big( \cyl,i_0,i_1,p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. Let \ar{a_{0},a_{1},j} be a cofibration equipped with a cleavage with respect to $\cylinder$ which is moreover a normally cloven cofibration with respect to $\cylinder$. 

We will typically refer to $j$ as a normally cloven cofibration without explicitly mentioning its cleavage. \end{terminology}

\begin{notn} Let $\cocylinder = \big( \cocyl,e_0,e_1 \big)$ be a co-cylinder in $\cl{A}$. Let \ar{a_{1},a_{2},f} be an arrow of $\cl{A}$ and let $a_{0}$ be an object of $\cl{A}$. 

Let $\Sigma_{f,a_{0}}^{\cocylinder}$ denote the set of pairs $(g,h)$ consisting of an arrow \ar{a_{0},a_{1},g} of $\cl{A}$, and a homotopy \ar{a_{0},\cocyl(a_{2}),h} with respect to $\cocylinder$, such that the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{a_{0},a_{1},\cocyl(a_{2}),a_{2},g,f,h,e_{0}(a_{2})} 

Let $\Upsilon_{f,a_{0}}^{\cocylinder}$ denote the set of homotopies \ar{a_{0},\cocyl(a_{1}),k} with respect to $\cocylinder$ such that the following diagram in $\cl{A}$ commutes. \pullback[{4,3,-1}]{\cocyl(a_{1}),a_{1},\cocyl(a_{2}),a_{2},a_{0},e_{0}(a_{1}),f,\cocyl(f),e_{0}(a_{2}),g,h,k} \end{notn}

\begin{defn} Let $\cocylinder = \big( \cocyl,e_0,e_1 \big)$ be a co-cylinder in $\cl{A}$. A {\em fibration equipped with a cleavage} with respect to $\cocylinder$ is an arrow \ar{a_{1},a_{2},f} of $\cl{A}$ together with a map \ar{\Sigma_{f,a_{0}}^{\cocylinder},\Upsilon_{f,a_{0}}^{\cocylinder},k_{a_{0}}} for every object $a_{0}$ of $\cl{A}$. \end{defn}   

\begin{rmk} \label{CleavageDualityCofibCylinderFibCoCylinderRemark} Let $\cocylinder = \big( \cocyl,e_0,e_1 \big)$ be a co-cylinder in $\cl{A}$, and let \ar{a_{1},a_{2},f} be a fibration equipped with a cleavage with respect to $\cocylinder$.  Let $a_{0}$ be an object of $\cl{A}$, and let \ar{\Sigma_{f,a_{0}}^{\cocylinder},\Upsilon_{f,a_{0}}^{\cocylinder},k_{a_{0}}} denote the corresponding map of the cleavage. 

Associating to a pair $(g^{op},h^{op})$ in $\Sigma_{f^{op},a_{0}}^{\cocylinder^{op}}$ the arrow $\big(k_{a_{0}}(g,h)\big)^{op}$ of $\cl{A}^{op}$ defines a map \ar{\Sigma_{f^{op},a_{0}}^{\cocylinder^{op}},{\Upsilon_{f^{op},a_{0}}^{\cocylinder^{op}}}} which we will denote by $(k^{op})_{a_{0}}$. Thus a cleavage with respect to $f$ and $\cocylinder$ gives rise to a cleavage with respect to $f^{op}$ and the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$. \end{rmk} 

\begin{defn} \label{NormallyClovenFibrationCoCylinderDefinition} Let $\cocylinder = \big( \cocyl,e_0,e_1,c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. A {\em normally cloven fibration} with respect to $\cocylinder$ is an arrow \ar{a_{1},a_{2},f} of $\cl{A}$ which is a fibration equipped with a cleavage with respect to $\cocylinder$ such that $f^{op}$ equipped with the cleavage of Remark \ref{CleavageDualityCofibCylinderFibCoCylinderRemark} is a normally cloven cofibration with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$ equipped with the contraction structure $c^{op}$. \end{defn}

\begin{terminology} \label{NormallyClovenFibrationCoCylinderTerminology} Let $\cocylinder = \big( \cocyl,e_0,e_1,c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Let \ar{a_{1},a_{2},f} be a fibration equipped with a cleavage with respect to $\cocylinder$ which is moreover a normally cloven fibration with respect to $\cocylinder$. 

We will typically refer to $f$ as a normally cloven fibration without explicitly mentioning its cleavage. \end{terminology}

\begin{defn} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. An arrow \ar{a_{0},a_{1},j} of $\cl{A}$ is a {\em trivial normally cloven cofibration} with respect to $\cylinder$ if it is both a normally cloven cofibration and a homotopy equivalence with respect to $\cylinder$. \end{defn}

\begin{defn} Let $\cocylinder = \big( \cocyl, e_0, e_1, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. An arrow \ar{a_{0},a_{1},f} of $\cl{A}$ is a {\em trivial normally cloven fibration} with respect to $\cocylinder$ if it is both a normally cloven fibration and a homotopy equivalence with respect to $\cocylinder$. \end{defn}

\begin{notn} \label{CleavageFibrationCylinderNotation} Let $\cylinder = \big( \cyl,i_0,i_1 \big)$ be a cylinder in $\cl{A}$. Let \ar{a_{1},a_{2},f} be an arrow of $\cl{A}$, and let $a_{0}$ be an object of $\cl{A}$. 

Let $\Delta_{f,a_{0}}^{\cylinder}$ denote the set of pairs $(g,h)$ consisting of an arrow \ar{a_{0},a_{1},g} of $\cl{A}$ and a homotopy \ar{\cyl(a_{0}),a_{1},h} with respect to $\cylinder$, such that the following diagram in $\cl{A}$ commutes. \sq{a_{0},a_{1},\cyl(a_{0}),a_{2},g,f,i_{0}(a_{0}),h} 

Let $\Omega_{f,a_{0}}^{\cylinder}$ denote the set of homotopies \ar{\cyl(a_{0}),a_{1},l} with respect to $\cylinder$ such that the following diagram in $\cl{A}$ commutes. \liftingsquare{a_{0},a_{1},\cyl(a_{0}),a_{2},g,f,i_{0}(a_{0}),h,l} \end{notn}

\begin{defn} Let $\cylinder = \big( \cyl,i_0,i_1 \big)$ be a cylinder in $\cl{A}$. A {\em fibration equipped with a cleavage} with respect to $\cylinder$ is an arrow \ar{a_{1},a_{2},f} of $\cl{A}$ together with a map \ar{\Delta_{f,a_{0}}^{\cylinder},\Omega_{f,a_{0}}^{\cylinder},l_{a_{0}}} for every object $a_{0}$ of $\cl{A}$. \end{defn}   

\begin{defn} \label{NormallyClovenFibrationCylinderDefinition} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. A {\em normally cloven fibration} with respect to $\cylinder$ is an arrow \ar{a_{1},a_{2},f} of $\cl{A}$ which is a fibration equipped with a cleavage with respect to $\cylinder$ satisfying the following conditions, for which we denote by \ar{\Delta_{f,a_{0}}^{\cylinder},\Omega_{f,a_{0}}^{\cylinder},l_{a_{0}}} the map of the cleavage corresponding to an object $a_{0}$ of $\cl{A}$.  

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] Suppose that we have a commutative diagram in $\cl{A}$ as follows. \triother{a_{0},a_{1},a_{2},g_{1},f,g_{2}} Then the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{0}),a_{0},a_{1},p(a_{0}),g_{1},{l_{a_{0}}(g_{1},g_{2} \circ p(a_{0}))}} 

\item[(ii)] Suppose that we have a commutative diagram in $\cl{A}$ as follows. \sq{a_{0},a_{1},\cyl(a_{0}),a_{2},g_{1},f,i_{0}(a_{0}),h} Then for any arrow \ar{a_{-1},a_{0},g_{0}} of $\cl{A}$ the following diagram in $\cl{A}$ commutes. \tri[{4,3}]{\cyl(a_{-1}),\cyl(a_{0}),a_{1},\cyl(g_{0}),{l_{a_{0}}(g_{1},h)},{l_{a_{-1}}\big(g_{1} \circ g_{0}, h \circ \cyl(g_{0})\big)}} 

\end{itemize} 

\end{defn}

\begin{terminology} Let $\cylinder = \big( \cyl,i_0,i_1,p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $c$. Let \ar{a_{1},a_{2},f} be a fibration equipped with a cleavage with respect to $\cylinder$ which is moreover a normally cloven fibration with respect to $\cylinder$. 

We will typically refer to $f$ as a normally cloven fibration without explicitly mentioning its cleavage. \end{terminology}

\begin{notn} Let $\cocylinder = \big( \cocyl,e_0,e_1 \big)$ be a co-cylinder in $\cl{A}$. Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$, and let $a_{2}$ be an object of $\cl{A}$. 

Let $\Delta_{j,a_{2}}^{\cocylinder}$ denote the set of pairs $(g,h)$ consisting of an arrow \ar{a_{1},a_{2},g} of $\cl{A}$ and a homotopy \ar{a_{0},\cocyl(a_{2}),h} with respect to $\cocylinder$ such that the following diagram in $\cl{A}$ commutes. \sq{a_{0},\cocyl(a_{2}),a_{1},a_{2},h,e_{0}(a_{2}),j,g} 

Let $\Omega_{j,a_{2}}^{\cocylinder}$ denote the set of homotopies \ar{a_{1},\cocyl(a_{2}),l} with respect to $\cocylinder$ such that the following diagram in $\cl{A}$ commutes. \liftingsquare{a_{0},\cocyl(a_{2}),a_{1},a_{2},h,e_{0}(a_{2}),j,l} \end{notn}

\begin{defn} Let $\cocylinder = \big( \cocyl,e_0,e_1 \big)$ be a co-cylinder in $\cl{A}$. A {\em cofibration equipped with a cleavage} with respect to $\cocylinder$ is an arrow \ar{a_{0},a_{1},j} of $\cl{A}$ together with a map \ar{\Delta_{j,a_{2}}^{\cocylinder},\Omega_{j,a_{2}}^{\cocylinder},l_{a_{2}}} for every object $a_{2}$ of $\cl{A}$. \end{defn}  

\begin{rmk} \label{CleavageDualityFibCylinderCofibCoCylinderRemark} Let $\cocylinder = \big( \cocyl,e_0,e_1 \big)$ be a co-cylinder in $\cl{A}$, and let \ar{a_{0},a_{1},j} be a cofibration equipped with a cleavage with respect to $\cocylinder$.  Let $a_{2}$ be an object of $\cl{A}$, and let \ar{\Delta_{j,a_{2}}^{\cocylinder},\Omega_{j,a_{2}}^{\cocylinder},l_{a_{2}}} denote the corresponding map of the cleavage. 

Associating to a pair $(g^{op},h^{op})$ in $\Delta_{j^{op},a_{2}}^{\cocylinder^{op}}$ the arrow $\big(l_{a_{2}}(g,h)\big)^{op}$ of $\cl{A}^{op}$ defines a map \ar{\Delta_{j^{op},a_{2}}^{\cocylinder^{op}},{\Omega_{j^{op},a_{2}}^{\cocylinder^{op}},}} which we will denote by $(l^{op})_{a_{2}}$. Thus a cleavage with respect to $j$ and $\cocylinder$ gives rise to a cleavage with respect to $j^{op}$ and the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$. \end{rmk}

\begin{defn} Let $\cocylinder = \big( \cocyl,e_0,e_1,c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. A {\em normally cloven cofibration} with respect to $\cocylinder$ is an arrow \ar{a_{0},a_{1},j} of $\cl{A}$ which is a cofibration equipped with a cleavage with respect to $\cocylinder$ such that $j^{op}$ equipped with the cleavage of Remark \ref{CleavageDualityFibCylinderCofibCoCylinderRemark} is a normally cloven fibration with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$ equipped with the contraction structure $c^{op}$. \end{defn}

\begin{terminology} Let $\cocylinder = \big( \cocyl,i_0,i_1,c \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $c$. Let \ar{a_{0},a_{1},j} be a cofibration equipped with a cleavage with respect to $\cocylinder$ which is moreover a normally cloven cofibration with respect to $\cocylinder$. 

We will typically refer to $j$ as a normally cloven fibration without explicitly mentioning its cleavage. \end{terminology}

\begin{prpn} \label{NormallyClovenFibrationCoCylinderIffNormallyClovenFibrationCylinderProposition} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, and let $\cocylinder = \big( \cocyl, e_0, e_1, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. 

Suppose that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. Then an arrow \ar{a_{1},a_{2},f} of $\cl{A}$ is a normally cloven fibration with respect to $\cocylinder$ if and only if it is a normally cloven fibration with respect to $\cylinder$. \end{prpn}

\begin{proof} We first prove that if $f$ is a normally cloven fibration with respect to $\cocylinder$, then it is a normally cloven fibration with respect to $\cylinder$. Adopting the shorthand of Recollection \ref{AdjunctionRecollection}, let \ar{{\mathsf{Hom}_{\cl{A}}\big(\cyl(-),- \big)},{\mathsf{Hom}_{\cl{A}}\big(-,\cocyl(-) \big)},\mathsf{adj}} denote the natural isomorphism which the adjunction between $\cyl$ and $\cocyl$ gives rise to. Suppose that we have a commutative diagram in $\cl{A}$ as follows. \sq{a,a_{1},\cyl(a),a_{2},g_{1},f,i_{0}(a),h} %
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As in the proof of Proposition \ref{FibrationCoCylinderIffFibrationCylinderProposition} we have that the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{a,a_{1},a_{2},\cocyl(a_{2}),g_{1},f,\mathsf{adj}(h),e_{0}(a_{2})} %
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Let \ar{\Sigma_{f,a}^{\cocylinder},\Upsilon_{f,a}^{\cylinder},k_{a}} denote the map of the cleavage with which $f$ is equipped corresponding to the object $a$ of $\cl{A}$. Then the homotopy \ar[4]{a,\cocyl(a_{1}),k_{a}(g_{1}),\mathsf{adj}(h))} with respect to $\cocylinder$ fits into a commutative diagram in $\cl{A}$ as follows. \pullback[{4,4,-1}]{\cocyl(a_{1}),a_{1},\cocyl(a_{2}),a_{2},a,e_{0}(a_{1}),f,\cocyl(f),e_{0}(a_{2}),g_{1},{\mathsf{adj}(h)},{k_{a}\big(g_{1},\mathsf{adj}(h)\big)}} %
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Let $l_{a}(g_{1},h)$ denote the arrow \ar[9]{\cyl(a),a_{1},{\mathsf{adj}^{-1}(k_{a}(g_{1},h))}} of $\cl{A}$. Following again the proof of Proposition \ref{FibrationCoCylinderIffFibrationCylinderProposition} we have that the following diagram in $\cl{A}$ commutes. \liftingsquare[{9,3}]{a,a_{1},\cyl(a),a_{2},g_{1},f,i_{0}(a),h,{l_{a}(g_{1},h)}} %
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We claim that the cleavage given by the maps \ar{\Delta_{f,a}^{\cylinder},\Omega_{f,a}^{\cylinder},l_{a}} defined by $(g_{1},h) \mapsto l_{a}(g_{1},h)$ for an object $a$ of $\cl{A}$ equips $f$ with the structure of a normally cloven fibration with respect to $\cylinder$. 

Indeed suppose that we have a commutative triangle in $\cl{A}$ as follows. \tri{\cyl(a_{0}),a_{0},a_{2},p(a_{0}),g_{2},h} Then \[ \mathsf{adj}(h) = \mathsf{adj}\big( g_{2} \circ p(a_{0}) \big). \] %
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Moreover the following diagram in $\cl{A}$ commutes since the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. \tri{a_{0},a_{2},\cocyl(a_{2}),g_{2},c(a_{2}),\mathsf{adj}\big(g_{2} \circ p(a_{0})\big)} %
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We deduce that the following diagram in $\cl{A}$ commutes. \tri{a_{0},a_{2},\cocyl(a_{2}),g_{2},c(a_{2}),\mathsf{adj}(h)} Hence the following diagram in $\cl{A}$ commutes since $f$ satisfies the lifting of identities condition of Definition \ref{NormallyClovenFibrationCoCylinderDefinition}. \tri{a_{0},a_{1},\cocyl(a_{1}),g_{1},c(a_{1}),{k_{a_{0}}\big(g_{1},\mathsf{adj}(h)\big)}} %
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Thus we have that \[ l_{a_{0}}(g_{1},h)  = \mathsf{adj}^{-1}\big( c(a_{1}) \circ g_{1} \big). \] %
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Moreover the following diagram in $\cl{A}$ commutes since the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. \tri{\cyl(a_{0}),a_{0},a_{1},p(a_{0}), g_{1}, \mathsf{adj}^{-1}(c(a_{1}) \circ g_{1})} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{0}),a_{0},a_{1},p(a_{0}),g_{1},{l_{a_{0}}(g_{1},h)}} %
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This proves that $f$ satisfies the lifting of identities condition of Definition \ref{NormallyClovenFibrationCylinderDefinition}. 

Instead let $h$ be arbitrary and let \ar{a_{-1},a_{0},g_{0}} be an arrow of $\cl{A}$. The following diagram in $\cl{A}$ commutes since $f$ satisfies the compatibility of lifts condition of Definition \ref{NormallyClovenFibrationCoCylinderDefinition}. \tri{a_{-1},a_{0},\cocyl(a_{1}),g_{0},{k_{a_{0}}\big(g_{1},\mathsf{adj}(h)\big)}, {k_{a_{-1}}\big(g_{1} \circ g_{0}, \mathsf{adj}(h) \circ g_{0}\big)}} %
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We also have that the following diagram in $\cl{A}$ commutes by the naturality of $\mathsf{adj}$. \tri{a_{-1},a_{0},\cocyl(a_{2}),g_{0},\mathsf{adj}(h),\mathsf{adj}\big(h \circ \cyl(g_{0}) \big)} %
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Putting the last two observations together, we have that \[ l_{a_{-1}}\big(g_{1} \circ g_{0}, h \circ \cyl(g_{0})\big) = \mathsf{adj}^{-1}\Big(k_{a_{0}}\big(g_{1},\mathsf{adj}(h)\big) \circ g_{0}\Big). \] %
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Moreover the following diagram in $\cl{A}$ commutes by the naturality of $\mathsf{adj}^{-1}$. \tri[{4,3}]{\cyl(a_{-1}),\cyl(a_{0}),a_{1},\cyl(g_{0}),{\mathsf{adj}^{-1}\Big(k_{a_{0}}\big(g_{1},\mathsf{adj}(h)\big) \Big)},{\mathsf{adj}^{-1}\Big(k_{a_{0}}\big(g_{1},\mathsf{adj}(h)\big) \circ g_{0}\Big)}} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \tri[{4,3}]{\cyl(a_{-1}),\cyl(a_{0}),a_{1},\cyl(g_{0}),{l_{a_{0}}(g_{1},h)},{l_{a_{-1}}\big(g_{1} \circ g_{0}, h \circ \cyl(g_{0}) \big)}} %
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This proves that $f$ satisfies the compatibility of liftings condition of Definition \ref{NormallyClovenFibrationCylinderDefinition} and concludes this direction of the proof. 

We now prove that if $f$ is a normally cloven fibration with respect to $\cylinder$ then $f$ is a normally cloven fibration with respect to $\cocylinder$. Suppose that we have a commutative diagram in $\cl{A}$ as follows. \sq{a,a_{1},\cocyl(a_{2}),a_{2},g_{1},f,h,e_{0}(a_{2})}%
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As in the proof of Proposition \ref{FibrationCoCylinderIffFibrationCylinderProposition} we have that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{a,a_{1},\cyl(a),a_{2},g_{1},f,i_{0}(a_{0}),\mathsf{adj}^{-1}(h)} %
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Let \ar{\Delta_{f,a}^{\cylinder},\Omega_{f,a}^{\cylinder},l_{a}} denote the map of the cleavage with which $f$ is equipped corresponding to the object $a$ of $\cl{A}$. Then the homotopy \ar[8]{\cyl(a),a_{1},{l_{a}\big(g_{1},\mathsf{adj}^{-1}(h)\big)}} with respect to $\cylinder$ fits into a commutative diagram in $\cl{A}$ as follows. \liftingsquare[{15,4}]{a,a_{1},\cyl(a),a_{2},g_{1},f,i_{0}(a),\mathsf{adj}^{-1}(h),{l_{a}\big(g_{1},\mathsf{adj}^{-1}(h)\big)}} %
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Let $k_{a}(g_{1},h)$ denote the arrow \ar[11]{a,\cocyl(a_{1}),{\mathsf{adj}\Big(l_{a}\big(g_{1},\mathsf{adj}^{-1}(h)\big)\Big)}} of $\cl{A}$. We claim that the cleavage defined by the maps \ar{\Sigma_{f,a}^{\cocylinder},\Upsilon_{f,a}^{\cocylinder},k_{a}} defined by $(g_{1},h) \mapsto k_{a}(g_{1},h)$ for an object $a$ of $\cl{A}$ equips $f$ with the structure of a normally cloven fibration with respect to $\cocylinder$. 

Indeed suppose that we have a commutative triangle in $\cl{A}$ as follows. \tri{a_{0},a_{2},\cocyl(a_{2}),g_{2},c(a_{2}),h} %
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Then we have \[ \mathsf{adj}^{-1}(h) = \mathsf{adj}^{-1}\big( c(a_{2}) \circ g_{2} \big). \] %
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Moreover the following diagram in $\cl{A}$ commutes since the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. \tri{\cyl(a_{0}),a_{0},a_{2},p(a_{0}),g_{2},\mathsf{adj}^{-1}\big(c(a_{2}) \circ g_{2}\big)} %
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We deduce that the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{0}),a_{0},a_{2},p(a_{0}),g_{2},\mathsf{adj}^{-1}(h)} %
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Hence the following diagram in $\cl{A}$ commutes since $f$ satisfies the lifting of identities condition of Definition \ref{NormallyClovenFibrationCylinderDefinition}. \tri{\cyl(a_{0}),a_{0},a_{1},p(a_{0}),g_{1},{l_{a_{0}}\big(g_{1},\mathsf{adj}^{-1}(h)\big)}} %
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Thus we have that \[ k_{a_{0}}(g_{1},h)  = \mathsf{adj}\big( g_{1} \circ p(a_{0})\big). \] %
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Moreover the following diagram in $\cl{A}$ commutes since the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. \tri{a_{0},a_{1},\cocyl(a_{1}),g_{1},c(a_{1}),\mathsf{adj}\big(g_{1} \circ p(a_{0})\big)} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \tri{a_{0},a_{1},\cocyl(a_{1}),g_{1},c(a_{1}),{k_{a_{0}}(g_{1},h)}} %
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This proves that $f$ satisfies the lifting of identities condition of Definition \ref{NormallyClovenFibrationCoCylinderDefinition}. 

Instead let $h$ be arbitrary and let \ar{a_{-1},a_{0},g_{0}} be an arrow of $\cl{A}$. The following diagram in $\cl{A}$ commutes since $f$ satisfies the compatibility of lifts condition of Definition \ref{NormallyClovenFibrationCylinderDefinition}. \tri[{4,3}]{\cyl(a_{-1}),\cyl(a_{0}),a_{1},\cyl(g_{0}),{l_{a_{0}}\big(g_{1},\mathsf{adj}^{-1}(h)\big)}, {l_{a_{-1}}\big(g_{1} \circ g_{0}, \mathsf{adj}^{-1}(h) \circ \cyl(g_{0})\big)}} %
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We also have that the following diagram in $\cl{A}$ commutes by the naturality of $\mathsf{adj}^{-1}$. \tri[{4,3}]{\cyl(a_{-1}),\cyl(a_{0}),a_{2},\cyl(g_{0}),\mathsf{adj}^{-1}(h),\mathsf{adj}^{-1}(h \circ g_{0})} %
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Putting the last two observations together we have that \[ k_{a_{-1}}\big(g_{1} \circ g_{0}, h \circ g_{0}\big) = \mathsf{adj}\Big(l_{a_{0}}\big(g_{1},\mathsf{adj}^{-1}(h)\big) \circ \cyl(g_{0}) \Big). \] %
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Moreover the following diagram in $\cl{A}$ commutes by the naturality of $\mathsf{adj}$. \tri{a_{-1},a_{0},\cocyl(a_{1}),g_{0},{\mathsf{adj}\Big(l_{a_{0}}\big(g_{1},\mathsf{adj}^{-1}(h)\big) \Big)},{\mathsf{adj}\Big(l_{a_{0}}\big(g_{1},\mathsf{adj}^{-1}(h)\big) \circ \cyl(g_{0}) \Big)}} %
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Thus the following diagram in $\cl{A}$ commutes. \tri{a_{-1},a_{0},\cocyl(a_{1}),g_{0},{k_{a_{0}}(g_{1},h)},{k_{a_{-1}}\big(g_{1} \circ g_{0}, h \circ g_{0} \big)}} %
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This proves that $f$ satisfies the compatibility of liftings condition of Definition \ref{NormallyClovenFibrationCoCylinderDefinition} and concludes this direction of the proof. \end{proof}

\begin{cor} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, and let $\cocylinder = \big( \cocyl, e_0, e_1, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

An arrow \ar{a_{0},a_{1},j} is a normally cloven cofibration with respect to $\cylinder$ if and only if it is a normally cloven cofibration with respect to $\cocylinder$. \end{cor}

\begin{proof} Follows immediately from Proposition \ref{NormallyClovenFibrationCoCylinderIffNormallyClovenFibrationCylinderProposition} by duality. \end{proof}

\begin{prpn} \label{RetractionNormallyClovenCofibrationIsNormallyClovenCofibrationProposition} Let $\cylinder = \big( \cyl, i_{0}, i_{1},p  \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$ which is a normally cloven cofibration with respect to $\cylinder$. 

Suppose that we have commutative diagrams \twosq{a_{2},a_{0},a_{3},a_{1},g_{0},j,j',g_{1},a_{0},a_{2},a_{1},a_{3},r_{0},j',j,r_{1}} in $\cl{A}$ such that $r_{0}$ is a retraction of $g_{0}$, and such that $r_{1}$ is a retraction of $g_{1}$. Then $j'$ is a normally cloven cofibration with respect to $\cylinder$. \end{prpn} 

\begin{proof} For any object $a_{4}$ of $\cl{A}$ let \ar{\Sigma^{\cylinder}_{j,a_{4}},\Upsilon^{\cylinder}_{j,a_{4}},k_{a_{4}}} denote the corresponding map of the cleavage with which $j$ is equipped. Suppose that we have a commutative diagram in $\cl{A}$ as follows. \sq{a_{2},\cyl(a_{2}),a_{3},a_{4},i_{0}(a_{2}),h,j',f} %
%
As in the proof of Proposition \ref{RetractionCofibrationIsCofibrationProposition} we have that the following diagram in $\cl{A}$ commutes. \sq{a_{0},\cyl(a_{0}),a_{1},a_{4},i_{0}(a_{0}),h \circ \cyl(r_{0}),j,f \circ r_{1}} %
%
Let us define a map \ar{\Sigma^{\cylinder}_{j',a_{4}},\Upsilon^{\cylinder}_{j',a_{4}},k'_{a_{4}}} by 

\begin{diagram} 

$(f,h)  \mapsto k_{a_{4}}\big(f \circ r_{1}, h \circ \cyl(r_{0}) \big) \circ \cyl(g_{1})$. 

\end{diagram} %
%
We claim that the maps $k'_{a}$ for $a$ an object of $\cl{A}$ define a cleavage exhibiting $j'$ to be a normally cloven cofibration with respect to $\cylinder$. 

After the proof of Proposition \ref{RetractionCofibrationIsCofibrationProposition} it remains to demonstrate that conditions (i) and (ii) of Definition \ref{NormallyClovenCofibrationCylinderDefinition} are satisfied. Let us first show that condition (i) holds. 

Suppose that we have a commutative diagram in $\cl{A}$ as follows. \triother{a_{2},a_{3},a_{4},j',f_{1},f_{0}} %
%
Then the following diagram in $\cl{A}$ commutes. \squareabovetriangle{a_{0},a_{1},a_{2},a_{3},a_{4},j,r_{1},r_{0},j',f_{1},f_{0}} %
%
Since the cleavage defined by the maps $k_{a}$ for $a$ an object of $\cl{A}$ exhibits $j$ to be a normally cloven cofibration with respect to $\cylinder$ it satisfies condition (i) of Definition \ref{NormallyClovenCofibrationCylinderDefinition}. We deduce that the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{1}),a_{1},a_{4},p(a_{1}),f_{1} \circ r_{1},{k_{a_{4}}\big( f_{1} \circ r_{1}, f_{0} \circ r_{0} \circ p(a_{0})\big)}} %
%
Thus by the commutativity of the diagram \sq{\cyl(a_{0}),a_{0},\cyl(a_{2}),a_{2},p(a_{0}),r_{0},\cyl(r_{0}),p(a_{2})} in $\cl{A}$ we have that the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{1}),a_{1},a_{4},p(a_{1}),f_{1} \circ r_{1},{k_{a_{4}}\big( f_{1} \circ r_{1}, f_{0} \circ p(a_{2}) \circ \cyl(r_{0}) \big)}} %
%
Moreover the following diagram in $\cl{A}$ commutes. \squareabovetriangle{\cyl(a_{3}),\cyl(a_{1}),a_{3}, a_{1}, a_{3},\cyl(g_{1}),p(a_{1}),p(a_{3}),g_{1},r_{1},id} %
%
Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \threetrianglestwo[{6,3,1}]{\cyl(a_{3}),\cyl(a_{1}),a_{3},a_{4},p(a_{3}),\cyl(g_{1}), r_{1} \circ p(a_{1}),f_{1},{k_{a_{4}}\big( f_{1} \circ r_{1},f_{0} \circ p(a_{2}) \circ \cyl(r_{0}) \big)}} %
%
Thus the following diagram in $\cl{A}$ commutes, as required. \tri{\cyl(a_{3}),a_{3},a_{4},p(a_{3}),f_{1},{k'_{a_{4}}\big(f_{1},f_{0} \circ p(a_{2})\big)}} 

Let us now prove that condition (ii) of Definition \ref{NormallyClovenCofibrationCylinderDefinition} holds. Let \ar{a_{4},a_{5},f'} be an arrow of $\cl{A}$. Since the cleavage defined by the maps $k_{a}$ for $a$ an object of $\cl{A}$ exhibits $j$ to be a normally cloven cofibration with respect to $\cylinder$ it satisfies condition (ii) of Definition \ref{NormallyClovenCofibrationCylinderDefinition}. We deduce that the following diagram in $\cl{A}$ commutes. \tri[{10,3}]{\cyl(a_{1}),a_{4},a_{5},{k_{a_{4}}\big(f \circ r_{1},h \circ \cyl(r_{0})\big)},f',{k_{a_{5}}\big( f' \circ f \circ r_{1},  f' \circ h \circ \cyl(r_{0}) \big)}} %
%
By definition of $k'_{a_{4}}(f,h)$ and $k'_{a_{5}}(f' \circ f, f' \circ h)$ the diagrams \tri{\cyl(a_{3}),a_{4},\cyl(a_{1}),\cyl(g_{1}),{k_{a_{4}}\big(f \circ r_{1}, h \circ \cyl(r_{0})\big)},{k_{a_{4}}'(f,h)}} and \tri{\cyl(a_{3}),\cyl(a_{1}),a_{5}, \cyl(g_{1}),{k_{a_{5}}\big(f' \circ f \circ r_{1},f' \circ h \circ \cyl(r_{0}) \big)},{k_{a_{5}}'(f' \circ f, f'\circ h)}} in $\cl{A}$ commute. %
%
Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes, as required. \tri[{4,3}]{\cyl(a_{3}),a_{4},a_{5},{k'_{a_{4}}(f,h)},f',{k'_{a_{5}}(f' \circ f, f' \circ h)}} 

\end{proof}

\begin{cor} \label{RetractionTrivialNormallyClovenCofibrationIsTrivialNormallyClovenCofibrationCorollary} Let $\cylinder = \big( \cyl, i_{0}, i_{1},p  \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$ which is a trivial normally cloven cofibration with respect to $\cylinder$. 

Suppose that we have commutative diagrams \twosq{a_{2},a_{0},a_{3},a_{1},g_{0},j,j',g_{1},a_{0},a_{2},a_{1},a_{3},r_{0},j',j,r_{1}} in $\cl{A}$ such that $r_{0}$ is a retraction of $g_{0}$, and such that $r_{1}$ is a retraction of $g_{1}$. Then $j'$ is a trivial normally cloven cofibration with respect to $\cylinder$. \end{cor}

\begin{proof} Follows immediately from Proposition \ref{RetractionNormallyClovenCofibrationIsNormallyClovenCofibrationProposition} and Proposition \ref{RetractionHomotopyEquivalenceIsHomotopyEquivalenceProposition}. \end{proof}  

\begin{cor} \label{RetractionNormallyClovenFibrationIsNormallyClovenFibrationCorollary} Let $\cocylinder = \big( \cocyl, e_{0}, e_{1}, c  \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$ which is a normally cloven fibration with respect to $\cocylinder$. 

Suppose that we have commutative diagrams \twosq{a_{2},a_{0},a_{3},a_{1},g_{0},f,f',g_{1},a_{0},a_{2},a_{1},a_{3},r_{0},f',f,r_{1}} in $\cl{A}$ such that $r_{0}$ is a retraction of $g_{0}$, and such that $r_{1}$ is a retraction of $g_{1}$. Then $f'$ is a normally cloven fibration with respect to $\cocylinder$. \end{cor} 

\begin{proof} Follows immediately from Proposition \ref{RetractionNormallyClovenCofibrationIsNormallyClovenCofibrationProposition} by duality. \end{proof}

\begin{cor} \label{RetractionTrivialNormallyClovenFibrationIsTrivialNormallyClovenFibrationCorollary} Let $\cocylinder = \big( \cocyl, e_{0}, e_{1}, c  \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$ which is a trivial normally cloven fibration with respect to $\cocylinder$. 

Suppose that we have commutative diagrams \twosq{a_{2},a_{0},a_{3},a_{1},g_{0},f,f',g_{1},a_{0},a_{2},a_{1},a_{3},r_{0},f',f,r_{1}} in $\cl{A}$ such that $r_{0}$ is a retraction of $g_{0}$, and such that $r_{1}$ is a retraction of $g_{1}$. Then $f'$ is a trivial normally cloven fibration with respect to $\cocylinder$. \end{cor} 

\begin{proof} Follows immediately from Corollary \ref{RetractionNormallyClovenFibrationIsNormallyClovenFibrationCorollary} and Proposition \ref{RetractionHomotopyEquivalenceIsHomotopyEquivalenceProposition}. \end{proof}
\end{section}

covering_homotopy_extension_property.tex
\begin{section}{Covering homotopy extension property} \label{CoveringHomotopyExtensionPropertySection}

We introduce the covering homotopy extension property with respect to a cylinder $\cylinder$ in a formal category $\cl{A}$ and an arrow $j$ of $\cl{A}$.  Suppose that $\cylinder$ is equipped with a contraction structure $p$. Let $\cocylinder$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$ and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

If an arrow $f$ of $\cl{A}$ has the covering homotopy extension property with respect to $\cylinder$ and $j$, and if $f$ is moreover a strong deformation retraction with respect to $\cocylinder$, we demonstrate that $f$ has the right lifting property with respect to $j$. This will be vital to us in \ref{LiftingAxiomsSection}.

The significance of the covering homotopy extension property for lifting goes back to the proof of Theorem 9 of the paper \cite{StromNoteOnCofibrationsII} of Str{\o}m. The idea is pursued in \S{6} of the paper \cite{HastingsFibrationsOfCompactlyGeneratedSpaces} of Hastings. The argument given in these two papers adapts directly to the proof in an abstract setting which we present here, as was already observed by Hastings. A proof in abstract homotopy theory is given towards the end of \S{3} of Chapter II of the book \cite{KampsPorterAbstractHomotopyAndSimpleHomotopyTheory} of Kamps and Porter.  
 
In the category of topological spaces Str{\o}m proved as Theorem 4 of \cite{StromNoteOnCofibrations} that fibrations have the covering homotopy extension property with respect to closed cofibrations. In Theorem 2.1 of \cite{HastingsFibrationsOfCompactlyGeneratedSpaces} Hastings proved that in the category of compactly generated Hausdorff spaces fibrations have the covering homotopy extension property with respect to arbitrary cofibrations. Related theorems go back to around 1960 if not further.

More recently fibrations in the category of topological spaces satisfying the covering homotopy extension property with respect to cofibrations have been studied in the paper \cite{SchwanzelVogtStrongCofibrationsAndFibrationsInEnrichedCategories} of Schw{\"a}nzel and Vogt, in which they are referred to as strong fibrations. They are also discussed in Chapter 4 of the book \cite{MaySigurdssonParametrizedHomotopyTheory} of May and Sigurdsson in which the same terminology is adopted. 

\begin{assum} Let $\cl{C}$ be a 2-category with a final object such that pushouts and pullbacks of 2-arrows of $\cl{C}$ give rise to pushouts and pullbacks in formal categories in the sense of Definition \ref{PushoutsPullbacks2ArrowsArePushoutsPullbacksInFormalCategoriesTerminology}. Let $\cl{A}$ be an object of $\cl{C}$. As before we view $\cl{A}$ as a formal category, writing of objects and arrows of $\cl{A}$. \end{assum} 

\begin{defn} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$. Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$, and suppose that $\big( a^{\cylinder}_{j}, d^{0}_{j}, d^{1}_{j} \big)$ defines a mapping cylinder of $j$ with respect to $\cylinder$. 

An arrow \ar{a_{2},a_{3},f} of $\cl{A}$ has the {\em covering homotopy extension property} with respect to $j$ and $\cylinder$ if for any commutative diagram \sq{a^{\cylinder}_{j},a_{2},\cyl(a_{1}),a_{3},g,f,m^{\cylinder}_{j},h} in $\cl{A}$ there is an arrow \ar{\cyl(a_{1}),a_{2},l} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \liftingsquare{a^{\cylinder}_{j},a_{2},\cyl(a_{1}),a_{3},g,f,m^{\cylinder}_{j},h,l} \end{defn}

\begin{rmk} This definition is equivalent to that given towards the end of \S{3} of Chapter II of the book \cite{KampsPorterAbstractHomotopyAndSimpleHomotopyTheory} of Kamps and Porter. We will not need this. \end{rmk}

\begin{prpn} \label{CHEPImpliesLiftingAxiomProposition} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, and let $\cocylinder = \big( \cocyl, e_{0}, e_{1}, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$, and suppose that $\big( a^{\cylinder}_{j}, d^{0}_{j}, d^{1}_{j} \big)$ defines a mapping cylinder of $j$ with respect to $\cylinder$. Let \ar{a_{2},a_{3},f} be an arrow of $\cl{A}$ which has the covering homotopy extension property with respect to $j$ and $\cylinder$. Moreover, let \ar{a_{3},a_{2},j'} be an arrow of $\cl{A}$ and suppose that $f$ is a strong deformation retraction of $j'$ with respect to $\cocylinder$. 

Then for any commutative diagram \sq{a_{0},a_{2},a_{1},a_{3},g_{0},f,j,g_{1}} in $\cl{A}$ there is an arrow \ar{a_{1},a_{2},l} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \liftingsquare{a_{0},a_{2},a_{1},a_{3},g_{0},f,j,g_{1},l} \end{prpn} 

\begin{proof} Since $f$ is a strong deformation retraction of $j'$ with respect to $\cocylinder$, we have by Proposition \ref{StrongDeformationRetractionCoCylinderCharacterisationProposition} that there is a homotopy \ar{\cyl(a_{2}),a_{2},h} over $a_{3}$ from $j'f$ to $id(a_{2})$ with respect to $\cylinder$ and $(f,f)$. The following diagram in $\cl{A}$ commutes. \trapeziumstwo[{3,3,1,0}]{a_{0},\cyl(a_{0}),a_{1},a_{2},\cyl(a_{2}),a_{3},a_{2},i_{0}(a_{0}),\cyl(g_{0}),h,j,g_{1},j',g_{0},i_{0}(a_{2}),f} %
%
Thus there is a canonical arrow \ar{a^{\cyl}_{j},a_{2},u} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),a_{1},a^{\cylinder}_{j},a_{2},i_{0}(a_{0}),d^{0}_{j},j,d^{1}_{j},h \circ \cyl(g_{0}),j' \circ g_{1},u} %
%
By definition of $h$ as a homotopy over $a_{3}$ with respect to $\cylinder$ and $(f,f)$ the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{2}),a_{2},\cyl(a_{3}),a_{3},h,f,\cyl(f),p(a_{3})} Moreover the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{2}),\cyl(a_{3}),a_{2},a_{3},\cyl(f),p(a_{3}),p(a_{2}),f} Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{2}),a_{2},a_{2},a_{3},h,f,p(a_{2}),f} %
%
We now have that the following diagram in $\cl{A}$ commutes. 

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=3 em, row sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| \cyl(a_{0}) \&                     \& |(2-0)| a^{\cylinder}_{j} \\ 
                    \& |(1-1)| \cyl(a_{2}) \& |(2-1)| a_{2}        \\
                    \& |(1-2)| a_{2}       \&                     \\
|(0-3)| a_{0}       \& |(1-3)| a_{1}       \& |(2-3)| a_{3}       \\
};
	
\draw[->] (0-0) to node[auto] {$d^{0}_{j}$} (2-0);
\draw[->] (0-0) to node[auto,swap] {$p(a_{0})$} (0-3);
\draw[->] (0-0) to node[auto] {$\cyl(g_{0})$} (1-1);
\draw[->] (1-1) to node[auto] {$h$} (2-1);
\draw[->] (1-1) to node[auto,swap] {$p(a_{2})$} (1-2);
\draw[->] (1-2) to node[auto] {$f$} (2-3);
\draw[->] (0-3) to node[auto] {$g_{0}$} (1-2);
\draw[->] (2-0) to node[auto] {$u$} (2-1);
\draw[->] (2-1) to node[auto] {$f$} (2-3);
\draw[->] (0-3) to node[auto,swap] {$j$} (1-3);
\draw[->] (1-3) to node[auto,swap] {$g_{1}$} (2-3);
\end{tikzpicture} 

\end{diagram} %
%
The following diagram in $\cl{A}$ also commutes. \squareabovetriangle{a_{1},a^{\cylinder}_{j},a_{3},a_{2},a_{3},d^{1}_{j},u,g_{1},j',f,id}%
%
Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \pushout[{4,3,0}]{a_{0},\cyl(a_{0}),a_{1},a^{\cylinder}_{j},a_{3},i_{0}(a_{0}),d^{0}_{j},j,d^{1}_{j},g_{1} \circ j \circ p(a_{0}), g_{1}, f \circ u} %
%
The following diagram in $\cl{A}$ also commutes.  

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=6 em, row sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| \cyl(a_{0}) \& |(1-0)| a^{\cylinder}_{j} \\ 
                    \& |(1-1)| \cyl(a_{1})   \\
|(0-2)| a_{0}       \& |(1-2)| a_{1}        \\
};
	
\draw[->] (0-0) to node[auto] {$d^{0}_{j}$} (1-0);
\draw[->] (0-0) to node[auto,swap] {$p(a_{0})$} (0-2);
\draw[->] (1-0) to node[auto] {$m^{\cylinder}_{j}$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$\cyl(j)$} (1-1);
\draw[->] (1-1) to node[auto] {$p(a_{1})$} (1-2);
\draw[->] (0-2) to node[auto,swap] {$j$} (1-2);
\end{tikzpicture} 

\end{diagram} %
%
Hence the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{0}),a^{\cylinder}_{j},a_{3},d^{0}_{j},g_{1} \circ p(a_{1}) \circ m^{\cylinder}_{j},g_{1} \circ j \circ p(a_{0})} %
%
Moreover the following diagram in $\cl{A}$ commutes.   

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=7 em, row sep=4 em, nodes={anchor=center}]
{ 
|(0-0)| a_{1} \& |(1-0)| a^{\cylinder}_{j} \\ 
|(0-1)| a_{1} \& |(1-1)| \cyl(a_{1})   \\
};
	
\draw[->] (0-0) to node[auto] {$d^{1}_{j}$} (1-0);
\draw[->] (0-0) to node[auto,swap] {$id$} (0-1);
\draw[->] (1-0) to node[auto] {$m^{\cylinder}_{j}$} (1-1);
\draw[->] (1-1) to node[auto] {$p(a_{1})$} (0-1);
\draw[->] (0-0) to node[auto,swap] {$i_{0}(a_{1})$} (1-1);
\end{tikzpicture} 

\end{diagram} %
%
Thus the following diagram in $\cl{A}$ commutes. \tri{a_{1},a^{\cylinder}_{j},a_{3},d^{1}_{j},g_{1} \circ p(a_{1}) \circ m^{\cylinder}_{j},g_{1}} %
%
We now have that the following diagram in $\cl{A}$ commutes. \pushout[{4,3,3}]{a_{0},\cyl(a_{0}),a_{1},a^{\cylinder}_{j},a_{3},i_{0}(a_{0}),d^{0}_{j},j,d^{1}_{j},g_{1} \circ j \circ p(a_{0}), g_{1}, g_{1} \circ p(a_{1}) \circ m^{\cylinder}_{j}} %
%
Appealing to the universal property of $a^{\cylinder}_{j}$ it follows that the diagram in $\cl{A}$ below commutes. \sq[{5,3}]{a^{\cylinder}_{j},a_{2},\cyl(a_{1}),a_{3},u,f,m^{\cylinder}_{j},g_{1} \circ p(a_{1})} % 
%
Since $f$ has the covering homotopy extension property with respect to $j$ we deduce there is an arrow \ar{\cyl(a_{1}),a_{2},l} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \liftingsquare[{5,3.5}]{a^{\cylinder}_{j},a_{2},\cyl(a_{1}),a_{3},u,f,m^{\cylinder}_{j},g_{1} \circ p(a_{1}),l} 

Let $x$ denote the arrow \ar[5]{a_{1},a_{2},l \circ i_{1}(a_{1})} of $\cl{A}$. We claim that the following diagram in $\cl{A}$ commutes. \liftingsquare{a_{0},a_{2},a_{1},a_{3},g_{0},f,j,g_{1},x} % 
%
Firstly note that the following diagram in $\cl{A}$ commutes. \liftingsquare[{4,4}]{\cyl(a_{0}),\cyl(a_{1}),a^{\cylinder}_{j},a_{2},\cyl(j),l,d^{0}_{j},u,m^{\cylinder}_{j}} Hence the following diagram in $\cl{A}$ commutes. \trapeziumsthree[{4,3.5,-1,0}]{a_{0},a_{1},\cyl(a_{0}),\cyl(a_{1}),a^{\cylinder}_{j},a_{2},j,x,i_{1}(a_{0}),d^{0}_{j},u,i_{1}(a_{1}),\cyl(j),l} %
%
Moreover the following diagram in $\cl{A}$ commutes by appeal to the definition of $u$ and $h$. 

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=3 em, row sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| a_{0}  \&[1em]                     \& |(2-0)| \cyl(a_{0}) \\
|(0-1)| a_{2}  \& |(1-1)| \cyl(a_{2}) \& |(2-1)| a^{\cylinder}_{j} \\
               \&                     \& |(2-2)| a_{2} \\
};
	
\draw[->] (0-0) to node[auto] {$i_{1}(a_{0})$} (2-0);
\draw[->] (0-0) to node[auto,swap] {$g_{0}$} (0-1);
\draw[->] (0-1) to node[auto,swap] {$i_{1}(a_{2})$} (1-1);
\draw[->] (2-0) to node[auto,swap] {$\cyl(g_{0})$} (1-1);
\draw[->] (2-0) to node[auto] {$d^{0}_{j}$} (2-1);
\draw[->,bend right=45] (0-1) to node[auto,swap] {$id$} (2-2);
\draw[->] (1-1) to node[auto,swap] {$h$} (2-2);
\draw[->] (2-1) to node[auto] {$u$} (2-2);
\end{tikzpicture} 

\end{diagram} %
%
Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \tri{a_{0},a_{1},a_{2},j,x,g_{0}} 

Secondly the following diagram in $\cl{A}$ commutes. \squareofthreetrianglestwo[{3,3.5,0,0}]{a_{1},a_{2},\cyl(a_{1}),a_{1},a_{3},x,f,id,g_{1},i_{1}(a_{1}),l,p(a_{1})} %
%
This completes the proof of the claim.

\end{proof}

\end{section}


cylindrical_adjunctions.tex
\begin{section}{Cylindrical adjunctions} \label{CylindricalAdjunctionsSection} 

We introduce a notion of adjunction between a cylinder and a co-cylinder. It is discussed for example in \S{3} of \cite{KampsPorterAbstractHomotopyAndSimpleHomotopyTheory}. In \ref{StructuresUponAnIntervalSection} we will explain that an interval in a category gives to rise to a cylinder and co-cylinder which are adjoint. 

In \ref{StructuresUponACylinderOrACoCylinderSection} we will define homotopy with respect to a cylinder or a co-cylinder. If we have both a cylinder and a co-cylinder it will be vital for us to know that the corresponding notions of homotopy equivalence coincide. That the cylinder is left adjoint to the co-cylinder will exactly ensure that this holds.

Furthermore we will in \ref{CofibrationsAndFibrationsSection} define cofibrations with respect to a cylinder, and dually define fibrations with respect to a co-cylinder. If we have both a cylinder $\cylinder$ and a co-cylinder $\cocylinder$ with $\cylinder$ left adjoint to $\cocylinder$ we will be able to characterise fibrations with respect to $\cocylinder$ via a homotopy lifting property with respect to $\cylinder$, and will be able to characterise cofibrations with respect to $\cylinder$ via a homotopy lifting property with respect to $\cocylinder$. 

We refer the reader to Recollection \ref{AdjunctionRecollection} for the notion of an adjunction between 1-arrows of $\cl{C}$.

\begin{assum} Let $\cl{C}$ be a 2-category with a final object, and let $\cl{A}$ be an object of $\cl{C}$. \end{assum}

\begin{defn} \label{CylindricalAdjunctionDefinition}  Let $\cylinder = \big( \cyl, i_{0}, i_{1} \big)$ be a cylinder in $\cl{A}$, and let $\cocylinder = \big( \cocyl, e_{0}, e_{1} \big)$ be a co-cylinder in $\cl{A}$. Then $\cylinder$ is {\em left adjoint} to $\cocylinder$ if the following conditions are satisfied. 

\begin{itemize}[itemsep=1em,topsep=1em] 

\item[(i)]  $\cyl$ is left adjoint to $\cocyl$. 

\item[(ii)] Suppose that (i) holds. Let \ar{{\mathsf{Hom}_{\cl{A}}\big( \cyl(-),- \big)},{\mathsf{Hom}_{\cl{A}}\big(-,\cocyl(-)\big)},\mathsf{adj}} denote the corresponding natural isomorphism, adopting the shorthand of Recollection \ref{AdjunctionRecollection}. We require that for every arrow \ar{\cyl(a_{0}),a_{1},h} of $\cl{A}$ the following diagrams in $\cl{A}$ commute. \twosq{a_{0},\cyl(a_{0}),\cocyl(a_{1}),a_{1},i_{0}(a_{0}),h,\mathsf{adj}(h),e_{0}(a_{1}),a_{0},\cyl(a_{0}),\cocyl(a_{1}),a_{1},i_{1}(a_{0}),h,\mathsf{adj}(h),e_{1}(a_{1})} 

\end{itemize} 
\end{defn}

\begin{defn} \label{CylindricalAdjunctionCompatibleWithContractionAndExpansionDefinition} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, and let $\cocylinder = \big( \cocyl, e_{0}, e_{1}, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$. 

Let \ar{{\mathsf{Hom}_{\cl{A}}\big( \cyl(-),- \big)},{\mathsf{Hom}_{\cl{A}}\big(-,\cocyl(-)\big)},\mathsf{adj}} denote the corresponding natural isomorphism, adopting the shorthand of Recollection \ref{AdjunctionRecollection}. The adjunction between $\cyl$ and $\cocyl$ is {\em compatible with $p$ and $c$} if for every arrow \ar{a_{0},a_{1},f} of $\cl{A}$ the following diagram in $\cl{A}$ commutes. \tri{a_{0},a_{1},\cocyl(a_{1}),f,c(a_{1}),\mathsf{adj}\big(f \circ p(a_{0}) \big)} \end{defn}

\begin{rmk} Given a cylinder $\cylinder$ in $\cl{A}$ and a 1-arrow \ar[4]{\cl{A},\cl{A},\cocyl} of $\cl{C}$ which is left adjoint to $\cyl$, one can always equip $\cocyl$ with the structure of a co-cylinder $\cocylinder$ in $\cl{A}$ via the adjunction. 

Moreover one can transfer structures upon $\cylinder$ across the adjunction to structures upon $\cocylinder$. This is explained for example in \S{1.8} of the paper \cite{GrandisMacDonaldHomotopyStructuresForAlgebrasOverAMonad} of Grandis and MacDonald. It goes back at least to \S{7} of the paper \cite{KampsKanBedingungenUndAbstrakteHomotopietheorie} of Kamps. \end{rmk}

\end{section}



cylindrical_model_structures.tex
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\newcommand{\cl}[1]{\mathcal{#1}}
\newcommand{\cyl}{\mathsf{Cyl}\xspace}
\newcommand{\cocyl}{\mathsf{co}\textnormal{-}\mathsf{Cyl}\xspace}
\newcommand{\cylinder}{\underline{\mathsf{Cyl}}\xspace}
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|(0-0)| \nth{#2}{1} \& |(1-0)| \nth{#2}{2} \&[\coladjustment em] \\ 
|(0-1)| \nth{#2}{3} \& |(1-1)| \nth{#2}{4} \& \\
\& \& |(2-2)| \nth{#2}{5} \\
};
	
\draw[->] (0-0) to node[auto] {$\nth{#2}{6}$} (1-0);
\draw[->] (1-0) to node[auto] {$\nth{#2}{7}$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$\nth{#2}{8}$} (0-1);
\draw[->] (0-1) to node[auto,swap] {$\nth{#2}{9}$} (1-1);
\draw[->,bend left=45] (1-0) to node[auto] {$\nth{#2}{10}$} (2-2);
\draw[->, bend right=45] (0-1) to node[auto,swap] {$\nth{#2}{11}$} (2-2);
\draw[->] (1-1) to node[auto,swap] {$\nth{#2}{12}$} (2-2);

\end{tikzpicture} 

\end{diagram} }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Pullback diagram #2={*1,...,*12} as below. The optional parameter #1 specifies the values of column sep, row sep, and the adjustment to the width of the first column - the default is #1={3,3,0}.
%
%           *10
%   *5 -------------         
%    |\             |
%    | \            |
%    |  \ *12       |
%    |   \          |
%*11 |    ↘   *6    ↓
%    |    *1 ----> *2
%    |     |        |
%    |  *8 |        | *7 
%    |     ↓        ↓
%     --> *3 ----> *4
%             *9
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\pullback}[2][{3,3,0}]{

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\pgfmathsetmacro\colsep{{#1}[0]}
\pgfmathsetmacro\rowsep{{#1}[1]}
\pgfmathsetmacro\coladjustment{{#1}[2]}

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=\colsep em, row sep=\colsep em, nodes={anchor=center}] 
{ 
|(0-0)| \nth{#2}{5} \&[\coladjustment em]  \&                     \\
                    \& |(1-1)| \nth{#2}{1} \& |(2-1)| \nth{#2}{2} \\ 
                    \& |(1-2)| \nth{#2}{3} \& |(2-2)| \nth{#2}{4} \\
};
	
\draw[->] (0-0) to node[auto] {$\nth{#2}{12}$} (1-1);
\draw[->, bend left=45] (0-0) to node[auto] {$\nth{#2}{10}$} (2-1);
\draw[->, bend right=45] (0-0) to node[auto,swap] {$\nth{#2}{11}$} (1-2);
\draw[->] (1-1) to node[auto] {$\nth{#2}{6}$} (2-1);
\draw[->] (1-1) to node[auto,swap] {$\nth{#2}{8}$} (1-2);
\draw[->] (2-1) to node[auto] {$\nth{#2}{7}$} (2-2);
\draw[->] (1-2) to node[auto,swap] {$\nth{#2}{9}$} (2-2);

\end{tikzpicture} 

\end{diagram} }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Double homotopy #2={*1,...,*9} as below. The optional parameter #1 specifies the values of column sep and row sep - the default is #1={1,1}. To omit the vertices, just omit the last four entries of the list.
%
%        *1
%   *6 -----> *7
%    |         |
% *3 |   *5    | *2 
%    ↓         ↓
%   *8 -----> *9
%        *4
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\doublehomotopy}[2][{1,1}]{

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\pgfmathsetmacro\colsep{{#1}[0]}
\pgfmathsetmacro\rowsep{{#1}[1]}

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=\colsep em, row sep=\rowsep em, nodes={anchor=center}] 
{ 
|(0-0)| \nth{#2}{6} \& \& |(2-0)| \nth{#2}{7} \\ 
\& \nth{#2}{5} \& \\
|(0-2)| \nth{#2}{8} \& \& |(2-2)| \nth{#2}{9} \\
};
	
\draw[->] (0-0) to node[auto] {$\nth{#2}{1}$} (2-0);
\draw[->] (2-0) to node[auto] {$\nth{#2}{2}$} (2-2);
\draw[->] (0-0) to node[auto,swap] {$\nth{#2}{3}$} (0-2);
\draw[->] (0-2) to node[auto,swap] {$\nth{#2}{4}$} (2-2);

\end{tikzpicture} 

\end{diagram} }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%'Lifting property' square #2={*1,...,*9} as below. The optional parameter #1 specifies the values of column sep and row sep, with default #1={3,3}.
%
%       *5
%   *1 ----> *2
%    |     ↗  |
%    |    /   |
% *7 |   / *9 | *6
%    |  /     |
%    ↓ /      ↓
%   *3 ----> *4
%       *8
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\liftingsquare}[2][{3,3}]{

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\pgfmathsetmacro\colsep{{#1}[0]}
\pgfmathsetmacro\rowsep{{#1}[1]}

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=\colsep em, row sep=\rowsep em, nodes={anchor=center}] 
{ 
|(0-0)| \nth{#2}{1} \& |(1-0)| \nth{#2}{2} \\ 
|(0-1)| \nth{#2}{3} \& |(1-1)| \nth{#2}{4} \\
};
	
\draw[->] (0-0) to node[auto] {$\nth{#2}{5}$} (1-0);
\draw[->] (1-0) to node[auto] {$\nth{#2}{6}$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$\nth{#2}{7}$} (0-1);
\draw[->] (0-1) to node[auto,swap] {$\nth{#2}{8}$} (1-1);
\draw[->] (0-1) to node[auto,swap] {$\nth{#2}{9}$} (1-0);

\end{tikzpicture} 

\end{diagram} }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Square with diagonal #2={*1,...,*9} as below. The optional parameter #1 specifies the values of column sep and row sep, with default #1={3,3}.
%
%       *5
%   *1 ----> *2
%    | \      |
%    |  \     |
% *7 |   \ *9 | *6
%    |    \   |
%    ↓     ↘  ↓
%   *3 <---- *4
%        *8
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\squarewithdiagonal}[2][{3,3}]{

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\pgfmathsetmacro\colsep{{#1}[0]}
\pgfmathsetmacro\rowsep{{#1}[1]}

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=\colsep em, row sep=\rowsep em, nodes={anchor=center}] 
{ 
|(0-0)| \nth{#2}{1} \& |(1-0)| \nth{#2}{2} \\ 
|(0-1)| \nth{#2}{3} \& |(1-1)| \nth{#2}{4} \\
};
	
\draw[->] (0-0) to node[auto] {$\nth{#2}{5}$} (1-0);
\draw[->] (1-0) to node[auto] {$\nth{#2}{6}$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$\nth{#2}{7}$} (0-1);
\draw[->] (1-1) to node[auto] {$\nth{#2}{8}$} (0-1);
\draw[->] (0-0) to node[auto,swap] {$\nth{#2}{9}$} (1-1);

\end{tikzpicture} 

\end{diagram} }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Square with diagonal #2={*1,...,*9} as below. The optional parameter #1 specifies the values of column sep and row sep, with default #1={3,3}.
%
%       *5
%   *1 ----> *2
%    |     /  |
%    |    /   |
% *7 |   / *9 | *6
%    |  /     |
%    ↓ ↙      ↓
%   *3 ----> *4
%       *8
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\squarewithdiagonaltwo}[2][{3,3}]{

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\pgfmathsetmacro\colsep{{#1}[0]}
\pgfmathsetmacro\rowsep{{#1}[1]}

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=\colsep em, row sep=\rowsep em, nodes={anchor=center}] 
{ 
|(0-0)| \nth{#2}{1} \& |(1-0)| \nth{#2}{2} \\ 
|(0-1)| \nth{#2}{3} \& |(1-1)| \nth{#2}{4} \\
};
	
\draw[->] (0-0) to node[auto] {$\nth{#2}{5}$} (1-0);
\draw[->] (1-0) to node[auto] {$\nth{#2}{6}$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$\nth{#2}{7}$} (0-1);
\draw[->] (0-1) to node[auto,swap] {$\nth{#2}{8}$} (1-1);
\draw[->] (1-0) to node[auto] {$\nth{#2}{9}$} (0-1);

\end{tikzpicture} 

\end{diagram} }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Square with diagonal #3={*1,...,*9} as below. The optional parameter #1 specifies the values of column sep and row sep, with default #1={3,3}.
%
%       *5
%   *1 ----> *2
%    | \      |
%    |  \     |
% *7 |   \ *9 | *6
%    |    \   |
%    ↓     ↘  ↓
%   *3 ----> *4
%        *8

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\squarewithdiagonalthree}[2][{3,3}]{

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\pgfmathsetmacro\colsep{{#1}[0]}
\pgfmathsetmacro\rowsep{{#1}[1]}

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=\colsep em, row sep=\rowsep em, nodes={anchor=center}] 
{ 
|(0-0)| \nth{#2}{1} \& |(1-0)| \nth{#2}{2} \\ 
|(0-1)| \nth{#2}{3} \& |(1-1)| \nth{#2}{4} \\
};
	
\draw[->] (0-0) to node[auto] {$\nth{#2}{5}$} (1-0);
\draw[->] (1-0) to node[auto] {$\nth{#2}{6}$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$\nth{#2}{7}$} (0-1);
\draw[->] (0-1) to node[auto,swap] {$\nth{#2}{8}$} (1-1);
\draw[->] (0-0) to node[auto] {$\nth{#2}{9}$} (1-1);

\end{tikzpicture} 

\end{diagram} }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%'Two trapeziums' #2={*1,...,*14} as below. The optional parameter #1 specifies the values of column sep, row sep, column sep adjustment, and row sep adjustment, with default #1={3,3,0,0}.
%
%             *7
%    *1 -------------> *2
%     | \               |
%     |  \              |
%     |   \ *12         | *8
%     |    \            |
%     |     ↘           ↓
% *10 |     *3 -------> *4
%     |     /    *13    |
%     |    /            |
%     |   / *14         | *9
%     |  /              |
%     ↓ ↙               ↓
%    *5 -------------> *6
%            *11
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\trapeziums}[2][{3,3,0,0}]{

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\pgfmathsetmacro\colsep{{#1}[0]}
\pgfmathsetmacro\rowsep{{#1}[1]}
\pgfmathsetmacro\coladjustment{{#1}[2]}
\pgfmathsetmacro\rowadjustment{{#1}[3]}

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=\colsep em, row sep=\rowsep em, nodes={anchor=center}] 
{ 
|(0-0)| \nth{#2}{1} \&                     \&[\coladjustment em] |(2-0)| \nth{#2}{2} \\ 
                    \& |(1-1)| \nth{#2}{3} \& |(2-1)| \nth{#2}{4} \\[\rowadjustment em]
|(0-2)| \nth{#2}{5} \&                     \& |(2-2)| \nth{#2}{6} \\
};
	
\draw[->] (0-0) to node[auto] {$\nth{#2}{7}$} (2-0);
\draw[->] (2-0) to node[auto] {$\nth{#2}{8}$} (2-1);
\draw[->] (2-1) to node[auto] {$\nth{#2}{9}$} (2-2);
\draw[->] (0-0) to node[auto,swap] {$\nth{#2}{10}$} (0-2);
\draw[->] (0-2) to node[auto,swap] {$\nth{#2}{11}$} (2-2);
\draw[->] (0-0) to node[auto] {$\nth{#2}{12}$} (1-1);
\draw[->] (1-1) to node[auto] {$\nth{#2}{13}$} (2-1);
\draw[->] (1-1) to node[auto] {$\nth{#2}{14}$} (0-2);

\end{tikzpicture} 

\end{diagram} }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%'A square above a triangle' #2={*1,...,*11} as below. The optional parameter #1 specifies the values of column sep, row sep, and row sep adjustment, with default #1={3,3,0}.
%
%         *6
%    *1 ------> *2
%     |          |
%  *8 |          | *7
%     ↓   *9     ↓
%    *3 ------> *4
%      \         |
%       \        |
%        \       |
%     *11 \      | *10
%          \     |
%           \    |
%            \   |
%             ↘  ↓
%               *5
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\squareabovetriangle}[2][{3,3,0}]{

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\pgfmathsetmacro\colsep{{#1}[0]}
\pgfmathsetmacro\rowsep{{#1}[1]}
\pgfmathsetmacro\rowadjustment{{#1}[2]}

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=\colsep em, row sep=\rowsep em, nodes={anchor=center}] 
{ 
|(0-0)| \nth{#2}{1} \& |(1-0)| \nth{#2}{2} \\
|(0-1)| \nth{#2}{3} \& |(1-1)| \nth{#2}{4} \\[\rowadjustment em]
                    \& |(1-2)| \nth{#2}{5} \\
};
	
\draw[->] (0-0) to node[auto] {$\nth{#2}{6}$} (1-0);
\draw[->] (1-0) to node[auto] {$\nth{#2}{7}$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$\nth{#2}{8}$} (0-1);
\draw[->] (0-1) to node[auto] {$\nth{#2}{9}$} (1-1);
\draw[->] (1-1) to node[auto] {$\nth{#2}{10}$} (1-2);
\draw[->] (0-1) to node[auto,swap] {$\nth{#2}{11}$} (1-2);

\end{tikzpicture} 

\end{diagram} }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%'A square beneath a triangle' #2={*1,...,*11} as below. The optional parameter #1 specifies the values of column sep, row sep, and row sep adjustment, with default #1={3,3,0}.
%
%    *1
%     | \
%     |  \
%     |   \
%     |    \
%  *6 |     \ *8
%     |      \
%     |       \
%     |        \
%     ↓         ↘
%    *2 ------> *3
%     |   *7     |
% *10 |          | *9
%     ↓          ↓
%    *4 ------> *5
%        *11
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\squarebeneathtriangle}[2][{3,3,0}]{

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\pgfmathsetmacro\colsep{{#1}[0]}
\pgfmathsetmacro\rowsep{{#1}[1]}
\pgfmathsetmacro\rowadjustment{{#1}[2]}

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=\colsep em, row sep=\rowsep em, nodes={anchor=center}] 
{ 
|(0-0)| \nth{#2}{1} \& \\
|(0-1)| \nth{#2}{2} \& |(1-1)| \nth{#2}{3} \\[\rowadjustment em]
|(0-2)| \nth{#2}{4} \& |(1-2)| \nth{#2}{5} \\
};
	
\draw[->] (0-0) to node[auto,swap] {$\nth{#2}{6}$} (0-1);
\draw[->] (0-1) to node[auto,swap] {$\nth{#2}{7}$} (1-1);
\draw[->] (0-0) to node[auto] {$\nth{#2}{8}$} (1-1);
\draw[->] (1-1) to node[auto] {$\nth{#2}{9}$} (1-2);
\draw[->] (0-1) to node[auto,swap] {$\nth{#2}{10}$} (0-2);
\draw[->] (0-2) to node[auto,swap] {$\nth{#2}{11}$} (1-2);

\end{tikzpicture} 

\end{diagram} }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%'Two trapeziums' #2={*1,...,*16} as below. The optional parameter #1 specifies the values of column sep, row sep, column sep adjustment, and row sep adjustment, with default #1={3,3,0,0}.
%
%             *8
%    *1 -------------> *2
%     | \               |
%     |  \              |
% *11 |   \ *14         | *9
%     |    \            |
%     ↓     ↘           ↓
%    *3     *4 -------> *5
%     |     /    *15    |
%     |    /            |
% *12 |   / *16         | *10
%     |  /              |
%     ↓ ↙               ↓
%    *6 -------------> *7
%            *13
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\trapeziumstwo}[2][{3,3,0,0}]{

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\pgfmathsetmacro\colsep{{#1}[0]}
\pgfmathsetmacro\rowsep{{#1}[1]}
\pgfmathsetmacro\coladjustment{{#1}[2]}
\pgfmathsetmacro\rowadjustment{{#1}[3]}

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=\colsep em, row sep=\rowsep em, nodes={anchor=center}] 
{ 
|(0-0)| \nth{#2}{1} \&                     \&[\coladjustment em] |(2-0)| \nth{#2}{2} \\ 
|(0-1)| \nth{#2}{3} \& |(1-1)| \nth{#2}{4} \& |(2-1)| \nth{#2}{5} \\[\rowadjustment em]
|(0-2)| \nth{#2}{6} \&                     \& |(2-2)| \nth{#2}{7} \\
};
	
\draw[->] (0-0) to node[auto] {$\nth{#2}{8}$} (2-0);
\draw[->] (2-0) to node[auto] {$\nth{#2}{9}$} (2-1);
\draw[->] (2-1) to node[auto] {$\nth{#2}{10}$} (2-2);
\draw[->] (0-0) to node[auto,swap] {$\nth{#2}{11}$} (0-1);
\draw[->] (0-1) to node[auto,swap] {$\nth{#2}{12}$} (0-2);
\draw[->] (0-2) to node[auto,swap] {$\nth{#2}{13}$} (2-2);
\draw[->] (0-0) to node[auto] {$\nth{#2}{14}$} (1-1);
\draw[->] (1-1) to node[auto] {$\nth{#2}{15}$} (2-1);
\draw[->] (1-1) to node[auto] {$\nth{#2}{16}$} (0-2);

\end{tikzpicture} 

\end{diagram} }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%'Two trapeziums' #2={*1,...,*14} as below. The optional parameter #1 specifies the values of column sep, row sep, column sep adjustment, and row sep adjustment, with default #1={3,3,0,0}.
%
%             *7
%    *1 -------------> *2
%     |              /  |
%     |             /   |
%  *9 |        *12 /    |
%     |           /     |
%     ↓          ↙      |
%    *3 -----> *4       | *8
%     |  *13     \      |
%     |           \     |
% *10 |        *14 \    | 
%     |             \   |
%     ↓              ↘  ↓
%    *5 -------------> *6
%            *11
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\trapeziumsthree}[2][{3,3,0,0}]{

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\pgfmathsetmacro\colsep{{#1}[0]}
\pgfmathsetmacro\rowsep{{#1}[1]}
\pgfmathsetmacro\coladjustment{{#1}[2]}
\pgfmathsetmacro\rowadjustment{{#1}[3]}

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=\colsep em, row sep=\rowsep em, nodes={anchor=center}] 
{ 
|(0-0)| \nth{#2}{1} \&                     \&[\coladjustment em] |(2-0)| \nth{#2}{2} \\ 
|(0-1)| \nth{#2}{3} \& |(1-1)| \nth{#2}{4} \&  \\[\rowadjustment em]
|(0-2)| \nth{#2}{5} \&                     \& |(2-2)| \nth{#2}{6} \\
};
	
\draw[->] (0-0) to node[auto] {$\nth{#2}{7}$} (2-0);
\draw[->] (2-0) to node[auto] {$\nth{#2}{8}$} (2-2);
\draw[->] (0-0) to node[auto,swap] {$\nth{#2}{9}$} (0-1);
\draw[->] (0-1) to node[auto,swap] {$\nth{#2}{10}$} (0-2);
\draw[->] (0-2) to node[auto,swap] {$\nth{#2}{11}$} (2-2);
\draw[->] (2-0) to node[auto,swap] {$\nth{#2}{12}$} (1-1);
\draw[->] (0-1) to node[auto,swap] {$\nth{#2}{13}$} (1-1);
\draw[->] (1-1) to node[auto,swap] {$\nth{#2}{14}$} (2-2);

\end{tikzpicture} 

\end{diagram} }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%'Two trapeziums' #2={*1,...,*15} as below. The optional parameter #1 specifies the values of column sep, row sep, column sep adjustment, and row sep adjustment, with default #1={3,3,0,0}.
%
%                *7
%    *1 ---------------------->  *2
%     | \                     /   |
%     |  \                   /    |
%     |   \ *11         *12 /     |
%     |    \               /      |
%     |     ↘             ↙       |
%  *9 |     *3 ---------> *4      | *8
%     |     /     *13     \       |
%     |    /               \      |
%     |   / *14         *15 \     |
%     |  /                   \    |
%     ↓ ↙                     ↘   ↓
%    *5 -----------------------> *6
%               *10
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\trapeziumsfive}[2][{3,3,0,0,0}]{

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\pgfmathsetmacro\colsep{{#1}[0]}
\pgfmathsetmacro\rowsep{{#1}[1]}
\pgfmathsetmacro\coladjustmentone{{#1}[2]}
\pgfmathsetmacro\coladjustmenttwo{{#1}[3]}
\pgfmathsetmacro\rowadjustment{{#1}[4]}

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=\colsep em, row sep=\rowsep em, nodes={anchor=center}] 
{ 
|(0-0)| \nth{#2}{1} \&                     \&[\coladjustmentone em]  \&[\coladjustmenttwo em] |(3-0)| \nth{#2}{2} \\ 
                    \& |(1-1)| \nth{#2}{3} \& |(2-1)| \nth{#2}{4}    \& \\[\rowadjustment em]
|(0-2)| \nth{#2}{5} \&                     \&                        \& |(3-2)| \nth{#2}{6} \\
};
	
\draw[->] (0-0) to node[auto] {$\nth{#2}{7}$} (3-0);
\draw[->] (3-0) to node[auto] {$\nth{#2}{8}$} (3-2);
\draw[->] (0-0) to node[auto,swap] {$\nth{#2}{9}$} (0-2);
\draw[->] (0-2) to node[auto,swap] {$\nth{#2}{10}$} (3-2);
\draw[->] (0-0) to node[auto] {$\nth{#2}{11}$} (1-1);
\draw[->] (3-0) to node[auto,swap] {$\nth{#2}{12}$} (2-1);
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%'Three triangles' #2={*1,...,*9} as below. The optional parameter #1 specifies the values of column sep, row sep, and row sep adjustment, with default #1={3,3,0}.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\title{Cylindrical model structures}
\date{\today}
\thanks{Supported September 2007 - October 2011 at Oxford by an EPSRC Research Studentship, and January 2012 - March 2012 at NTNU by Forskningsr{\aa}det (The Research Council of Norway) project {\em Topology}, code 185335/V30.} 
\author{Richard Williamson}
\address{Institutt for matematiske fag, NTNU, Sentralbygg 2, Alfred Getz Vei 1, 7491 Trondheim, Norway}
\email{richard.williamson@math.ntnu.no}
\dedicatory{Dedicated to Kari Chard}

\begin{abstract} We build a model structure from the simple point of departure of a structured interval in a monoidal category --- more generally, a structured cylinder and a structured co-cylinder in a category. \end{abstract}

\maketitle

\tableofcontents

\include{introduction}
\include{formal_category_theory_preliminaries}
\include{structures_upon_a_cylinder_or_a_co-cylinder}
\include{cylindrical_adjunctions}
\include{monoidal_category_theory_preliminaries}
\include{structures_upon_an_interval}
\include{homotopy_and_relative_homotopy}
\include{cofibrations_and_fibrations}
\include{mapping_cylinders_and_mapping_co-cylinders}
\include{covering_homotopy_extension_property}
\include{dold-s_theorem}
\include{lifting_axioms}
\include{factorisation_axioms}
\include{model_category_recollections}
\include{model_structure}
\include{example--categories_and_groupoids} 

\bibliography{ref}
\bibliographystyle{siam}

\end{document}

dold-s_theorem.tex
\begin{section}{Dold's theorem on homotopy equivalences under or over an object} \label{DoldTheoremSection}

Let $\cylinder$ be a cylinder in a formal category $\cl{A}$ equipped with a contraction structure, an involution structure compatible with contraction, a subdivision structure compatible with contraction, and an upper left connection structure. Suppose that we have a commutative diagram in $\cl{A}$ as follows in which $j_{0}$ and $j_{1}$ are fibrations with respect to $\cylinder$. \triother{a_{0},a_{1},a,f,j_{1},j_{0}} We prove that if $f$ is a homotopy equivalence with respect to $\cylinder$, then $f$ is moreover a homotopy equivalence over $a$ with respect to $\cylinder$ and $(j_{0},j_{1})$. 

For topological spaces this is due to Dold --- it is Theorem 3.1 of \cite{DoldPartitionsOfUnityInTheTheoryOfFibrations}. Our proof is an abstraction of a hybrid of Dold's proof and a proof presented in \S{5} of Chapter 6 of the book \cite{MayAConciseCourseInAlgebraicTopology} of May. Both proofs exhibit a double homotopy \doublehomotopy{h_{0},h_{1},h_{2},h_{3},\sigma,f_{0},f_{1},f_{2},f_{3}} in which the boundary homotopies $h_{1}$, $h_{2}$, and $h_{3}$ can be proven to be homotopies over $a$. %
%
This allows us to construct a homotopy over $a$ from $f_{0}$ to $f_{1}$ by taking the indirect route around the square, reversing $h_{1}$. 

In the construction of a double homotopy with these properties in our proof the key role is played by the upper left connection structure. The reader may observe that the map \ar[12]{I^{2},{I,},{(t_{0},t_{1}) \mapsto t_{0} + (1-t_{0})t_{1}}} in which $I$ is the unit interval, underlies the construction of the double homotopy in \cite{MayAConciseCourseInAlgebraicTopology}. This map defines an upper left connection structure with respect to the topological interval.

Given a cylinder whose associated cubical set satisfies low dimensional Kan conditions a proof of Dold's theorem was given by Kamps in \S{6} of \cite{KampsKanBedingungenUndAbstrakteHomotopietheorie}. A variation is presented in \S{6} of Chapter I of \cite{KampsPorterAbstractHomotopyAndSimpleHomotopyTheory}. There is a fundamental difference between our proof and these two.

We demonstrate that the required double homotopy can be constructed if $\cylinder$ admits certain structures. Requiring that the cubical set associated to $\cylinder$ satisfies low dimensional Kan conditions ensures the existence of this double homotopy --- but the Kan conditions must themselves be proven to hold. To put it another way, the proof given that the Kan conditions hold can be thought of as a plan for the construction of the double homotopy --- the structures upon a cylinder which we assume allow us to carry out this plan.

A quite different proof of Dold's theorem in an abstract setting can be given if $\cylinder$ admits sufficient structure to ensure that the objects, arrows, and homotopies up to homotopy with respect to $\cylinder$ assemble into a 2-category. The argument is presented in \S{1} and \S{2} of Chapter IV of the book \cite{KampsPorterAbstractHomotopyAndSimpleHomotopyTheory} of Kamps and Porter.

Dold's theorem implies that a trivial cofibration admits a strong deformation retraction. As was observed by Dold as Satz 3.6 of \cite{DoldHalbexakteHomotopiefunktoren} the theorem also can immediately be dualised, giving that a trivial fibration is a strong deformation retraction. These observations will play a crucial role in our proof in \ref{LiftingAxiomsSection} that if we have strictness of identities then the lifting axioms for a model structure hold. 

When our cylinder has strictness of identities we will prove in \ref{LiftingAxiomsSection} that trivial cofibrations are exactly sections of strong deformation retractions. When our co-cylinder has strictness of identities we will dually prove in \ref{LiftingAxiomsSection} that trivial fibrations are exactly strong deformation retractions. 

\begin{assum} Let $\cl{C}$ be a 2-category with a final object such that pushouts and pullbacks of 2-arrows of $\cl{C}$ give rise to pushouts and pullbacks in formal categories in the sense of Definition \ref{PushoutsPullbacks2ArrowsArePushoutsPullbacksInFormalCategoriesTerminology}. Let $\cl{A}$ be an object of $\cl{C}$. As before we view $\cl{A}$ as a formal category, writing of objects and arrows of $\cl{A}$. \end{assum} 

\begin{lem} \label{DoldLemma1} Let $\cylinder = \big( \cyl, i_0, i_1, p, v, \subdiv, r_{0}, r_{1}, s \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $v$, and a subdivision structure $\big(\subdiv, r_{0}, r_{1}, s \big)$. 

Let \ar{a,a_{0},j_{0}} be an arrow of $\cl{A}$ which is a fibration with respect to $\cylinder$. Let \ar{a,a_{1},j_{1}} and \ar{a_{0},a_{1},f} be arrows of $\cl{A}$ such that the diagram \triother{a_{0},a_{1},a,f,j_{1},j_{0}} in $\cl{A}$ commutes, and such that $f$ is a homotopy equivalence with respect to $\cylinder$. 

Then there is an arrow \ar{a_{1},a_{0},g} of $\cl{A}$ and a homotopy from $fg$ to $id(a_{1})$ with respect to $\cylinder$ such that the following diagram in $\cl{A}$ commutes. \triother{a_{1},a_{0},a,g,j_{0},j_{1}} \end{lem} 

\begin{proof} Let \ar{\cyl(a_{1}),a_{1},h} be a homotopy from $ff^{-1}$ to $id(a_{1})$ with respect to $\cylinder$. The following diagram in $\cl{A}$ commutes. \squareabovetriangle{a_{1},\cyl(a_{1}),a_{0},a_{1},a,i_{0}(a_{1}),h,f^{-1},f,j_{1},j_{0}} %
%
Since $j_{0}$ is a fibration with respect to $\cylinder$ we deduce that there is an arrow \ar{\cyl(a_{1}),a_{0},k} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \liftingsquare[{4,3}]{a_{1},a_{0},\cyl(a_{1}),a,{f^{-1}},j_{0},i_{0}(a_{1}),j_{1} \circ h,k} %
%
Let $g$ denote the  arrow \ar[5]{a_{1},a_{0},k \circ i_{1}(a_{1})} of $\cl{A}$. The following diagram in $\cl{A}$ commutes. \squarebeneathtriangle{a_{1},\cyl(a_{1}),a_{1},a_{1},a,i_{1}(a_{1}),h,id,j_{1},k,j_{0}} %
%
Thus the following diagram in $\cl{A}$ commutes. \triother{a_{1},a_{0},a,g,j_{0},j_{1}} 

It remains to construct a homotopy from $fg$ to $id(a_{1})$ with respect to $\cylinder$. Firstly the following diagram in $\cl{A}$ commutes. 

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\end{tikzpicture} 

\end{diagram} %
%
We also have that the diagram \sq{a_{1},\cyl(a_{1}),a_{0},a_{1},i_{0}(a_{1}),h,f^{-1},f} in $\cl{A}$ commutes. Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{a_{1},\cyl(a_{1}),\cyl(a_{1}),a_{1},i_{0}(a_{1}),h,i_{1}(a_{1}),f \circ k \circ v(a_{1})} 

Let us denote by \ar{\cyl(a_{1}),a_{1},l} the homotopy $\big(f \circ k \circ v(a_{1}) \big) + h$ with respect to $\cylinder$. The following diagram in $\cl{A}$ commutes. 

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=6 em, row sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| a_{1} \& |(1-0)| \cyl(a_{1}) \\
|(0-1)| a_{0} \& |(1-1)| \cyl(a_{1}) \\ 
};
	
\draw[->] (0-0) to node[auto] {$i_{0}(a_{1})$} (1-0);
\draw[->] (0-0) to node[auto,swap] {$g$} (0-1);
\draw[->] (0-0) to node[auto,swap] {$i_{1}(a_{1})$} (1-1);
\draw[->] (1-0) to node[auto] {$v(a_{1})$} (1-1);
\draw[->] (1-1) to node[auto] {$k$} (0-1);
\end{tikzpicture} 

\end{diagram} %
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Thus since the diagram \sq{a_{1},\cyl(a_{1}),\cyl(a_{1}),a_{1},i_{0}(a_{1}),l,i_{0}(a_{1}),h} in $\cl{A}$ commutes we have that the following diagram in $\cl{A}$ commutes. \tri{a_{1},\cyl(a_{1}),a_{1},i_{0}(a_{1}),l,fg} 

In addition since the diagram \sq{a_{1},\cyl(a_{1}),\cyl(a_{1}),a_{1},i_{1}(a_{1}),l,i_{1}(a_{1}),h} in $\cl{A}$ commutes and since the diagram \tri{a_{1},\cyl(a_{1}),a_{1},i_{1}(a_{1}),h,id} in $\cl{A}$ commutes we have that the following diagram in $\cl{A}$ commutes. \tri{a_{1},\cyl(a_{1}),a_{1},i_{1}(a_{1}),l,id}

\end{proof}

\begin{lem} \label{DoldLemma2} Let $\cylinder = \big( \cyl, i_0, i_1, p, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{ul} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $v$, a subdivision structure $\big(\subdiv, r_{0}, r_{1}, s \big)$, and an upper left connection structure $\Gamma_{ul}$. Suppose that $\cyl$ preserves subdivision with respect to $\cylinder$. 

Let \ar{a,a_{0},j_{0}} be an arrow of $\cl{A}$ which is a fibration with respect to $\cylinder$. Let \ar{a,a_{1},j_{1}} and \ar{a_{0},a_{1},f} be arrows of $\cl{A}$ such that the diagram \triother{a_{0},a_{1},a,f,j_{1},j_{0}} in $\cl{A}$ commutes and such that $f$ is a homotopy equivalence with respect to $\cylinder$. 

Let \ar{a_{1},a_{0},g} and \ar{\cyl(a_{1}),a_{1},l} denote the arrows of $\cl{A}$ constructed in Lemma \ref{DoldLemma1} such that the diagram \triother{a_{1},a_{0},a,g,j_{0},j_{1}} in $\cl{A}$ commutes, and such that $l$ defines a homotopy from $fg$ to $id(a_{1})$ with respect to $\cylinder$. 

Then there is an arrow \ar{\cyl^{2}(a_{1}),a,\tau} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),a_{1},a,i_{0}\big(\cyl(a_{1})\big),\tau,l,j_{1}} \end{lem} 

\begin{proof} Let \ar{\cyl(a_{1}),a_{0},k} denote the arrow of $\cl{A}$ constructed in the proof of Lemma \ref{DoldLemma1}. In particular the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{1}),a_{0},a_{1},a,k,j_{0},h,j_{1}} %
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By definition of $\Gamma_{ul}$ as an upper left connection structure the following diagram in $\cl{A}$ also commutes. \tri[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),\cyl(a_{1}),\cyl\big(i_{0}(a_{1})\big),\Gamma_{ul}(a_{1}),id} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \tri[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),a,\cyl\big(i_{0}(a_{1})\big),j_{1} \circ h \circ \Gamma_{ul}(a_{1}),j_{0} \circ k} %
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By definition of $v$ as an involution structure the following diagram in $\cl{A}$ commutes. \tri{a_{1},\cyl(a_{1}),\cyl(a_{1}),i_{1}(a_{1}),v(a_{1}),i_{0}(a_{1})} Thus appealing once more to the commutativity of the diagram \tri[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),\cyl(a_{1}),\cyl\big(i_{0}(a_{1})\big),\Gamma_{ul}(a_{1}),id} in $\cl{A}$ we have that the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{1}),\cyl^{2}(a_{1}),a,\cyl(a_{1}),\cyl\big(i_{1}(a_{1})\big),\Gamma_{ul}(a_{1}),\cyl\big(v(a_{1})\big),id} %
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Hence the following diagram in $\cl{A}$ commutes. \tri[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),a,\cyl\big(i_{1}(a_{1})\big),j_{0} \circ k \circ \Gamma_{ul}(a_{1}) \circ \cyl\big(v(a_{1})\big),j_{0} \circ k} %
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Putting everything together we have now shown that the following diagram in $\cl{A}$ commutes. \sq[{11,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),\cyl^{2}(a_{1}),a,\cyl\big(i_{0}(a_{1})\big),j_{1} \circ h \circ \Gamma_{ul}(a_{1}),\cyl\big(i_{1}(a_{1})\big),j_{0} \circ k \circ \Gamma_{ul}(a_{1}) \circ \cyl\big(v(a_{1})\big)} %
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Since $\cyl$ preserves subdivision with respect to $\cylinder$ the following diagram in $\cl{A}$ is co-cartesian. \sq[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),\cyl^{2}(a_{1}),\cyl\big(\subdiv(a_{1})\big),\cyl\big(i_{0}(a_{1})\big),\cyl\big(r_{0}(a_{1})\big),\cyl\big(i_{1}(a_{1})\big),\cyl\big(r_{1}(a_{1})\big)} %
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Thus there is an arrow \ar{\cyl\big(\subdiv(a_{1})\big),a,u} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout[{5,3,0}]{\cyl(a_{1}),\cyl^{2}(a_{1}),\cyl^{2}(a_{1}),\cyl\big(\subdiv(a_{1})\big),a,\cyl\big(i_{0}(a_{1})\big),\cyl\big(r_{0}(a_{1})\big),\cyl\big(i_{1}(a_{1})\big),\cyl\big(r_{1}(a_{1})\big),j_{1} \circ h \circ \Gamma_{ul}(a_{1}),j_{0} \circ k \circ \Gamma_{ul}(a_{1}) \circ \cyl\big(v(a_{1})\big),u} 

Let \ar{\cyl^{2}(a_{1}),a,\tau} denote the arrow $u \circ \cyl\big(s(a_{1})\big)$ of $\cl{A}$. We claim that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),a_{1},a,i_{0}\big(\cyl(a_{1})\big),\tau,l,j_{1}} %
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We have that the following diagram in $\cl{A}$ commutes. \trapeziumstwo[{3,3.5,2,0}]{\cyl(a_{1}),\subdiv(a_{1}),\cyl(a_{1}),\cyl^{2}(a_{1}),\cyl\big(\subdiv(a_{1})\big),\cyl(a_{1}),a,r_{1}(a_{1}),i_{0}\big(\subdiv(a_{1})\big),u,v(a_{1}),i_{0}\big(\cyl(a_{1})\big),j_{0} \circ k \circ \Gamma_{ul}(a_{1}),i_{0}\big(\cyl(a_{1})\big),\cyl\big(r_{1}(a_{1})\big),\cyl\big(v(a_{1})\big)} %  
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Since the diagram \tri[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),\cyl(a_{1}),i_{0}\big(\cyl(a_{1})\big),\Gamma_{ul}(a_{1}),id} in $\cl{A}$ commutes we deduce that the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{1}),\subdiv(a_{1}),a,r_{1}(a_{1}),u \circ i_{0}\big(\subdiv(a_{1})\big),j_{0} \circ k \circ v(a_{1})} %
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We also have that the following diagram in $\cl{A}$ commutes. \trapeziums[{3,3.5,2,0}]{\cyl(a_{1}),\subdiv(a_{1}),\cyl^{2}(a_{1}),\cyl\big(\subdiv(a_{1})\big),\cyl(a_{1}),a,r_{0}(a_{1}),i_{0}\big(\subdiv(a_{1})\big),u,id,j_{1} \circ h,i_{0}\big(\cyl(a_{1})\big),\cyl\big(r_{0}(a_{1})\big),\Gamma_{ul}(a_{1})} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \pushout{a_{1},\cyl(a_{1}),\cyl(a_{1}),\subdiv(a_{1}),a,i_{0}(a_{1}),r_{0}(a_{1}),i_{1}(a_{1}),r_{1}(a_{1}),j_{1} \circ h,j_{0} \circ k \circ v(a_{1}),u \circ i_{0}\big(\subdiv(a_{1})\big)} %
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In the proof of Lemma \ref{DoldLemma1} we showed that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{a_{1},\cyl(a_{1}),\cyl(a_{1}),a_{1},i_{0}(a_{1}),h,i_{1}(a_{1}),f \circ k \circ v(a_{1})} %
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Thus there is an arrow \ar{\subdiv(a_{1}),a_{1},r} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout{a_{1},\cyl(a_{1}),\cyl(a_{1}),\subdiv(a_{1}),a_{1},i_{0}(a_{1}),r_{0}(a_{1}),i_{1}(a_{1}),r_{1}(a_{1}),h,f \circ k \circ v(a_{1}),r} %
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In particular since the diagram \tri{\cyl(a_{1}),\subdiv(a_{1}),a_{1},r_{0}(a_{1}),r,h} in $\cl{A}$ commutes we have that the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{1}),\subdiv(a_{1}),a,r_{0}(a_{1}),j_{1} \circ r, j_{1} \circ h} %
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Moreover the following diagram in $\cl{A}$ commutes. \squareabovetriangle{\cyl(a_{1}),\subdiv(a_{1}),a_{0},a_{1},a,r_{1}(a_{1}),r,k \circ v(a_{1}),f,j_{1},j_{0}} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \pushout{a_{1},\cyl(a_{1}),\cyl(a_{1}),\subdiv(a_{1}),a,i_{0}(a_{1}),r_{0}(a_{1}),i_{1}(a_{1}),r_{1}(a_{1}),j_{1} \circ h,j_{0} \circ k \circ v(a_{1}),j_{1} \circ r} %
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Appealing to the universal property of $\subdiv(a_{1})$ we deduce that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\subdiv(a_{1}),\cyl\big(\subdiv(a_{1})\big),a_{1},a,i_{0}\big(\subdiv(a_{1})\big),u,r,j_{1}} % 
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Hence the following diagram in $\cl{A}$ commutes.\trapeziumsthree[{5,3.5,-2,0}]{\cyl(a_{1}),\cyl^{2}(a_{1}),\subdiv(a_{1}),\cyl\big(\subdiv(a_{1})\big),a_{1},a,i_{0}\big(\subdiv(a_{1})\big),\tau,s(a_{1}),r,j_{1},\cyl\big(s(a_{1})\big),i_{0}\big(\cyl(a_{1})\big),u} %
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By definition of the homotopy $l$ with respect to $\cylinder$ constructed in the proof of Lemma \ref{DoldLemma1} the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{1}),\subdiv(a_{1}),a_{1},s(a_{1}),r,l} % 
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),a_{1},a,i_{0}\big(\cyl(a_{1})\big),\tau,l,j_{1}} This completes the proof of the claim.

\end{proof} 

\begin{lem} \label{DoldLemma3} Let $\cylinder = \big( \cyl, i_0, i_1, p, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{ul} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $v$ compatible with $p$, a subdivision structure $\big(\subdiv, r_{0}, r_{1}, s \big)$ compatible with $p$, and an upper left connection structure $\Gamma_{ul}$. Suppose that $\cyl$ preserves subdivision with respect to $\cylinder$. 

Let \ar{a,a_{0},j_{0}} and \ar{a,a_{1},j_{1}} be arrows of $\cl{A}$ which are fibrations with respect to $\cylinder$. Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$ such that the diagram \triother{a_{0},a_{1},a,f,j_{1},j_{0}} in $\cl{A}$ commutes and such that $f$ is a homotopy equivalence with respect to $\cylinder$. 

Let \ar{a_{1},a_{0},g} and \ar{\cyl(a_{1}),a_{1},l} denote the arrows of $\cl{A}$ constructed in Lemma \ref{DoldLemma1} such that the diagram \triother{a_{1},a_{0},a,g,j_{0},j_{1}} in $\cl{A}$ commutes and such that $l$ defines a homotopy from $fg$ to $id(a_{1})$ with respect to $\cylinder$. 

Then there is an arrow \ar{\cyl^{2}(a_{1}),a,\sigma} of $\cl{A}$ with the following properties. 

\begin{itemize}[topsep=1em,itemsep=1em]

\item[(i)] The following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),a_{1},a,i_{0}\big(\cyl(a_{1})\big),\sigma,l,j_{1}}

\item[(ii)] Let $h_{1}$, $h_{2}$, and $h_{3}$ denote the right, left, and bottom boundary homotopies of $\sigma$ respectively, so that we may depict $\sigma$ as follows in the pictorial notation of Remark \ref{DoubleHomotopyPictorialNotationRemark}. \doublehomotopy{l,h_{1},h_{2},h_{3},\sigma} %
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Then the following diagrams in $\cl{A}$ commute. \twosq{\cyl(a_{1}),a_{1},\cyl(a),a,h_{1},j_{1},\cyl(j_{1}),p(a),\cyl(a_{1}),a_{1},\cyl(a),a,h_{2},j_{1},\cyl(j_{1}),p(a)} %
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\sq{\cyl(a_{1}),a_{1},\cyl(a),a,h_{3},j_{1},\cyl(j_{1}),p(a)}

\end{itemize}

\end{lem} 
 
\begin{proof} Let \ar{\cyl^{2}(a_{1}),a,\tau} denote the double homotopy with respect to $\cylinder$ constructed in the proof of Lemma \ref{DoldLemma2}. In particular the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{1}),a_{1},\cyl^{2}(a_{1}),a,l,\tau,i_{0}\big(\cyl(a_{1})\big),j_{1}} %
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Since $j_{1}$ is a fibration with respect to $\cylinder$ there is thus an arrow \ar{\cyl^{2}(a_{1}),a_{1},\sigma} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \liftingsquare[{5,3}]{\cyl(a_{1}),a_{1},\cyl^{2}(a_{1}),a,l,\tau,i_{0}\big(\cyl(a_{1})\big),j_{1},\sigma} %
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Let $h_{1}$, $h_{2}$, and $h_{3}$ denote the right, left, and bottom boundary homotopies of $\sigma$ respectively, so that we may depict $\sigma$ as follows in the pictorial notation of Remark \ref{DoubleHomotopyPictorialNotationRemark}. \doublehomotopy{l,h_{1},h_{2},h_{3},\sigma}

Firstly let us prove that the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{1}),a_{1},\cyl(a),a,h_{1},j_{1},\cyl(j_{1}),p(a)} %
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By definition of $\Gamma_{ul}$ as an upper left connection structure the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),a_{1},\cyl(a_{1}),\cyl\big(i_{1}(a_{1})\big),\Gamma_{ul}(a_{1}),p(a_{1}),i_{1}(a_{1})} %
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Let \ar{\cyl\big(\subdiv(a_{1})\big),a,u} denote the arrow of $\cl{A}$ constructed in the proof of Lemma \ref{DoldLemma2}. We have that the following diagram in $\cl{A}$ commutes. \trapeziumstwo[{3,3.5,2,0}]{\cyl(a_{1}),\cyl^{2}(a_{1}),a_{1},\cyl^{2}(a_{1}),\cyl\big(\subdiv(a_{1})\big),\cyl(a_{1}),a,\cyl\big(i_{1}(a_{1})\big),\cyl\big(s(a_{1})\big),u,p(a_{1}),i_{1}(a_{1}),j_{1} \circ h,\cyl\big(i_{1}(a_{1})\big),\cyl\big(r_{0}(a_{1})\big),\Gamma_{ul}(a_{1})} %
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We also have that the following diagram in $\cl{A}$ commutes. \tri{a_{1},\cyl(a_{1}),a_{1},i_{1}(a_{1}),h,id} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),a_{1},a,\cyl\big(i_{1}(a_{1})\big),u \circ \cyl\big(s(a_{1})\big),p(a_{1}),j_{1}} %
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Moreover by definition of $\tau$ the following diagram in $\cl{A}$ commutes. \tri[{5,3}]{\cyl^{2}(a_{1}),\cyl\big(\subdiv(a_{1})\big),a,\cyl\big(s(a_{1})\big),u,\tau} %
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Thus we have that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),a_{1},a,\cyl\big(i_{1}(a_{1})\big),\tau,p(a_{1}),j_{1}} %
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By definition of $h_{1}$ the following diagram in $\cl{A}$ commutes.\tri[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),a,\cyl\big(i_{1}(a_{1})\big),\sigma,h_{1}} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes \squareofthreetrianglesthree[{4,3.5,0,-0.5}]{\cyl(a_{1}),a,\cyl^{2}(a_{1}),a_{1},a,h_{1},j_{1},p(a_{1}),j_{1},\cyl\big(i_{1}(a_{1})\big),\sigma,\tau} %
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We also have that the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{1}),a_{1},\cyl(a),a,p(a_{1}),j_{1},\cyl(j_{1}),p(a)} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes, as required.  \sq{\cyl(a_{1}),a_{1},\cyl(a),a,h_{1},j_{1},\cyl(j_{1}),p(a)} 

Secondly let us prove that the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{1}),a_{1},\cyl(a),a,h_{2},j_{1},\cyl(j_{1}),p(a)} %
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Let \ar{\cyl(a_{1}),a_{0},k} denote the arrow of $\cl{A}$ of the proof of Lemma \ref{DoldLemma1}. We have that the following diagram in $\cl{A}$ commutes. \trapeziums[{3,3.5,2,0}]{\cyl(a_{1}),\cyl^{2}(a_{1}),\cyl^{2}(a_{1}),\cyl\big(\subdiv(a_{1})\big),\cyl^{2}(a_{1}),a,\cyl\big(i_{0}(a_{1})\big),\cyl\big(s(a_{1})\big),u,\cyl\big(i_{1}(a_{1})\big),j_{0} \circ k \circ \Gamma_{ul}(a_{1}),\cyl\big(i_{0}(a_{1})\big),\cyl\big(r_{1}(a_{1})\big),\cyl\big(v(a_{1})\big)} %
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By definition of $\Gamma_{ul}$ as an upper left connection structure the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),a_{1},\cyl(a_{1}),\cyl\big(i_{1}(a_{1})\big),\Gamma_{ul}(a_{1}),p(a_{1}),i_{1}(a_{1})} %
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Again we also have that the following diagram in $\cl{A}$ commutes by definition of $\tau$. \tri[{5,3}]{\cyl^{2}(a_{1}),\cyl\big(\subdiv(a_{1})\big),a,\cyl\big(s(a_{1})\big),u,\tau} %
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Putting the last three observations together we have that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),a_{1},a,\cyl\big(i_{0}(a_{1})\big),\tau,p(a_{1}),j_{0} \circ k \circ i_{1}(a_{1})} %
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By definition of $g$ the following diagram in $\cl{A}$ also commutes. \tri{a_{1},\cyl(a_{1}),a_{0},i_{1}(a_{1}),k,g} %
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Moreover we have that the following diagram in $\cl{A}$ commutes. \tri{a_{1},a_{0},a,g,j_{0},j_{1}} %
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Putting the last three observations together we have that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),a_{1},a,\cyl\big(i_{0}(a_{1})\big),\tau,p(a_{1}),j_{1}} %
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By definition of $h_{2}$ the following diagram in $\cl{A}$ also commutes.\tri[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),a,\cyl\big(i_{0}(a_{1})\big),\sigma,h_{2}} %
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Putting the last two observations together, we have that the following diagram in $\cl{A}$ commutes. \squareofthreetrianglesthree[{4,3.5,0,-0.5}]{\cyl(a_{1}),a,\cyl^{2}(a_{1}),a_{1},a,h_{2},j_{1},p(a_{1}),j_{1},\cyl\big(i_{0}(a_{1})\big),\sigma,\tau} %
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Appealing to the commutativity of the diagram \sq{\cyl(a_{1}),a_{1},\cyl(a),a,p(a_{1}),j_{1},\cyl(j_{1}),p(a)} in $\cl{A}$ we deduce that the following diagram in $\cl{A}$ commutes, as required. \sq{\cyl(a_{1}),a_{1},\cyl(a),a,h_{2},j_{1},\cyl(j_{1}),p(a)} 

Thirdly let us prove that the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{1}),a_{1},\cyl(a),a,h_{3},j_{1},\cyl(j_{1}),p(a)} %
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The following diagram in $\cl{A}$ commutes. \trapeziumstwo[{3,3.5,2,0}]{\cyl(a_{1}),\subdiv(a_{1}),\cyl(a_{1}),\cyl^{2}(a_{1}),\cyl\big(\subdiv(a_{1})\big),\cyl^{2}(a_{1}),a,r_{1}(a_{1}),i_{1}\big(\subdiv(a_{1})\big),u,v(a_{1}),i_{1}\big(\cyl(a_{1})\big),j_{0} \circ k \circ \Gamma_{ul}(a_{1}),i_{1}\big(\cyl(a_{1})\big),\cyl\big(r_{1}(a_{1})\big),\cyl\big(v(a_{1})\big)} %
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By definition of $\Gamma_{ul}$ as an upper left connection structure the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),a_{1},\cyl(a_{1}),i_{1}\big(\cyl(a_{1})\big),\Gamma_{ul}(a_{1}),p(a_{1}),i_{1}(a_{1})} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{\cyl(a_{1}),\subdiv(a_{1}),a_{1},a,r_{1}(a_{1}),u \circ i_{1}\big(\subdiv(a_{1})\big),p(a_{1}) \circ v(a_{1}),j_{0} \circ k \circ i_{1}(a_{1})} %
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Moreover as earlier in the proof the following diagram in $\cl{A}$ commutes. \tri{a_{1},\cyl(a_{1}),a,i_{1}(a_{1}),k \circ j_{0},j_{1}} %
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In addition since $v$ is compatible with $p$ the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{1}),\cyl(a_{1}),a_{1},v(a_{1}),p(a_{1}),p(a_{1})} %
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Putting the last three observations together we have that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl(a_{1}),\subdiv(a_{1}),a_{1},a,r_{1}(a_{1}),u \circ i_{1}\big(\subdiv(a_{1})\big),p(a_{1}),j_{1}} %
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Let \ar{\cyl(a_{1}),a_{1},h} denote the homotopy from $ff^{-1}$ to $id(a_{1})$ of the proof of Lemma \ref{DoldLemma1}. In particular the following diagram in $\cl{A}$ commutes. \tri{a_{1},\cyl(a_{1}),a_{1},i_{1}(a_{1}),h,id} %
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We also have that the following diagram in $\cl{A}$ commutes. \trapeziumstwo[{3,3.5,2,0}]{\cyl(a_{1}),\subdiv(a_{1}),a_{1},\cyl^{2}(a_{1}),\cyl\big(\subdiv(a_{1})\big),\cyl(a_{1}),a,r_{0}(a_{1}),i_{1}\big(\subdiv(a_{1})\big),u,p(a_{1}),i_{1}(a_{1}),j_{1} \circ h,i_{1}\big(\cyl(a_{1})\big),\cyl\big(r_{0}(a_{1})\big),\Gamma_{ul}(a_{1})} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{1}),\subdiv(a_{1}),a_{1},a,r_{0}(a_{1}),u \circ i_{1}\big(\subdiv(a_{1})\big),p(a_{1}),j_{1}} %
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Putting everything together we have now shown that the following diagram in $\cl{A}$ commutes. \pushout[{3,3,1}]{a_{1},\cyl(a_{1}),\cyl(a_{1}),\subdiv(a_{1}),a,i_{0}(a_{1}),r_{0}(a_{1}),i_{1}(a_{1}),r_{1}(a_{1}),j_{1} \circ p(a_{1}),j_{1} \circ p(a_{1}),u \circ i_{1}\big(\subdiv(a_{1})\big)} %
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Let \ar{\subdiv,\id_{\cl{A}},\overline{p}} denote the 2-arrow of $\cl{C}$ of Definition \ref{SubdivisionCompatibleWithContractionDefinition}. By definition the following diagram in $\cl{A}$ commutes.  \pushout{a_{1},\cyl(a_{1}),\cyl(a_{1}),\subdiv(a_{1}),a_{1},i_{0}(a_{1}),r_{0}(a_{1}),i_{1}(a_{1}),r_{1}(a_{1}),p(a_{1}),p(a_{1}),\overline{p}(a_{1})} %
%
Hence the following diagram in $\cl{A}$ commutes. \pushout{a_{1},\cyl(a_{1}),\cyl(a_{1}),\subdiv(a_{1}),a,i_{0}(a_{1}),r_{0}(a_{1}),i_{1}(a_{1}),r_{1}(a_{1}),j_{1} \circ p(a_{1}),j_{1} \circ p(a_{1}),j_{1} \circ \overline{p}(a_{1})} %
%
Appealing to the universal property of $\subdiv(a_{1})$ we deduce that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\subdiv(a_{1}),\cyl\big(\subdiv(a_{1})\big),a_{1},a,i_{1}\big(\subdiv(a_{1})\big),u,\overline{p}(a_{1}),j_{1}} %
%
Since the subdivision structure $\big(\subdiv,r_{0},r_{1},s\big)$ is compatible with $p$ we also have that the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{1}),\subdiv(a_{1}),a_{1},s(a_{1}),\overline{p}(a_{1}),p(a_{1})} %
%
Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \trapeziums[{3,3.5,2,0}]{\cyl(a_{1}),\subdiv(a_{1}),\subdiv(a_{1}),\cyl\big(\subdiv(a_{1})\big),a_{1},a,i_{1}\big(\cyl(a_{1})\big),\cyl\big(s(a_{1})\big),u,p(a_{1}),j_{1},s(a_{1}),i_{1}\big(\subdiv(a_{1})\big),\overline{p}(a_{1})} %
%
Once more we also have that the following diagram in $\cl{A}$ commutes by definition of $\tau$. \tri[{5,3}]{\cyl^{2}(a_{1}),\cyl\big(\subdiv(a_{1})\big),a,\cyl\big(s(a_{1})\big),u,\tau} %
%
Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl(a_{1}),\subdiv(a_{1}),a_{1},a,i_{1}\big(\cyl(a_{1})\big),\tau,p(a_{1}),j_{1}} %
%
By definition of $h_{3}$ the following diagram in $\cl{A}$ also commutes.\tri[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),a,i_{1}\big(\cyl(a_{1})\big),\sigma,h_{3}} %
%
Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \squareofthreetrianglesthree[{4,3.5,0,-0.5}]{\cyl(a_{1}),a,\cyl^{2}(a_{1}),a_{1},a,h_{3},j_{1},p(a_{1}),j_{1},i_{0}\big(\cyl(a_{1})\big),\sigma,\tau} %
%
Appealing to the commutativity of the diagram \sq{\cyl(a_{1}),a_{1},\cyl(a),a,p(a_{1}),j_{1},\cyl(j_{1}),p(a)} in $\cl{A}$ we deduce that the following diagram in $\cl{A}$ commutes, as required. \sq{\cyl(a_{1}),a_{1},\cyl(a),a,h_{3},j_{1},\cyl(j_{1}),p(a)} %

\end{proof}

\begin{lem} \label{DoldRightOverHomotopyInverseLemma} Let $\cylinder = \big( \cyl, i_0, i_1, p, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{ul} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $v$ compatible with $p$, a subdivision structure $\big(\subdiv,r_{0},r_{1},s \big)$ compatible with $p$, and an upper left connection structure $\Gamma_{ul}$. Suppose that $\cyl$ preserves subdivision with respect to $\cylinder$. 

Let \ar{a,a_{0},j_{0}} and \ar{a,a_{1},j_{1}} be arrows of $\cl{A}$ which are fibrations with respect to $\cylinder$. Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$ such that the diagram \triother{a_{0},a_{1},a,f,j_{1},j_{0}} in $\cl{A}$ commutes and such that $f$ is a homotopy equivalence with respect to $\cylinder$. 

Then there is an arrow \ar{a_{1},a_{0},g} of $\cl{A}$ such that the diagram \triother{a_{1},a_{0},a,g,j_{0},j_{1}} in $\cl{A}$ commutes and such that there is a homotopy over $a$ from $fg$ to $id(a_{1})$ with respect to $\cylinder$ and $(j_{1},j_{1})$.  \end{lem}

\begin{proof} Let \ar{a_{1},a_{0},g} and \ar{\cyl(a_{1}),a_{1},l} denote the arrows of $\cl{A}$ constructed in Lemma \ref{DoldLemma1}. Let \ar{\cyl^{2}(a_{1}),a_{1},\sigma} denote the arrow of $\cl{A}$ constructed in Lemma \ref{DoldLemma3}. In the pictorial notation of Remark \ref{DoubleHomotopyPictorialNotationRemark} the boundary of $\sigma$ is as follows, where $h_{1}$, $h_{2}$, and $h_{3}$ are all homotopies over $a$ with respect to $\cylinder$ and $(j_{1},j_{1})$. \doublehomotopy{l,h_{1},h_{2},h_{3},\sigma} %
%
By Proposition \ref{ReverseHomotopyIsOverHomotopyProposition} and Proposition \ref{CompositeHomotopyIsOverHomotopyProposition} the following diagram in $\cl{A}$ commutes. \sq[{7,3}]{\cyl(a_{1}),a_{1},\cyl(a),a,{(h_{2} + h_{3}) + h_{1}^{-1}},j_{1},\cyl(j_{1}),p(a)} %
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The following diagram in $\cl{A}$ also commutes. \sq{a_{1},\cyl(a_{1}),\cyl(a_{1}),a_{1},i_{0}(a_{1}),{(h_{2}) + h_{3}) + h_{1}^{-1}},i_{0}(a_{1}),l} %
%
Since the diagram \tri{a_{1},\cyl(a_{1}),a_{1},i_{0}(a_{1}),l,gf} in $\cl{A}$ commutes we deduce that the following diagram in $\cl{A}$ commutes. \tri{a_{1},\cyl(a_{1}),a_{1},i_{0}(a_{1}),{(h_{2} + h_{3}) + h_{1}^{-1}},gf} %
%
Moreover the following diagram in $\cl{A}$ commutes. \sq{a_{1},\cyl(a_{1}),\cyl(a_{1}),a_{1},i_{1}(a_{1}),{(h_{2}) + h_{3}) + h_{1}^{-1}},i_{1}(a_{1}),l} %
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Since the diagram \tri{a_{1},\cyl(a_{1}),a_{1},i_{1}(a_{1}),l,id} in $\cl{A}$ commutes we deduce that the following diagram in $\cl{A}$ commutes. \tri{a_{1},\cyl(a_{1}),a_{1},i_{1}(a_{1}),{(h_{2} + h_{3}) + h_{1}^{-1}},id} %
%
Putting everything together we have that $(h_{2} + h_{3}) + h_{1}^{-1}$ defines a homotopy over $a$ from $gf$ to $id(a_{1})$ with respect to $\cylinder$ and $(j_{1},j_{1})$. \end{proof}

\begin{prpn} \label{DoldOverHomotopiesCylinderProposition} Let $\cylinder = \big( \cyl, i_0, i_1, p, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{ul} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $v$ compatible with $p$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ compatible with $p$, and an upper left connection structure $\Gamma_{ul}$. Suppose that $\cyl$ preserves subdivision with respect to $\cylinder$. 

Let \ar{a,a_{0},j_{0}} and \ar{a,a_{1},j_{1}} be arrows of $\cl{A}$ which are fibrations with respect to $\cylinder$. Let \ar{a_{0},a_{1},f} of $\cl{A}$ be an arrow of $\cl{A}$ such that the diagram \triother{a_{0},a_{1},a,f,j_{1},j_{0}} in $\cl{A}$ commutes, and such that $f$ is a homotopy equivalence with respect to $\cylinder$. Then $f$ is moreover a homotopy equivalence over $a$ with respect to $\cylinder$ and $(j_{0},j_{1})$. \end{prpn}

\begin{proof} By Lemma \ref{DoldRightOverHomotopyInverseLemma} there is an arrow \ar{a_{1},a_{0},g} of $\cl{A}$ such that the diagram \triother{a_{1},a_{0},a,g,j_{0},j_{1}} in $\cl{A}$ commutes together with a homotopy \ar{\cyl(a_{1}),a_{1},h} over $a$ from $fg$ to $id(a_{1})$ with respect to $\cylinder$ and $(j_{1},j_{1})$. It remains to construct a homotopy over $a$ from $gf$ to $id(a_{0})$ with respect to $\cylinder$ and $(j_{0},j_{0})$. 

By Lemma \ref{RightHomotopyInverseIsHomotopyInverseLemma} $g$ is a homotopy equivalence with respect to $\cylinder$. Thus $g$ satisfies the hypotheses of Lemma \ref{DoldRightOverHomotopyInverseLemma}. We deduce that there is an arrow \ar{a_{0},a_{1},g'} of $\cl{A}$ such that the diagram \triother{a_{0},a_{1},a,g',j_{1},j_{0}} in $\cl{A}$ commutes and such that there is a homotopy \ar{\cyl(a_{0}),a_{0},h'} over $a$ from $gg'$ to $id(a_{0})$ with respect to $\cylinder$ and $(j_{0},j_{0})$.

By Corollary \ref{ReverseHomotopyIsOverHomotopyCorollary} we have that $(h')^{-1}$ defines a homotopy over $a$ from $id(a_{0})$ to $gg'$ with respect to $\cylinder$ and $(j_{0},j_{0})$. Thus by Lemma \ref{HomotopyPreAndPostCompositionIsOverHomotopyLemma} the arrow \ar[8]{\cyl(a_{1}),a_{1},{(h')^{-1}} \circ \cyl(gf)} of $\cl{A}$ defines a homotopy over $a$ from $gf$ to $gfgg'$ with respect to $\cylinder$ and $(j_{0},j_{0})$. 

Appealing again to Lemma \ref{HomotopyPreAndPostCompositionIsOverHomotopyLemma}, we also have that the arrow \ar[7]{\cyl(a_{1}),a_{1},g \circ h \circ \cyl(g')} of $\cl{A}$ defines a homotopy over $a$ from $gfgg'$ to $gg'$ with respect to $\cylinder$ and $(j_{0},j_{0})$. %
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By virtue of Corollary \ref{CompositeHomotopyIsOverHomotopyCorollary}, we have that the arrow \ar[17]{\cyl(a_{1}),a_{1},\Big( (h')^{-1} \circ \cyl(gf) \Big) + \Big( g \circ h \circ \cyl(g') \Big)} of $\cl{A}$ defines a homotopy over $a$ from $gf$ to $gg'$ with respect to $\cylinder$ and $(j_{0},j_{0})$. %
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Let us denote it by $k$ for brevity. Appealing to Corollary \ref{CompositeHomotopyIsOverHomotopyCorollary} once more we have that the arrow \ar[4]{\cyl(a_{1}),a_{1},k + h'} of $\cl{A}$ defines a homotopy over $a$ from $gf$ to $id(a_{0})$. \end{proof}

\begin{cor} \label{DoldUnderHomotopiesCoCylinderCorollary} Let $\cocylinder = \big( \cocyl, e_0, e_1, c, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{ul} \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$, an involution structure $v$ compatible with $c$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ compatible with $c$, and an upper left connection structure $\Gamma_{ul}$. Suppose that $\cocyl$ preserves subdivision with respect to $\cocylinder$. 

Let \ar{a,a_{0},j_{0}} and \ar{a,a_{1},j_{1}} be arrows of $\cl{A}$ which are cofibrations with respect to $\cocylinder$. Let \ar{a_{0},a_{1},f} of $\cl{A}$ be an arrow of $\cl{A}$ such that the diagram \tri{a,a_{0},a_{1},j_{0},f,j_{1}} in $\cl{A}$ commutes, and such that $f$ is a homotopy equivalence with respect to $\cocylinder$. Then $f$ is moreover a homotopy equivalence under $a$ with respect to $\cocylinder$ and $(j_{0},j_{1})$. \end{cor}

\begin{proof} Follows immediately from Proposition \ref{DoldOverHomotopiesCylinderProposition} by duality. \end{proof}

\begin{cor} \label{DoldUnderHomotopiesCylinderCorollary} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. Let $\cocylinder = \big( \cocyl, e_0, e_1, c, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{ul}  \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$, an involution structure $v$ compatible with $c$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ compatible with $c$, and an upper left connection structure $\Gamma_{ul}$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

Let \ar{a,a_{0},j_{0}} and \ar{a,a_{1},j_{1}} be arrows of $\cl{A}$ which are cofibrations with respect to $\cylinder$. Let \ar{a_{0},a_{1},f} of $\cl{A}$ be an arrow of $\cl{A}$ such that the diagram \tri{a,a_{0},a_{1},j_{0},f,j_{1}} in $\cl{A}$ commutes, and such that $f$ is a homotopy equivalence with respect to $\cylinder$. Then $f$ is moreover a homotopy equivalence under $a$ with respect to $\cylinder$ and $(j_{0},j_{1})$. \end{cor}

\begin{proof} Since $\cylinder$ is left adjoint to $\cocylinder$ we have that $\cocylinder$ preserves subdivision. By Proposition \ref{CofibrationCoCylinderIffCofibrationCylinderCorollary} we have that $j_{0}$ and $j_{1}$ are cofibrations with respect to $\cocylinder$. %
%
By Proposition \ref{HomotopyEquivalenceCylinderIffHomotopyEquivalenceCoCylinderProposition} we have that $f$ is a homotopy equivalence with respect to $\cocylinder$. 

We deduce by Corollary \ref{DoldUnderHomotopiesCoCylinderCorollary} that $f$ is moreover a homotopy equivalence under $a$ with respect to $\cocylinder$ and $(j_{0},j_{1})$. Hence by Corollary \ref{UnderHomotopyCoCylinderGivesUnderHomotopyCylinderCorollary} $f$ is a homotopy equivalence under $a$ with respect to $\cylinder$ and $(j_{0},j_{1})$. \end{proof}

\begin{rmk} Assuming that $\cylinder$ preserves subdivision but not necessarily that it is left adjoint to a co-cylinder it is possible to prove Corollary \ref{DoldUnderHomotopiesCylinderCorollary} directly for a cylinder $\cylinder = \big(\cyl, i_{0}, i_{1}, p, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{ul} \big)$ equipped with a contraction structure $p$, an involution structure $v$ compatible with $p$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ compatible with $p$, and an upper left connection structure $\Gamma_{ul}$. 

For this we need that if an arrow $j$ of $\cl{A}$ is a cofibration then so is $\cyl(j)$. This can be proven, and is the approach taken in the book \cite{MayAConciseCourseInAlgebraicTopology} of May and the book \cite{KampsPorterAbstractHomotopyAndSimpleHomotopyTheory} of Kamps and Porter. %
%
However the proof relies upon the assumption that $\cylinder$ admits a {\em transposition} structure, namely a 2-arrow \ar{\cyl^{2},\cyl^{2},t} of $\cl{C}$ such that the following diagrams in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commute. \twotriangles[{4,3,-1}]{\cyl,\cyl^{2},\cyl^{2},i_{0} \cdot \cyl,t,\cyl \cdot i_0,\cyl,\cyl^{2},\cyl^{2},\cyl \cdot i_0, t, i_0 \cdot \cyl} %
%
We might also require that the two analogous diagrams involving $i_{1}$ commute, but this is not necessary for the proof of Dold's theorem.

For example suppose that we are working in the 2-category of categories and that $\cylinder$ arises from an interval $\widehat{I}$ in a braided monoidal category $\cl{A}$. Then the arrow \ar{I^{2},I^{2}} of $\cl{A}$ which defines the braiding gives a transposition structure with respect to $\cylinder$. 

We will not need a transposition structure anywhere else in this work. Moreover it will later be indispensable for us to assume that we have an adjoint cylinder and co-cylinder. For these reasons we have chosen to give a different proof. \end{rmk}

\begin{cor} \label{DoldOverHomotopiesCoCylinderCorollary} Let $\cylinder = \big( \cyl, i_0, i_1, p, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{ul} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $v$ compatible with $p$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s, \big)$ compatible with $p$, and an upper left connection structure $\Gamma_{ul}$. Let $\cocylinder = \big( \cocyl, e_0, e_1, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

Let \ar{a,a_{0},j_{0}} and \ar{a,a_{1},j_{1}} be arrows of $\cl{A}$ which are fibrations with respect to $\cocylinder$. Let \ar{a_{0},a_{1},f} of $\cl{A}$ be an arrow of $\cl{A}$ such that the diagram \triother{a_{0},a_{1},a,f,j_{1},j_{0}} in $\cl{A}$ commutes, and such that $f$ is a homotopy equivalence with respect to $\cocylinder$. Then $f$ is moreover a homotopy equivalence over $a$ with respect to $\cocylinder$ and $(j_{0},j_{1})$. \end{cor}

\begin{proof} Follows immediately from Corollary \ref{DoldUnderHomotopiesCylinderCorollary} by duality. \end{proof}

\begin{cor} \label{TrivialCofibrationAdmitsStrongDeformationRetractionCorollary} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. Let $\cocylinder = \big( \cocyl, e_0, e_1, c, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{ul}  \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$, an involution structure $v$ compatible with $c$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ compatible with $p$, and an upper left connection structure $\Gamma_{ul}$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$ which is a trivial cofibration with respect to $\cylinder$. Then $j$ admits a strong deformation retraction with respect to $\cylinder$. \end{cor}

\begin{proof} By Proposition \ref{IdentityIsCofibrationProposition} $id(a_{0})$ is a cofibration with respect to $\cylinder$. Moreover the following diagram in $\cl{A}$ commutes. \tri{a_{0},a_{0},a_{1},id,j,j} %
%
Thus by Corollary \ref{DoldUnderHomotopiesCylinderCorollary} $j$ defines a homotopy equivalence under $a_{0}$ with respect to $\cylinder$ and $\big( id(a_{0}),j \big)$. This means exactly that there is an arrow \ar{a_{1},a_{0},j'} of $\cl{A}$ which is a strong deformation retraction of $j$ with respect to $\cylinder$.  \end{proof}

\begin{cor} \label{TrivialFibrationIsStrongDeformationRetractionCorollary} Let $\cylinder = \big( \cyl, i_0, i_1, p, v, \subdiv, r_0, r_1, s, \Gamma_{ul} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $v$ compatible with $p$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ compatible with $p$, and an upper left connection structure $\Gamma_{ul}$. Let $\cocylinder = \big( \cocyl, e_0, e_1, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$ which is a trivial fibration with respect to $\cocylinder$. Then there is an arrow \ar{a_{0},a_{1},j} of $\cl{A}$ such that $f$ is a strong deformation retraction of $j$ with respect to $\cocylinder$. \end{cor}

\begin{proof} Follows immediately from Corollary \ref{TrivialCofibrationAdmitsStrongDeformationRetractionCorollary} by duality. \end{proof}

\end{section}

example--categories_and_groupoids.tex
\begin{section}{Example --- categories and groupoids} \label{ExampleSection}

We define an interval in the category $\mathsf{Cat}$ of categories equipped with its cartesian monoidal structure. It admits all the structures of \ref{StructuresUponAnIntervalSection} and has strictness of identities. By \ref{ModelStructureSection} we thus obtain two model structures upon $\mathsf{Cat}$. In a non-constructive setting both model structures can be proven to coincide with folk model structure. 

In the same way we obtain two model structures upon the category $\mathsf{Grpd}$ of groupoids. Again both may be demonstrated by a non-constructive argument to coincide with the folk model structure.

The folk model structure on $\mathsf{Cat}$ was constructed by Joyal and Tierney in \cite{JoyalTierneyStrongStacksAndClassifyingSpaces}. Independently a construction was given by Rezk in \cite{RezkAModelCategoryForCategories}. 

The folk model structure on $\mathsf{Grpd}$ appeared in the literature earlier. It was first described by Anderson in \S{5} of \cite{AndersonFibrationsAndGeometricRealizations}, and is also discussed in \S{14.1} of the paper \cite{BousfieldHomotopySpectralSequencesAndObstructions} of Bousfield. A detailed construction is given in \S{6.1} of the article \cite{StricklandKnLocalDualityForFiniteGroupsAndGroupoids} of Strickland, built upon in \S{3} of the thesis \cite{HollanderAHomotopyTheoryForStacks} of Hollander. 

The folk model structure on groupoids can also be seen to arise as the restriction to groupoids of the model structure on $\mathsf{Cat}$ constructed by Thomason in \cite{ThomasonCatAsAClosedModelCategory}. This is observed for example in \S{1} of the paper \cite{CasacubertaGolasinskiTonksHomotopyLocalizationOfGroupoids} of Casacuberta, Golasi{\'n}ski, and Tonks.

In all these works the non-constructive characterisation of equivalences of categories as functors which are fully faithful and essentially surjective is essential. From this point of view the folk model structure on $\mathsf{Cat}$ or $\mathsf{Grpd}$ is akin to the Serre model structure on topological spaces which was first constructed in \S{II.3} of \cite{QuillenHomotopicalAlgebra}. 

The conceptual approach we have taken is significantly different. Our two model structures are akin to the model structure on topological spaces which was constructed by Str{\o}m in \cite{StromTheHomotopyCategoryIsAHomotopyCategory}. The fact that we may non-constructively identify the two model structures on categories or groupoids which we construct with the folk model structure might reasonably we think be viewed as something of a coincidence.

\begin{notn} Let $\mathsf{Cat}$ denote the category of categories and let $\mathsf{Grpd}$ denote the category of groupoids. We denote by $1$ the final object of $\mathsf{Cat}$ and $\mathsf{Grpd}$, the category with a unique object $\bullet$ and a unique arrow. \end{notn}

\begin{rmk} We will regard $\mathsf{Cat}$ and $\mathsf{Grpd}$ as equipped with their cartesian monoidal structures. These monoidal structures are closed and thus Requirement \ref{SubdivisionRequirement} is satisfied. \end{rmk}

\begin{notn} Let $\cl{I}$ denote the free groupoid on the following directed graph. \ar{0,1} \end{notn}

\begin{notn} Let \ar{1,\cl{I},i_{0}} denote the unique functor which maps $\bullet$ to $0$. \end{notn}

\begin{notn} Let \ar{1,\cl{I},i_{1}} denote the unique functor which maps $\bullet$ to $1$. \end{notn}

\begin{notn} Let $\interval$ denote the interval $\big( \cl{I},i_{0},i_{1} \big)$ in $\mathsf{Cat}$ or $\mathsf{Grpd}$. \end{notn}

\begin{observation} The canonical functor \ar{\cl{I},1,p} equips $\interval$ with a contraction structure. \end{observation} 

\begin{notn} Let \ar{\cl{I},\cl{I},v} denote the unique functor which maps the arrow \ar{0,1} of $\cl{I}$ to the arrow \ar{1,0} of $\cl{I}$. \end{notn}

\begin{observation} The functor $v$ equips $\interval$ with an involution structure which is compatible with $p$. \end{observation}

\begin{notn} Let $\cl{S}$ denote the free groupoid on the following directed graph. \tri{0,1,2} \end{notn}

\begin{notn} Let \ar{\cl{I},\cl{S},r_{0}} denote the unique functor which maps the arrow \ar{0,1} of $\cl{I}$ to the arow \ar{0,1} of $\cl{S}$. \end{notn}

\begin{notn} Let \ar{\cl{I},\cl{S},r_{1}} denote the unique functor which maps the arrow \ar{0,1} of $\cl{I}$ to the arow \ar{1,2} of $\cl{S}$. \end{notn}

\begin{notn} Let \ar{\cl{I},\cl{S},s} denote the unique functor which maps the arrow \ar{0,1} of $\cl{I}$ to the arow \ar{0,2} of $\cl{S}$. \end{notn}

\begin{observation} The data $\big( \cl{S}, r_{0}, r_{1}, s \big)$ equips $\interval$ with a subdivision structure which is compatible with $p$.
\end{observation}

\begin{observation} With respect to the involution structure $v$ and the subdivision structure $\big( S, r_{0}, r_{1}, s \big)$ the interval $\interval$ has strictness of identities and strictness of left inverses. \end{observation}

\begin{observation} The groupoid $\cl{I}^{2}= \cl{I} \times \cl{I}$ is the unique groupoid with objects and arrows as follows excluding the four identity arrows. 

\begin{diagram}

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\draw[->] (2-1) to node[auto] {} (2-0);
\draw[->] (3-1) to node[auto] {} (2-1);
\draw[->] (0-0) to node[auto,swap] {} (1-1);
\draw[->] (3-1) to node[auto] {} (2-0);

\end{tikzpicture} 

\end{diagram} 

\end{observation}

\begin{notn} Let \ar{\cl{I}^{2},\cl{I},\Gamma_{ul}} denote the unique functor with the following properties: 

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] the arrow \ar{{(0,0)},{(1,0)}} of $\cl{I}^{2}$ maps to the arrow \ar{0,1} of $\cl{I}$. 

\item[(ii)] the arrow \ar{{(0,0)},{(0,1)}} of $\cl{I}^{2}$ maps to the arrow \ar{0,1} of $\cl{I}$. 

\item[(iii)] the arrow \ar{{(1,0)},{(1,1)}} of $\cl{I}^{2}$ maps to the arrow \ar{1,1} of $\cl{I}$. 

\item[(iv)] the arrow \ar{{(0,1)},{(1,1)}} of $\cl{I}^{2}$ maps to the arrow \ar{1,1} of $\cl{I}$.

\end{itemize}

\end{notn} 

\begin{observation} The functor $\Gamma_{ul}$ equips $\interval$ with an upper left connection structure. \end{observation}

\begin{notn} Let \ar{\cl{I}^{2},\cl{I},\Gamma_{lr}} denote the unique functor with the following properties: 

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] the arrow \ar{{(0,0)},{(1,0)}} of $\cl{I}^{2}$ maps to the arrow \ar{0,0} of $\cl{I}$. 

\item[(ii)] the arrow \ar{{(0,0)},{(0,1)}} of $\cl{I}^{2}$ maps to the arrow \ar{0,0} of $\cl{I}$. 

\item[(iii)] the arrow \ar{{(1,0)},{(1,1)}} of $\cl{I}^{2}$ maps to the arrow \ar{0,1} of $\cl{I}$. 

\item[(iv)] the arrow \ar{{(0,1)},{(1,1)}} of $\cl{I}^{2}$ maps to the arrow \ar{0,1} of $\cl{I}$.

\end{itemize}

\end{notn} 

\begin{observation} The functor $\Gamma_{lr}$ equips $\interval$ with a lower right connection structure which is compatible with $p$. \end{observation}

\begin{notn} Let \ar{\cl{I}^{2},\cl{I},\Gamma_{ur}} denote the unique functor with the following properties: 

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] the arrow \ar{{(0,0)},{(1,0)}} of $\cl{I}^{2}$ maps to the arrow \ar{0,1} of $\cl{I}$. 

\item[(ii)] the arrow \ar{{(0,0)},{(0,1)}} of $\cl{I}^{2}$ maps to the arrow \ar{0,0} of $\cl{I}$. 

\item[(iii)] the arrow \ar{{(1,0)},{(1,1)}} of $\cl{I}^{2}$ maps to the arrow \ar{1,0} of $\cl{I}$. 

\item[(iv)] the arrow \ar{{(0,1)},{(1,1)}} of $\cl{I}^{2}$ maps to the arrow \ar{0,0} of $\cl{I}$.

\end{itemize}

\end{notn} 

\begin{observation} The functor $\Gamma_{ur}$ equips $\interval$ with an upper right connection structure. We have that $\Gamma_{lr}$ and $\Gamma_{ur}$ are compatible with $\big(\cl{S},r_{0},r_{1},s\big)$. \end{observation}

\begin{observation} A functor is a homotopy equivalence with respect to $\cylinderinterval$ if and only if it is an equivalence of categories. \end{observation}  

\begin{recollection} An {\em iso-fibration} is a functor \ar{\cl{A}_{0},\cl{A}_{1},F} with the property that for every commutative diagram \sq{1,\cl{A}_{0},\cl{I},\cl{A}_{1},a,F,i_{0},g} in $\mathsf{Cat}$ there is a functor \ar{\cl{I},\cl{A}_{0},l} such that the following diagram in $\mathsf{Cat}$ commutes. \liftingsquare{1,\cl{A}_{0},\cl{I},\cl{A}_{1},a,F,i_{0},g,l} \end{recollection}

\begin{recollection} A {\em normally cloven iso-fibration} is a functor \ar{\cl{A}_{0},\cl{A}_{1},F} with the property that to every commutative diagram \sq{1,\cl{A}_{0},\cl{I},\cl{A}_{1},a,F,i_{0},g} in $\mathsf{Cat}$ we can associate a functor \ar{\cl{I},\cl{A}_{0},l} such that the following hold.

\begin{itemize}[itemsep=1em,topsep=1em] 

\item[(i)] The diagram \liftingsquare{1,\cl{A}_{0},\cl{I},\cl{A}_{1},a,F,i_{0},g,l} in $\mathsf{Cat}$ commutes.

\item[(ii)] If the diagram \tri{\cl{I},1,\cl{A}_{0},p,F(a),g} in $\mathsf{Cat}$ commutes then the diagram \tri{\cl{I},1,\cl{A}_{1},p,a,l} in $\mathsf{Cat}$ commutes.

\end{itemize}
\end{recollection}

\begin{observation} A functor \ar{\cl{A}_{0},\cl{A}_{1},F} is a fibration with respect to $\cylinderinterval$ if and only if it is an iso-fibration. This goes back to Proposition 2.1 of the paper \cite{BrownFibrationsOfGroupoids} of Brown. 

Moreover $F$ is a normally cloven fibration with respect to $\cylinderinterval$ if and only if it is a normally cloven iso-fibration. \end{observation}

\begin{defn} An {\em iso-cofibration} is a functor \ar{\cl{A}_{0},\cl{A}_{1},j} such that $j$ is a cofibration with respect to $\cylinderinterval$. \end{defn} 

\begin{rmk} Non-constructively it is possible to characterise an iso-cofibration as a functor which is injective on objects. \end{rmk}

\begin{defn} A {\em normally cloven iso-cofibration} is a functor \ar{\cl{A}_{0},\cl{A}_{1},j} such that $j$ is a normally cloven cofibration with respect to $\cylinderinterval$. \end{defn} 

\begin{thm} We obtain a model structure on $\mathsf{Cat}$ and $\mathsf{Grpd}$ by taking:

\begin{itemize}[itemsep=1em, topsep=1em]

\item[(i)] weak equivalences to be equivalences of categories,

\item[(ii)] fibrations to be iso-fibrations,

\item[(iii)] cofibrations to be normally cloven iso-cofibrations.

\end{itemize}
\end{thm}

\begin{proof} Follows immediately from Corollary \ref{NormallyClovenCofibrationsFibrationsIntervalModelStructureCorollary}. \end{proof}

\begin{thm} We obtain a model structure on $\mathsf{Cat}$ and $\mathsf{Grpd}$ by taking:

\begin{itemize}[itemsep=1em, topsep=1em]

\item[(i)] weak equivalences to be equivalences of categories,

\item[(ii)] fibrations to be normally cloven iso-fibrations,

\item[(iii)] cofibrations to be iso-cofibrations.

\end{itemize}
\end{thm}

\begin{proof} Follows immediately from Corollary \ref{CofibrationsNormallyClovenFibrationsIntervalModelStructureCorollary}. \end{proof}

\end{section}

factorisation_axioms.tex
\begin{section}{Factorisation axioms} \label{FactorisationAxiomsSection}

Let $\cylinder$ be a cylinder in a formal category $\cl{A}$. In \ref{MappingCylindersAndMappingCoCylindersSection} we showed that if $\cylinder$ admits sufficient structure satisfying strictness of right identities then the mapping cylinder of an arrow $f$ of $\cl{A}$ with respect to $\cylinder$ yields a factorisation into a normally cloven cofibration followed by a strong deformation retraction.

We now prove that if $\cylinder$ has strictness of left identities then a strong deformation retraction with respect to $\cylinder$ is a trivial fibration with respect to $\cylinder$. Thus if $\cylinder$ has strictness of both left and right identities, the mapping cylinder of $f$ with respect to $\cylinder$ yields a factorisation of $f$ into a normally cloven cofibration followed by a trivial fibration. 

Dually if $\cocylinder$ is a co-cylinder in $\cl{A}$ which admits sufficient structure satisfying strictness of identities we see that the mapping co-cylinder of $f$ with respect to $\cocylinder$ yields a factorisation of $f$ into a trivial cofibration followed by a normally cloven fibration. 

Building upon these two facts we moreover construct a factorisation of $f$ into a cofibration followed by a trivial normally cloven fibration and into a trivial normally cloven cofibration followed by a fibration.  

Neither strictness of left identities nor strictness of right identities holds for the usual cylinder and co-cylinder in topological space defined by means of the unit interval. The glueing of a path $g$ to a constant path is homotopic to $g$, but does not recover $g$ itself.

Whilst the mapping cylinder of a map between topological spaces gives a factorisation into a cofibration followed by a homotopy equivalence, this homotopy equivalence is not necessarily a fibration. The mapping cylinder of the inclusion of the circle into the disk is an example. 

Dually whilst the mapping co-cylinder of a map between topological spaces gives a factorisation into a homotopy equivalence followed by a fibration, this homotopy equivalence is not necessarily a cofibration. The mapping co-cylinder of any inclusion of spaces which is not closed is an example.

We refer the reader to the introduction \ref{IntroductionSection} to this work for a discussion of a way around this. A guiding example in which strictness of identities does hold is that of the homotopy theory of categories or groupoids which we explore in \ref{ExampleSection}. 

\begin{assum} Let $\cl{C}$ be a 2-category with a final object such that pushouts and pullbacks of 2-arrows of $\cl{C}$ give rise to pushouts and pullbacks in formal categories in the sense of Definition \ref{PushoutsPullbacks2ArrowsArePushoutsPullbacksInFormalCategoriesTerminology}. Let $\cl{A}$ be an object of $\cl{C}$. As before we view $\cl{A}$ as a formal category, writing of objects and arrows of $\cl{A}$. \end{assum} 

\begin{prpn} \label{StrongDeformationRetractionIsFibrationProposition} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p, \subdiv, r_{0}, r_{1}, s \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$. Suppose that $\cylinder$ has strictness of left identities. 

Let \ar{a_{1},a_{2},j} be an arrow of $\cl{A}$, and let \ar{a_{2},a_{1},f} be an arrow of $\cl{A}$ which is a retraction of $j$. Suppose that \ar{\cyl(a_{2}),a_{2},h} defines a homotopy over $a_{1}$ from $id(a_{2})$ to $jf$ with respect to $\cylinder$ and $(f,f)$. Then $f$ is a fibration with respect to $\cylinder$. \end{prpn}

\begin{proof} Suppose that we have a commutative diagram in $\cl{A}$ as follows. \sq{a_{0},a_{2},\cyl(a_{0}),a_{1},g,f,i_{0}(a_{0}),k} %
%
By definition of $h$ the following diagram in $\cl{A}$ commutes. \tri{a_{2},\cyl(a_{2}),a_{2},i_{1}(a_{2}),h,j \circ f} %
%
Appealing to the commutativity of the diagram \sq{a_{0},\cyl(a_{0}),a_{2},\cyl(a_{2}),i_{1}(a_{0}),\cyl(g),g,i_{1}(a_{2})} in $\cl{A}$ we deduce that the following diagram in $\cl{A}$ commutes. \tri{a_{0},\cyl(a_{0}),a_{2},i_{1}(a_{0}),h \circ \cyl(g),j \circ f \circ g} %
%
Moreover the following diagram in $\cl{A}$ commutes. \tri{a_{0},\cyl(a_{0}),a_{2},i_{0}(a_{0}),j \circ k,j \circ f \circ g} %
%
Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{a_{0},\cyl(a_{0}),\cyl(a_{0}),a_{2},i_{0}(a_{0}), j \circ k,i_{1}(a_{0}),h \circ \cyl(g)}

Let \ar{\cyl(a_{0}),a_{2},l} denote the homotopy $\big(h \circ \cyl(g) \big) + (j \circ k)$ with respect to $\cylinder$. We claim that the following diagram in $\cl{A}$ commutes. \liftingsquare{a_{0},a_{2},\cyl(a_{0}),a_{1},g,f,i_{0}(a_{0}),k,l} %
%
Firstly the following diagram in $\cl{A}$ commutes. \squareabovetriangle{a_{0},\cyl(a_{0}),a_{2},\cyl(a_{2}),a_{2},i_{0}(a_{0}),\cyl(g),g,i_{0}(a_{2}),h,id} %
%
By definition of $l$ we also have that the following diagram in $\cl{A}$ commutes. \tri{a_{0},\cyl(a_{0}),a_{2},i_{0}(a_{0}),l,h \circ \cyl(g)} %
%
Hence the following diagram in $\cl{A}$ commutes, as required. \tri{a_{0},\cyl(a_{0}),a_{2},i_{0}(a_{0}),l,g} 

Secondly let us turn to proving that the diagram \tri{\cyl(a_{0}),a_{2},a_{1},l,f,k} in $\cl{A}$ commutes. Let \ar{\subdiv(a_{0}),a_{2},u} denote the canonical arrow of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),\cyl(a_{0}),\subdiv(a_{0}),a_{2},i_{0}(a_{0}),r_{0}(a_{0}),i_{1}(a_{0}),r_{1}(a_{0}),j \circ k, h \circ \cyl(g),u} %
%
By definition of $h$ the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{1}),a_{1},\cyl(a_{2}),a_{2},h,f,\cyl(f),p(a_{2})} %
%
Appealing to the commutativity of the diagram \tri{\cyl(a_{0}),\subdiv(a_{0}),a_{2},r_{1}(a_{0}),u,h \circ \cyl(g)} in $\cl{A}$ we deduce that the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{0}),\subdiv(a_{0}),a_{1},s(a_{0}),f \circ u, p(a_{1}) \circ \cyl(f \circ g)} %
%
We also have that the following diagram in $\cl{A}$ commutes. \trapeziumstwo[{3,3.5,2,0}]{\cyl(a_{0}),\cyl(a_{2}),a_{0},\cyl^{2}(a_{0}),\cyl(a_{1}),\cyl(a_{0}),a_{1},\cyl(g),\cyl(f),p(a_{1}),p(a_{0}),i_{0}(a_{0}),k,\cyl\big(i_{0}(a_{0})\big),\cyl(k),p\big(\cyl(a_{0})\big)} %
%
Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{0}),\subdiv(a_{0}),a_{1},s(a_{0}),f \circ u,k \circ i_{0}(a_{0}) \circ p(a_{0})} %
%
Moreover the following diagram in $\cl{A}$ commutes. \squareabovetriangle{\cyl(a_{0}),\subdiv(a_{0}),a_{1},a_{2},a_{1},r_{0}(a_{0}),u,k,j,f,id} %
%
Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),\cyl(a_{0}),\subdiv(a_{0}),a_{1},i_{0}(a_{0}),r_{0}(a_{0}),i_{1}(a_{0}),r_{1}(a_{0}),k,k \circ i_{0}(a_{0}) \circ p(a_{0}), f \circ u} %
%
Let \ar{\subdiv,\cyl,q_{l}} denote the canonical 2-arrow of $\cl{C}$ of Definition \ref{StrictnessLeftIdentitiesCylinderDefinition}. We have that the following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),\cyl(a_{0}),\subdiv(a_{0}),\cyl(a_{0}),i_{0}(a_{0}),r_{0}(a_{0}),i_{1}(a_{0}),r_{1}(a_{0}),id, i_{0}(a_{0}) \circ p(a_{0}), q_{l}(a_{0})} %
%
Then the following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),\cyl(a_{0}),\subdiv(a_{0}),a_{1},i_{0}(a_{0}),r_{0}(a_{0}),i_{1}(a_{0}),r_{1}(a_{0}),k, k \circ i_{0}(a_{0}) \circ p(a_{0}), k \circ q_{l}(a_{0})} %
%
Appealing to the universal property of $\subdiv(a_{0})$ we deduce that the following diagram in $\cl{A}$ commutes. \tri{\subdiv(a_{0}),\cyl(a_{0}),a_{1},q_{l}(a_{0}),k,f \circ u} %
%
By definition of $l$ the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{0}),\subdiv(a_{0}),a_{2},s(a_{0}),u,l} %
%
Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{0}),a_{2},a_{1},l,f,k \circ q_{l}(a_{0}) \circ s(a_{0})} %
%
Since $\cylinder$ has strictness of left identities the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{0}),\subdiv(a_{0}),\cyl(a_{0}),s(a_{0}),q_{l}(a_{0}),id} %
%
Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes, as required. \tri{\cyl(a_{0}),a_{2},a_{1},l,f,k}
\end{proof}

\begin{cor} \label{TrivialFibrationCylinderIffStrongDeformationRetractionCorollary} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p, \subdiv, r_{0}, r_{1}, s \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$. Suppose that $\cylinder$ has strictness of left identities. 

An arrow \ar{a_{1},a_{0},f} of $\cl{A}$ is a trivial fibration with respect to $\cylinder$ if and only if there is an arrow \ar{a_{0},a_{1},j} of $\cl{A}$ such that $f$ is a retraction of $j$ and such that there exists a homotopy over $a_{0}$ from $jf$ to $id(a_{1})$ with respect to $\cylinder$ and $(f,f)$. \end{cor} 

\begin{proof} Follows immediately from Proposition \ref{DoldOverHomotopiesCylinderProposition} and Proposition \ref{StrongDeformationRetractionIsFibrationProposition}. \end{proof}

\begin{cor} \label{TrivialFibrationCoCylinderIffStrongDeformationRetractionCorollary} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p, v, \subdiv, r_{0}, r_{1}, s \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $v$ compatible with $p$, and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$. Suppose that $\cylinder$ has strictness of left identities. 

Let $\cocylinder = \big( \cocyl, e_0, e_1, $c$ \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

An arrow \ar{a_{0},a_{1},f} of $\cl{A}$ is a trivial fibration with respect to $\cocylinder$ if and only if it is a strong deformation retraction with respect to $\cocylinder$. \end{cor} 

\begin{proof} Follows immediately from Proposition \ref{StrongDeformationRetractionCoCylinderCharacterisationProposition}, Proposition \ref{ReverseHomotopyIsOverHomotopyProposition}, and Corollary \ref{TrivialFibrationCylinderIffStrongDeformationRetractionCorollary}. \end{proof}

\begin{cor} \label{TrivialCofibrationCylinderIffStrongDeformationRetractionCorollary} Let $\cocylinder = \big( \cocyl, e_{0}, e_{1}, c, v, \subdiv, r_{0}, r_{1}, s \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$, an involution structure $v$ compatible with $c$, and a subdivision structure $\big(\subdiv, r_{0}, r_{1}, s \big)$. Suppose that $\cocylinder$ has strictness of left identities. 

Let $\cylinder = \big( \cyl, i_0, i_1,p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

An arrow \ar{a_{0},a_{1},f} of $\cl{A}$ is a trivial cofibration with respect to $\cylinder$ if and only if it admits a strong deformation retraction with respect to $\cylinder$. \end{cor} 

\begin{proof} Follows immediately from Corollary \ref{TrivialFibrationCoCylinderIffStrongDeformationRetractionCorollary} by duality. \end{proof}

\begin{cor} \label{NormallyClovenCofibrationFollowedByTrivialFibrationCorollary} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{lr} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $p$ compatible with $p$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$, and a lower right connection structure $\Gamma_{lr}$. Suppose that $\Gamma_{lr}$ is compatible with $p$, and that $\cylinder$ has strictness of identities. Suppose moreover that $\cyl$ preserves mapping cylinders with respect to $\cylinder$. 

Let $\cocylinder = \big( \cocyl, e_0, e_1, $c$ \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$, and let $\big( a^{\cylinder}_{f},d^{0}_{f},d^{1}_{f} \big)$ be a mapping cylinder of $f$ with respect to $\cylinder$. Let \tri{a_{0},a^{\cylinder}_{f},a_{1},j,g,f} denote the corresponding mapping cylinder factorisation of $f$. Then $j$ is a normally cloven cofibration with respect to $\cylinder$ and $g$ is a trivial fibration with respect to $\cocylinder$. \end{cor}

\begin{proof} Follows immediately from Proposition \ref{MappingCylinderFactorisationGivesNormallyClovenCofibrationProposition}, Corollary \ref{MappingCylinderFactorisationStrongDeformationRetractionWithRespectToCoCylinderCorollary}, and Corollary \ref{TrivialFibrationCoCylinderIffStrongDeformationRetractionCorollary}. \end{proof} 

\begin{cor} \label{TrivialCofibrationFollowedByNormallyClovenFibrationCorollary} Let $\cocylinder = \big( \cocyl, e_{0}, e_{1}, c, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{lr} \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$, an involution structure compatible with $c$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$, and a lower right connection structure $\Gamma_{lr}$. Suppose that $\Gamma_{lr}$ is compatible with $c$, and that $\cocylinder$ has strictness of identities. Suppose moreover that $\cocyl$ preserves mapping co-cylinders with respect to $\cocylinder$. 

Let $\cylinder = \big( \cyl, i_0, i_1,p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$.
 
Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$, and let $\big( a^{\cocylinder}_{f},d^{0}_{f},d^{1}_{f} \big)$ be a mapping co-cylinder of $f$ with respect to $\cocylinder$. Let \tri{a_{0},a^{\cocylinder}_{f},a_{1},j,g,f} denote the corresponding mapping co-cylinder factorisation of $f$. 

Then $j$ is a trivial cofibration with respect to $\cylinder$ and $g$ is a normally cloven fibration with respect to $\cocylinder$. \end{cor} 

\begin{proof} Follows immediately from Corollary \ref{NormallyClovenCofibrationFollowedByTrivialFibrationCorollary} by duality. \end{proof}

\begin{cor} \label{CofibrationFollowedByTrivialNormallyClovenFibrationCorollary} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{lr} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $p$ compatible with $p$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$, and a lower right connection structure $\Gamma_{lr}$. Suppose that $\Gamma_{lr}$ is compatible with $p$, and that $\cylinder$ has strictness of identities. Suppose moreover that $\cyl$ preserves mapping cylinders with respect to $\cylinder$.

Let $\cocylinder = \big( \cocyl, e_{0}, e_{1}, c, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{lr} \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$, an involution structure compatible with $c$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$, and a lower right connection structure $\Gamma_{lr}$. Suppose that $\Gamma_{lr}$ is compatible with $c$, and that $\cocylinder$ has strictness of identities. Suppose moreover that $\cocyl$ preserves mapping co-cylinders with respect to $\cocylinder$. 

Suppose that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$.

Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$. Then there is an object $a$ of $\cl{A}$, an arrow \ar{a_{0},a,j} of $\cl{A}$ which is a cofibration with respect to $\cylinder$, and an arrow \ar{a,a_{1},g} of $\cl{A}$ which is a trivial normally cloven fibration with respect to $\cocylinder$ such that the following diagram in $\cl{A}$ commutes. \tri{a_{0},a,a_{1},j,g,f} \end{cor}

\begin{proof} Let $\big( a^{\cylinder}_{f},d^{0}_{f},d^{1}_{f} \big)$ be a mapping cylinder of $f$ with respect to $\cylinder$. Let \tri{a_{0},a^{\cylinder}_{f},a_{1},j',g',f} denote the corresponding mapping cylinder factorisation of $f$. By Proposition \ref{MappingCylinderFactorisationGivesNormallyClovenCofibrationProposition} we have that $j'$ is a cofibration with respect to $\cylinder$. 

Let $\big( a^{\cocylinder}_{g'},d^{0}_{g'},d^{1}_{g'} \big)$ be a mapping co-cylinder of $g'$ with respect to $\cocylinder$. Let \tri{a^{\cylinder}_{f},a^{\cocylinder}_{g'},a_{1},j'',g,g'} denote the corresponding mapping co-cylinder factorisation of $g'$. Let us take $a$ to be $a^{\cocylinder}_{g'}$. By Corollary \ref{TrivialCofibrationFollowedByNormallyClovenFibrationCorollary} we have that $j''$ is a trivial cofibration with respect to $\cylinder$, and $g$ is a normally cloven fibration with respect to $\cocylinder$.  

Since $g'$ and $j''$ are homotopy equivalences with respect to $\cylinder$, it follows from Proposition \ref{TwoOutOfThreeHomotopyEquivalencesProposition} that $g$ is a homotopy equivalence with respect to $\cylinder$. Hence by Proposition \ref{HomotopyEquivalenceCylinderIffHomotopyEquivalenceCoCylinderProposition} $g$ is a homotopy equivalence with respect to $\cocylinder$. Thus $g$ is a trivial normally cloven fibration with respect to $\cocylinder$.

Let \ar{a_{0},a^{\cocylinder}_{g'},j} denote the arrow $j'' \circ j'$ of $\cl{A}$. Since both $j'$ and $j''$ are cofibrations with respect to $\cylinder$ we conclude that $j$ is a cofibration with respect to $\cylinder$ by Proposition \ref{CompositionOfCofibrationsIsACofibrationProposition}.

\end{proof}

\begin{cor} \label{TrivialNormallyClovenCofibrationFollowedByFibrationCorollary} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{lr} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $p$ compatible with $p$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$, and a lower right connection structure $\Gamma_{lr}$. Suppose that $\Gamma_{lr}$ is compatible with $p$, and that $\cylinder$ has strictness of identities. Suppose moreover that $\cyl$ preserves mapping cylinders with respect to $\cylinder$.

Let $\cocylinder = \big( \cocyl, e_{0}, e_{1}, c, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{lr} \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$, an involution structure compatible with $c$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$, and a lower right connection structure $\Gamma_{lr}$. Suppose that $\Gamma_{lr}$ is compatible with $c$, and that $\cocylinder$ has strictness of identities. Suppose moreover that $\cocyl$ preserves mapping co-cylinders with respect to $\cocylinder$. 

Suppose that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$.

Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$. Then there is an object $a$ of $\cl{A}$, an arrow \ar{a_{0},a,j} of $\cl{A}$ which is a trivial normally cloven cofibration with respect to $\cylinder$, and an arrow \ar{a,a_{1},g} of $\cl{A}$ which is a fibration with respect to $\cocylinder$ such that the following diagram in $\cl{A}$ commutes. \tri{a_{0},a,a_{1},j,g,f} \end{cor}

\begin{proof} Follows immediately from Corollary \ref{TrivialNormallyClovenCofibrationFollowedByFibrationCorollary} by duality. \end{proof}
\end{section}

formal_category_theory_preliminaries.tex
\begin{section}{Formal category theory preliminaries} \label{FormalCategoryTheoryPreliminariesSection}

In \ref{StructuresUponACylinderOrACoCylinderSection} we introduce various structures with which a cylinder or co-cylinder may be able to be equipped. The structures upon a co-cylinder are formally dual to those upon a cylinder, and this duality will manifest itself throughout this work. It is of a 2-categorical rather than a 1-categorical nature as was already observed by Gray in \cite{GrayFibredAndCofibredCategories}. 

In order to express the duality we will work throughout in a strict 2-category equipped with a strict final object. We will think of the objects of this 2-category as formal categories. We now introduce these ideas as far as we will need.

\begin{assum} Let $\cl{C}$ be a strict 2-category, and let $1$ be a final object of $\cl{C}$. \end{assum} 

\begin{defn} Let $\cl{A}$ be an object of $\cl{C}$. An {\em object} of $\cl{A}$ is a 1-arrow \ar{1,\cl{A}} of $\cl{C}$. \end{defn}

\begin{defn} Let $\cl{A}$ be an object of $\cl{C}$. Given objects $a_{0}$ and $a_{1}$ of $\cl{A}$, an {\em arrow} of $\cl{A}$ from $a_{0}$ to $a_{1}$ is  a 2-arrow \ar{a_{0},a_{1},} of $\cl{C}$. \end{defn}

\begin{rmk} Let $\underline{\mathsf{CAT}}$ denote the strict 2-category of large categories. The definitions above are motivated by the observation that for any category $\cl{A}$ the category $\underline{\mathsf{Hom}}_{\underline{\mathsf{CAT}}}(1,\cl{A})$ is isomorphic to $\cl{A}$.  \end{rmk}

\begin{defn} Let $\cl{A}$ be an object of $\cl{C}$, and let \ar{a_{0},a_{1},f_{0}} and \ar{a_{1},a_{2},f_{1}} be arrows of $\cl{A}$. The {\em composition} $f_{1} \circ f_{0}$ of $f_{1}$ and $f_{0}$ in $\cl{A}$ is their composition in $\underline{\mathsf{Hom}}_{\cl{C}}(1,\cl{A})$. \end{defn}

\begin{defn} A {\em diagram} in $\cl{A}$ is a diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(1,\cl{A})$. We similarly define a {\em commutative diagram} in $\cl{A}$, a {\em co-cartesian square} in $\cl{A}$, and a {\em cartesian square} in $\cl{A}$. \end{defn}

\begin{notn} Given 1-arrows 

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| \cl{A}_{0} \& |(1-0)| \cl{A}_{1} \& |(2-0)| \cl{A}_{2} \\ 
};
	
\draw[->] ({0-0}.north east) to node[auto] {$F_{0}$} ({1-0}.north west);
\draw[->] ({0-0}.south east) to node[auto,swap] {$F_{1}$} ({1-0}.south west);
\draw[->] (1-0) to node[auto] {$G$} (2-0); 

\end{tikzpicture} 

\end{diagram} of $\cl{C}$ and a 2-arrow \ar{F_{0},F_{1},\eta} of $\cl{C}$ we denote by $G \cdot \eta$ the 2-arrow \ar{GF_{0},GF_{1},} of $\cl{C}$ obtained by horizontal compositition of $id(G)$ and $\eta$. It is sometimes referred to as the {\em whiskering} of $G$ and $\eta$.

Similarly given 1-arrows

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| \cl{A}_{0} \& |(1-0)| \cl{A}_{1} \& |(2-0)| \cl{A}_{2} \\ 
};
	
\draw[->] ({1-0}.north east) to node[auto] {$F_{0}$} ({2-0}.north west);
\draw[->] ({1-0}.south east) to node[auto,swap] {$F_{1}$} ({2-0}.south west);
\draw[->] (0-0) to node[auto] {$G$} (1-0); 

\end{tikzpicture} 

\end{diagram} of $\cl{C}$ and a 2-arrow \ar{F_{0},F_{1},\eta} of $\cl{C}$ we denote by $\eta \cdot G$ the 2-arrow \ar{F_{0}G,F_{1}G,} of $\cl{C}$ obtained by horizontal composition of $\eta$ and $id(G)$. It is also sometimes referred to as the {\em whiskering} of $\eta$ and $G$. \end{notn}

\begin{notn} Let \ar{\cl{A}_{0},\cl{A}_{1},F} be a 1-arrow of $\cl{C}$. If $a$ is an object of $\cl{A}_{0}$ we denote by $F(a)$ the object of $\cl{A}_{1}$ defined by the 1-arrow $F \circ a$ of $\cl{C}$. 

If \ar{a_{0},a_{1},f} is an arrow of $\cl{A}_{0}$ we denote by $F(f)$ the arrow of $\cl{A}_{1}$ from $F(a_{0})$ to $F(a_{1})$ defined by the whiskered 2-arrow $F \cdot f$ of $\cl{C}$. \end{notn}

\begin{rmk} \label{1ArrowsAsFunctors} In this way we think of a 1-arrow \ar{\cl{A}_{0},\cl{A}_{1},F} of $\cl{C}$ as a functor from $\cl{A}_{0}$ to $\cl{A}_{1}$, corresponding to the functor \ar[6]{{\underline{\mathsf{Hom}}_{\cl{C}}(1,\cl{A}_{0})},{\underline{\mathsf{Hom}}_{\cl{C}}(1,\cl{A}_{1}).},{\underline{\mathsf{Hom}}_{\cl{C}}(1,F)}} \end{rmk}

\begin{notn} \label{2ArrowAsNaturalTransformationNotation} Let \pair{\cl{A}_{0},\cl{A}_{1},F_{0},F_{1}} be 1-arrows of $\cl{C}$, and let \ar{F_{0},F_{1},\eta} be a 2-arrow of $\cl{C}$. If $a$ is an object of $\cl{A}_{0}$ we denote by $\eta(a)$ the arrow of $\cl{A}_{1}$ from $F_{0}(a)$ to $F_{1}(a)$ defined by the whiskered 2-arrow $\eta \cdot a$ of $\cl{C}$. \end{notn}

\begin{rmk} Let \pair{\cl{A}_{0},\cl{A}_{1},F_{0},F_{1}} be 1-arrows of $\cl{C}$, thought of as functors from $\cl{A}_{0}$ to $\cl{A}_{1}$ as in Remark \ref{1ArrowsAsFunctors}. We think of a 2-arrow \ar{F_{0},F_{1},\eta} of $\cl{C}$ as a natural transformation from $F_{0}$ to $F_{1}$, corresponding to the natural transformation \ar[6]{{\underline{\mathsf{Hom}}_{\cl{C}}(1,F_{0})},{{\underline{\mathsf{Hom}}_{\cl{C}}(1,F_{1})}.},{\underline{\mathsf{Hom}}_{\cl{C}}(1,\eta)}} \end{rmk}

\begin{notn} We denote by $\cl{C}^{op}$ the 2-category obtained from $\cl{C}$ by reversing all 2-arrows. If $f$ is a 2-arrow of $\cl{C}$, we denote by $f^{op}$ the corresponding 2-arrow of $\cl{C}^{op}$. When viewing an object $\cl{A}$ of $\cl{C}$ as an object of $\cl{C}^{op}$ we denote it by $\cl{A}^{op}$. Thus $\underline{\mathsf{Hom}}_{\cl{C}^{op}}(1,\cl{A}^{op})$ is the opposite category of $\underline{\mathsf{Hom}}_{\cl{C}}(1,\cl{A})$. In particular if \ar{a_{0},a_{1},f} defines an arrow of $\cl{A}$ then the 2-arrow $f^{op}$ of $\cl{C}^{op}$ defines an arrow of $\cl{A}^{op}$ from $a_{1}$ to $a_{0}$. \end{notn}

\begin{recollection} \label{AdjunctionRecollection} An {\em adjunction} between a pair $(F,G)$ of 1-arrows \adjunction{\cl{A}_{0},\cl{A}_{1},F,G} of $\cl{C}$ is a 2-arrow \ar{id(\cl{A}_{0}),GF,\eta} of $\cl{C}$ and a 2-arrow \ar{FG,id(\cl{A}_{1}),\zeta} of $\cl{C}$ such that the diagrams \tri{F,FGF,F,F \eta, \zeta F, id} in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A}_{0},\cl{A}_{1})$ and \tri{G,GFG,G,\eta G, G \zeta,id} in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A}_{1},\cl{A}_{0})$ commute. We refer to $F$ as a {\em left adjoint} of $G$, and to $G$ as a {\em right adjoint} of $F$. 

If we have an adjunction between $F$ and $G$ then for any object $\cl{A}$ of $\cl{C}$ the natural transformations $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\eta)$ and $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\zeta)$ define an adjunction between the following pair of functors. \adjunction[7]{{\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A}_{0})},{\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A}_{1})},{\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},F)},{\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},G)}} %
%
In particular we have a natural isomorphism \ar{{\mathsf{Hom}_{\underline{\mathsf{Hom}}_{\cl{C}}(1,\cl{A}_{1})}\Big(\underline{\mathsf{Hom}}_{\cl{C}}\big(1,F(-)\big),-\Big)},{\mathsf{Hom}_{\underline{\mathsf{Hom}}_{\cl{C}}(1,\cl{A}_{0})}\Big(-,\underline{\mathsf{Hom}}_{\cl{C}}\big(1,G(-)\big)\Big)},\mathsf{adj}} of functors \ar{{\mathsf{Hom}_{\cl{C}}(1,\cl{A}_{0}) \times \mathsf{Hom}_{\cl{C}}(1,\cl{A}_{1})},{\mathsf{Set}.}} %
%
Adopting the shorthand 

\begin{diagram}

\begin{tikzpicture} [>=stealth, column 1/.style={anchor=east}, column 2/.style={anchor=west}]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=3 em, row sep=1em, nodes={anchor=center}]
{
|(0-0)| \cl{A} \& |(1-0)| \underline{\mathsf{Hom}}_{\cl{C}}(1,\cl{A}) \\  
|(0-1)| F   \& |(1-1)| \underline{\mathsf{Hom}}_{\cl{C}}\big(1,F \big) \\ 
|(0-2)| G   \& |(1-2)| \underline{\mathsf{Hom}}_{\cl{C}}\big(1,G \big) \\
};

\draw[<->] (0-0) to (1-0);
\draw[<->] (0-1) to (1-1);
\draw[<->] (0-2) to (1-2);

\end{tikzpicture}

\end{diagram} %
%
we will write the above natural isomorphism as \ar{{\mathsf{Hom}_{\cl{A}_{1}}\big(F(-),-\big)},{\mathsf{Hom}_{\cl{A}_{0}}\big(-,G(-)\big).},\mathsf{adj}} \end{recollection}

\begin{defn} \label{PushoutsPullbacks2ArrowsArePushoutsPullbacksInFormalCategoriesTerminology} If for any objects $\cl{A}_{0}$ and $\cl{A}_{1}$ of $\cl{C}$, any object $a$ of $\cl{A}_{0}$, and any co-cartesian (respectively cartesian) square \sq{F_{0},F_{1},F_{2},F_{3},\eta_{0},\eta_{1},\eta_{2},\eta_{3}} in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A}_{0},\cl{A}_{1})$ the square \sq{F_{0}(a),F_{1}(a),F_{2}(a),F_{3}(a),\eta_{0}(a),\eta_{1}(a),\eta_{2}(a),\eta_{3}(a)} in $\cl{A}_{1}$ is co-cartesian (respectively cartesian), we write that {\em pushouts (respectively pullbacks) of 2-arrows of $\cl{C}$ give rise to pushouts (respectively pullbacks) in formal categories}. \end{defn}

\begin{rmk} In both $\underline{\mathsf{CAT}}$ and $\underline{\mathsf{CAT}}^{op}$ pushouts and pullbacks of 2-arrows give rise to pushouts and pullbacks in formal categories more or less by definition. The author expects that colimits and limits of 2-arrows equally give rise to colimits and limits in formal categories in quite general 2-categories --- in 2-topoi, for instance. \end{rmk}
 
\begin{rmk} It would certainly be possible for us to work with weak rather than strict 2-categories, weakening all 2-categorical aspects appropriately. However, we are motivated by ordinary categories. In addition to $\underline{\mathsf{CAT}}$ the only 2-category of relevance to us is $\underline{\mathsf{CAT}}^{op}$, exactly in order to capture duality. Thus it is sufficient for us to work with strict 2-categories, and we do so in order to avoid the distraction of coherency. \end{rmk}

\end{section}

homotopy_and_relative_homotopy.tex
\begin{section}{Homotopy and relative homotopy} \label{HomotopyAndRelativeHomotopySection}

In \ref{StructuresUponACylinderOrACoCylinderSection} we introduced a notion of a cylinder or a co-cylinder in a formal category $\cl{A}$. We now explain that a cylinder gives rise to a notion of homotopy between arrows of $\cl{A}$. In a dual manner a co-cylinder gives rise to a notion of homotopy between arrows of $\cl{A}$. Thus a cylinder or a cylinder allows us to define a notion of homotopy equivalence in $\cl{A}$. 

If $\cl{A}$ admits both a cylinder and a co-cylinder as we will assume later in this work, it will be crucial for us to know that the corresponding notions of homotopy coincide. In \ref{CylindricalAdjunctionsSection} we mentioned that an adjunction between a cylinder and a co-cylinder exactly ensures that this holds. We will now be able to observe this.

Homotopy theory with respect to a cylinder or co-cylinder alone is rather spartan. The structures upon a cylinder or co-cylinder defined in \ref{StructuresUponACylinderOrACoCylinderSection} allow us to construct a much richer theory. This theory will for now be directly analogous to that of topological spaces. Later when we require in \ref{MappingCylindersAndMappingCoCylindersSection} for instance that the strictness hypotheses introduced in \ref{StructuresUponACylinderOrACoCylinderSection} hold the homotopy theory of topological spaces will no longer be our guide.

A contraction structure allows us to construct identity homotopies. An involution structure allows us to reverse homotopies. A subdivision structure allows to compose homotopies. In the presence of both an involution structure and a subdivision structure homotopy equivalences in $\cl{A}$ have the two-out-of-three property. 

Given a pair of commutative diagrams \twosq{a_{2},a_{0},a_{3},a_{1},g_{0},f,f',g_{1},a_{0},a_{2},a_{1},a_{3},r_{0},j',j,r_{1}} in $\cl{A}$ such that $r_{0}$ is a retraction of $g_{0}$, and such that $r_{1}$ is a retraction of $g_{1}$, we demonstrate that if $f$ is a homotopy equivalence then so is $f'$. We make a technical observation concerning homotopy inverses which will be appealed to in our proof in \ref{DoldTheoremSection} of Dold's theorem. 

We introduce the notion of a double homotopy with respect to a cylinder and explain a pictorial notation. Double homotopies will play an indispensable role throughout this work. Our three flavours of connection structures allow us to construct double homotopies with specific boundary homotopies.

With respect to a cylinder or a co-cylinder equipped with a contraction structure we define a notion of homotopy under or over an object of $\cl{A}$. If $\cl{A}$ admits both a cylinder $\cylinder$ equipped with a contraction structure $p$ and a co-cylinder $\cocylinder$ equipped with a contraction structure $c$ then an adjunction between $\cylinder$ and $\cocylinder$ which is compatible with $p$ and $c$ allows us to observe that the notion of a homotopy under (respectively over) an object with respect to $\cylinder$ coincides with that of a homotopy under (respectively over) an object with respect to $\cocylinder$.

Identity homotopies give rise to identity homotopies under or over an object. An involution structure which is compatible with contraction allows us to construct reverse homotopies under or over an object. A subdivision structure which is compatible with contraction allows us to compose homotopies under or over an object.  

We conclude by introducing the notion of a strong deformation retraction with respect to a cylinder or co-cylinder. All our involvement with homotopies under or over an object in this work will relate to strong deformation retractions.

As discussed in the introduction \ref{IntroductionSection} to this work homotopy with respect to a cylinder in an abstract setting was first considered by Kan in \cite{KanAbstractHomotopyTheoryII}. The insight that further structure upon a cylinder can give rise to a richer theory is due to Kamps, presented in works such as \cite{KampsKanBedingungenUndAbstrakteHomotopietheorie} around 1970.  

In a setting closer to that in which we are working the observation that a subdivision structure upon a cylinder allows one to compose homotopies was first explored to the author's knowledge by Grandis in  papers such as \cite{GrandisCategoricallyAlgebraicFoundationsForHomotopicalAlgebra} written in the 1990s. The abstract notion of homotopy under and over an object is discussed for example in the book \cite{KampsPorterAbstractHomotopyAndSimpleHomotopyTheory} of Kamps and Porter. This is also a good reference for most of the other ideas of this section.

\begin{assum} Let $\cl{C}$ be a 2-category with a final object such that pushouts and pullbacks of 2-arrows of $\cl{C}$ give rise to pushouts and pullbacks in formal categories in the sense of Definition \ref{PushoutsPullbacks2ArrowsArePushoutsPullbacksInFormalCategoriesTerminology}. Let $\cl{A}$ be an object of $\cl{C}$. Recall that we view $\cl{A}$ as a formal category, writing of objects and arrows of $\cl{A}$. This terminology and all other formal category theory preliminaries can be found in \ref{FormalCategoryTheoryPreliminariesSection}. \end{assum}

\begin{defn} \label{HomotopyCylinderDefinition} Let $\cylinder = \big( \cyl, i_0, i_1, \big)$ be a cylinder in $\cl{A}$, and let \pair{a_{0},a_{1},f_{0},f_{1}} be arrows of $\cl{A}$. A {\em homotopy} from $f_{0}$ to $f_{1}$ with respect to $\cylinder$ is an arrow \ar{\cyl(a_{0}),a_{1},h} of $\cl{A}$ such that the following diagrams in $\cl{A}$ commute. \twotriangles[{4,3,2}]{a_{0},\cyl(a_{0}),a_{1},i_{0}(a_{0}),h,f_{0},a_{0},\cyl(a_{0}),a_{1},i_{1}(a_{0}),h,f_{1}} \end{defn}

\begin{defn} Let $\cocylinder = \big( \cocyl, e_0, e_1 \big)$ be a co-cylinder in $\cl{A}$, and let \pair{a_{0},a_{1},f_{0},f_{1}} be arrows of $\cl{A}$. 

A {\em homotopy} from $f_{0}$ to $f_{1}$ with respect to $\cocylinder$ is an arrow \ar{a_{0},\cocyl(a_{1}),h} of $\cl{A}$ such that the following diagrams in $\cl{A}$ commute. \twotriangles[{4,3,2}]{a_{0},\cocyl(a_{1}),a_{1},h,e_{0}(a_{1}),f_{0},a_{0},\cocyl(a_{1}),a_{1},h,e_{1}(a_{1}),f_{1}} \end{defn} 

\begin{rmk} Let $\cocylinder = \big( \cocyl, e_0, e_1 \big)$ be a co-cylinder in $\cl{A}$, and let \pair{a_{0},a_{1},f_{0},f_{1}} be arrows of $\cl{A}$. An arrow \ar{a_{0},\cocyl(a_{1},h} of $\cl{A}$ is a homotopy from $f_{0}$ to $f_{1}$ if and only if $h^{op}$ is a homotopy from $f_{0}^{op}$ to $f_{1}^{op}$ with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$. \end{rmk}

\begin{prpn} \label{HomotopyCylinderGivesHomotopyCoCylinderProposition} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$, and let $\cocylinder = \big( \cocyl, e_0, e_1 \big)$ be a co-cylinder in $\cl{A}$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$. Let \ar{{\mathsf{Hom}_{\cl{A}}\big(\cyl(-),- \big)},{\mathsf{Hom}_{\cl{A}}\big(-,\cocyl(-) \big)},\mathsf{adj}} denote the corresponding natural isomorphism, adopting the shorthand of Recollection \ref{AdjunctionRecollection}. 

Let \pair{a_{0},a_{1},f_{0},f_{1}} be arrows of $\cl{A}$, and suppose that \ar{\cyl(a_{0}),a_{1},h} defines a homotopy from $f_{0}$ to $f_{1}$ with respect to $\cylinder$. Then the arrow \ar[4]{a_{0},\cocyl(a_{1}),\mathsf{adj}(h)} of $\cl{A}$ defines a homotopy from $f_{0}$ to $f_{1}$ with respect to $\cocylinder$. \end{prpn} 

\begin{proof} Follows immediately from the fact that $\cylinder$ is left adjoint to $\cocylinder$. \end{proof}

\begin{cor} \label{HomotopyCoCylinderGivesHomotopyCylinderCorollary} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$, and let $\cocylinder = \big( \cocyl, e_0, e_1 \big)$ be a co-cylinder in $\cl{A}$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$. Let \ar{{\mathsf{Hom}_{\cl{A}}\big(\cyl(-),- \big)},{\mathsf{Hom}_{\cl{A}}\big(-,\cocyl(-) \big)},\mathsf{adj}} denote the corresponding natural isomorphism, adopting the shorthand of Recollection \ref{AdjunctionRecollection}. 

Let \pair{a_{0},a_{1},f_{0},f_{1}} be arrows of $\cl{A}$, and suppose that \ar{a_{0},\cocyl(a_{1}),h} defines a homotopy from $f_{0}$ to $f_{1}$ with respect to $\cocylinder$. Then the arrow \ar[4]{\cyl(a_{0}),a_{1},\mathsf{adj}^{-1}(h)} of $\cl{A}$ defines a homotopy from $f_{0}$ to $f_{1}$ with respect to $\cylinder$. \end{cor} 

\begin{proof} Follows formally from Proposition \ref{HomotopyCylinderGivesHomotopyCoCylinderProposition} by duality. \end{proof}

\begin{prpn} \label{IdentityHomotopyProposition} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$. Then the arrow \ar[5]{\cyl(a_{0}), a_{1}, f \circ p(a_{0})} of $\cl{A}$ defines a homotopy from $f$ to itself with respect to $\cylinder$. \end{prpn}

\begin{proof} Follows immediately from that $p$ defines a contraction structure with respect to $\cylinder$. \end{proof}

\begin{rmk} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. Given an arrow $f$ of $\cl{A}$, we refer to the corresponding homotopy of Proposition \ref{IdentityHomotopyProposition} from $f$ to itself as the {\em identity homotopy} from $f$ to itself with respect to $\cylinder$. \end{rmk} 

\begin{prpn} \label{ReverseHomotopyProposition} Let $\cylinder = \big( \cyl, i_0, i_1, v \big)$ be a cylinder in $\cl{A}$ equipped with an involution structure $v$. Let \pair{a_{0}, a_{1}, f_{0}, f_{1}} be arrows of $\cl{A}$, and let \ar{\cyl(a_{0}), a_{1}, h} be a homotopy from $f_{0}$ to $f_{1}$ with respect to $\cylinder$. The arrow \ar[5]{\cyl(a_{0}),a_{1}, h \circ v(a_{0})} of $\cl{A}$ defines a homotopy from $f_{1}$ to $f_{0}$ with respect to $\cylinder$. \end{prpn}
 
\begin{proof} Follows immediately from the fact that $v$ defines an involution structure with respect to $\cylinder$. \end{proof}

\begin{rmk} Let $\cylinder = \big( \cyl, i_0, i_1, v \big)$ be a cylinder in $\cl{A}$ equipped with an involution structure $v$. Given a homotopy $h$ with respect to $\cylinder$ between a pair of arrows of $\cl{A}$, we refer to the corresponding homotopy of Proposition \ref{ReverseHomotopyProposition} as the {\em reverse} of $h$, and denote it by $h^{-1}$. \end{rmk} 

\begin{prpn} \label{CompositionHomotopiesProposition} Let $\cylinder = \big( \cyl, i_0, i_1, \subdiv, r_{0}, r_{1}, s \big)$ be an interval in $\cl{A}$ equipped with a subdivision structure $(\subdiv, r_{0}, r_{1}, s \big)$. Let \ar{\cyl(a_{0}), a_{1}, h} be a homotopy with respect to $\cylinder$ from an arrow \ar{a_{0},a_{1},f_{0}} of $\cl{A}$ to an arrow \ar{a_{0},a_{1},f_{1}}of $\cl{A}$. Let \ar{\cyl(a_{0}), a_{1}, k} be a homotopy with respect to $\cylinder$ from $f_{1}$ to a third arrow \ar{a_{0},a_{1},f_{2}} of $\cl{A}$. 

Let \ar{\subdiv(a_{0}), a_{1}, r} denote the canonical arrow of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout{a_{0}, \cyl(a_{0}), \cyl(a_{0}), \subdiv(a_{0}), a_{1}, i_{0}(a_{0}), r_{0}(a_{0}), i_{1}(a_{0}), r_{1}(a_{0}), k, h, r} Then the arrow \ar[5]{\cyl(a_{0}), a_{1}, r \circ s(a_{0})} of $\cl{A}$ defines a homotopy from $f_{0}$ to $f_{2}$ with respect to $\cyl$. \end{prpn}

\begin{proof} The following diagram in $\cl{A}$ commutes. 

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=6 em, row sep=4 em, nodes={anchor=center}]
{ 
|(0-0)| a_{0} \& |(1-0)| \cyl(a_{0}) \\ 
|(0-1)| \cyl(a_{0}) \& |(1-1)| \subdiv(a_{0}) \\[1em]
|(0-2)| a_{1} \& \\
};
	
\draw[->] (0-0) to node[auto] {$i_{0}(a_{0})$} (1-0);
\draw[->] (1-0) to node[auto] {$s(a_{0})$} (1-1);
\draw[->] (0-0) to node[auto] {$i_{0}(a_{0})$} (0-1);
\draw[->] (0-1) to node[auto,swap] {$r_{1}(a_{0})$} (1-1);
\draw[->] (1-1) to node[auto] {$r$} (0-2);
\draw[->] (0-1) to node[auto] {$h$} (0-2);
\draw[->, bend right=60] (0-0) to node[auto,swap] {$f_{0}$} (0-2);

\end{tikzpicture} 

\end{diagram} The following diagram in $\cl{A}$ also commutes.

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=6 em, row sep=4 em, nodes={anchor=center}]
{ 
|(0-0)| a_{0} \& |(1-0)| \cyl(a_{0}) \\ 
|(0-1)| \cyl(a_{0}) \& |(1-1)| \subdiv(a_{0}) \\[1em]
|(0-2)| a_{1} \& \\
};
	
\draw[->] (0-0) to node[auto] {$i_{1}(a_{0})$} (1-0);
\draw[->] (1-0) to node[auto] {$s(a_{0})$} (1-1);
\draw[->] (0-0) to node[auto] {$i_{1}(a_{0})$} (0-1);
\draw[->] (0-1) to node[auto,swap] {$r_{0}(a_{0})$} (1-1);
\draw[->] (1-1) to node[auto] {$r$} (0-2);
\draw[->] (0-1) to node[auto] {$k$} (0-2);
\draw[->, bend right=60] (0-0) to node[auto,swap] {$f_{2}$} (0-2);

\end{tikzpicture} 

\end{diagram} \end{proof}

\begin{rmk} Let $\cylinder = \big( \cyl, i_0, i_1, \subdiv, r_{0}, r_{1}, s \big)$ be a cylinder in $\cl{A}$ equipped with a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$. Let $f_{0}$, $f_{1}$, and $f_{2}$ be arrows of $\cl{A}$, let $h$ be a homotopy from $f_{0}$ to $f_{1}$ with respect to $\cylinder$, and let $k$ be a homotopy from $f_{1}$ to $f_{2}$ with respect to $\cylinder$. 

We denote by $h + k$ the corresponding homotopy of Proposition \ref{CompositionHomotopiesProposition} from $f_{0}$ to $f_{2}$ with respect to $\cylinder$, and refer to it as a {\em composite homotopy}. \end{rmk}

\begin{rmk} Thus if $\cylinder = \big( \cyl, i_0, i_1, p, v, \subdiv, r_{0}, r_{1}, s \big)$ is a cylinder in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $v$, and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ then homotopy with respect to $\cylinder$ defines an equivalence relation upon the arrows of $\cl{A}$. \end{rmk}

\begin{defn} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$, and let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$. A {\em homotopy inverse} of $f$ with respect to $\cylinder$ is an arrow \ar{a_{1},a_{0},f^{-1}} of $\cl{A}$ together with a homotopy from $f^{-1}f$ to $id(a_{0})$ with respect to $\cylinder$ and a homotopy from $ff^{-1}$ to $id(a_{1})$ with respect to $\cylinder$. \end{defn}

\begin{defn} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$. An arrow \ar{a_{0},a_{1},f} of $\cl{A}$ is a {\em homotopy equivalence} with respect to $\cylinder$ if it admits a homotopy inverse with respect to $\cylinder$. \end{defn} 

\begin{defn} Let $\cocylinder = \big( \cocyl, e_0, e_1 \big)$ be a co-cylinder in $\cl{A}$. An arrow \ar{a_{0},a_{1},f} of $\cl{A}$ is a {\em homotopy equivalence} with respect to $\cocylinder$ if $f^{op}$ is a homotopy equivalence with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$. \end{defn} 

\begin{prpn} \label{HomotopyEquivalenceCylinderIffHomotopyEquivalenceCoCylinderProposition} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$, and let $\cocylinder = \big( \cocyl, e_0, e_1 \big)$ be a co-cylinder in $\cl{A}$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$. Then an arrow \ar{a_{0},a_{1},f} of $\cl{A}$ is a homotopy equivalence with respect to $\cylinder$ if and only if it is a homotopy equivalence with respect to $\cocylinder$. \end{prpn}

\begin{proof} Follows immediately from Proposition \ref{HomotopyCylinderGivesHomotopyCoCylinderProposition} and Corollary \ref{HomotopyCoCylinderGivesHomotopyCylinderCorollary}. \end{proof} 

\begin{lem} \label{HomotopyPreAndPostCompositionLemma} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$. Suppose that we have four arrows

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=3 em, nodes={anchor=center}] 
{ 
|(0-0)| a_{0} \& |(1-0)| a_{1} \& |(2-0)| a_{2} \& |(3-0)| a_{3} \\ 
};
	
\draw[->] (0-0) to node[auto] {$g_{0}$} (1-0);
\draw[->] ({1-0}.north east) to node[auto] {$f_{0}$} ({2-0}.north west);
\draw[->] ({1-0}.south east) to node[auto,swap] {$f_{1}$} ({2-0}.south west);
\draw[->] (2-0) to node[auto] {$g_{1}$} (3-0);
\end{tikzpicture} 

\end{diagram} of $\cl{A}$ and a homotopy \ar{\cyl(a_{1}),a_{2},h}from $f_{0}$ to $f_{1}$ with respect to $\cylinder$. Then the arrow \ar[8]{\cyl(a_{0}),a_{3},g_{1} \circ h \circ \cyl(g_{0})} of $\cl{A}$ defines a homotopy from $g_{1}f_{0}g_{0}$ to $g_{1}f_{1}g_{0}$ with respect to $\cylinder$. 

\end{lem}

\begin{proof} The following diagram in $\cl{A}$ commutes.

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=4 em, row sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| a_{0} \& |(1-0)| \cyl(a_{0}) \\ 
|(0-1)| a_{1} \& |(1-1)| \cyl(a_{1}) \\
              \& |(1-2)| a_{2}       \\
};
	
\draw[->] (0-0) to node[auto] {$i_{0}(a_{0})$} (1-0);
\draw[->] (0-0) to node[auto,swap] {$g_{0}$} (0-1);
\draw[->] (1-0) to node[auto] {$\cyl(g_{0})$} (1-1);
\draw[->] (0-1) to node[auto] {$i_{0}(a_{1})$} (1-1);
\draw[->] (1-1) to node[auto] {$h$} (1-2);
\draw[->] (0-1) to node[auto,swap] {$f_{0}$} (1-2);
\end{tikzpicture} 

\end{diagram} Hence the following diagram in $\cl{A}$ commutes. \tri[{4,3}]{a_{0},\cyl(a_{0}),a_{3},i_{0}(a_{0}),g_{1} \circ h \circ \cyl(g_{0}),g_{1} \circ f_{0} \circ g_{0}} The following diagram in $\cl{A}$ also commutes.  
 
\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=4 em, row sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| a_{0} \& |(1-0)| \cyl(a_{0}) \\ 
|(0-1)| a_{1} \& |(1-1)| \cyl(a_{1}) \\
              \& |(1-2)| a_{2}       \\
};
	
\draw[->] (0-0) to node[auto] {$i_{1}(a_{0})$} (1-0);
\draw[->] (0-0) to node[auto,swap] {$g_{0}$} (0-1);
\draw[->] (1-0) to node[auto] {$\cyl(g_{0})$} (1-1);
\draw[->] (0-1) to node[auto] {$i_{1}(a_{1})$} (1-1);
\draw[->] (1-1) to node[auto] {$h$} (1-2);
\draw[->] (0-1) to node[auto,swap] {$f_{1}$} (1-2);
\end{tikzpicture} 

\end{diagram} Hence the following diagram in $\cl{A}$ commutes. \tri[{4,3}]{a_{0},\cyl(a_{0}),a_{3},i_{1}(a_{0}),g_{1} \circ h \circ \cyl(g_{0}),g_{1} \circ f_{1} \circ g_{0}} 
\end{proof}

\begin{lem} \label{TwoOutOfThreeHomotopyEquivalencesFirstLemma} Let $\cylinder = \big( \cyl, i_0, i_1, v, \subdiv, r_0, r_1, s \big)$ be a cylinder in $\cl{A}$ equipped with an involution structure $v$ and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s  \big)$. Suppose that we have a commutative diagram in $\cl{A}$ as follows. \tri{a_{0},a_{1},a_{2},f_{0},f_{1},f_{2}} If $f_{1}$ and $f_{2}$ are homotopy equivalences with respect to $\cylinder$, then $f_{0}$ is a homotopy equivalence with respect to $\cylinder$. \end{lem}

\begin{proof} Let $f_{1}^{-1}$ be a homotopy inverse of $f_{1}$ with respect to $\cylinder$, and let \ar{\cyl(a_{1}),a_{1},h_{1}} denote the corresponding homotopy from $f_{1}^{-1}f_{1}$ to $id(a_{1})$ with respect to $\cylinder$. Let $f_{2}^{-1}$ be a homotopy inverse of $f_{2}$ with respect to $\cylinder$, and let \ar{\cyl(a_{2}),a_{2},h_{2}} denote the corresponding homotopy from $f_{2}f_{2}^{-1}$ to $id(a_{2})$ with respect to $\cylinder$. We claim that the arrow \ar[5]{a_{1},a_{0},{f_{2}^{-1}} \circ f_{1}} of $\cl{A}$ defines a homotopy inverse to $f_{0}$ with respect to $\cylinder$.

Let \ar{\cyl(a_{1}),a_{1},k_{0}} denote the arrow $h_{1}^{-1} \circ \cyl(f_{0}f_{2}^{-1}f_{1})$ of $\cl{A}$. By Lemma \ref{HomotopyPreAndPostCompositionLemma}, we have that $k_{0}$ defines a homotopy from $f_{0}f_{2}^{-1}f_{1}$ to $f_{1}^{-1}f_{1}f_{0}f_{2}^{-1}f_{1}$ with respect to $\cylinder$. We have that \[ f_{1}^{-1}f_{1}f_{0}f_{2}^{-1}f_{1} = f_{1}^{-1}f_{2}f_{2}^{-1}f_{1}. \] 

Let \ar{\cyl(a_{1}),a_{1},k_{1}} denote the arrow $f_{1}^{-1} \circ h_{2} \circ \cyl(f_{1})$ of $\cl{A}$. Appealing again to Lemma \ref{HomotopyPreAndPostCompositionLemma} we have that $k_{1}$ defines a homotopy from $f_{1}^{-1}f_{2}f_{2}^{-1}f_{1}$ to $f_{1}^{-1}f_{1}$ with respect to $\cylinder$. 

We deduce that the arrow \ar[7]{\cyl(a_{1}),a_{1},{(k_{0} + k_{1}) + h_{1}}} of $\cl{A}$ defines a homotopy from $f_{0}f_{2}^{-1}f_{1}$ to $id(a_{1})$ with respect to $\cylinder$. 

We also have that $h_{2}^{-1}$ defines a homotopy from \[ f_{2}^{-1}f_{1}f_{0} = f_{2}^{-1}f_{2} \] to $id(a_{0})$ with respect to $\cylinder$. This completes the proof of the claim. \end{proof}

\begin{lem} \label{TwoOutOfThreeHomotopyEquivalencesSecondLemma} Let $\cylinder = \big( \cyl, i_0, i_1, v, \subdiv, r_0, r_1, s \big)$ be a cylinder in $\cl{A}$ equipped with an involution structure $v$ and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$. Suppose that we have a commutative diagram in $\cl{A}$ as follows. \tri{a_{0},a_{1},a_{2},f_{0},f_{1},f_{2}} If $f_{0}$ and $f_{2}$ are homotopy equivalences with respect to $\cylinder$, then $f_{1}$ is a homotopy equivalence with respect to $\cylinder$. \end{lem}

\begin{proof} Let $f_{0}^{-1}$ be a homotopy inverse of $f_{0}$ with respect to $\cylinder$, and let \ar{\cyl(a_{1}),a_{1},h_{0}} denote the corresponding homotopy from $f_{0}f_{0}^{-1}$ to $id(a_{1})$ with respect to $\cylinder$. Let $f_{2}^{-1}$ be a homotopy inverse of $f_{2}$ with respect to $\cylinder$, and let \ar{\cyl(a_{2}),a_{2},h_{2}} denote the corresponding homotopy from $f_{2}f_{2}^{-1}$ to $id(a_{2})$ with respect to $\cylinder$. We claim that the arrow \ar[5]{a_{2},a_{1},f_{0} \circ f_{1}^{-1}} of $\cl{A}$ defines a homotopy inverse to $f_{1}$ with respect to $\cylinder$.

Let \ar{\cyl(a_{1}),a_{1},k_{0}} denote the arrow $f_{0}f_{2}^{-1}f_{1} \circ h_{0}^{-1}$ of $\cl{A}$. By Lemma \ref{HomotopyPreAndPostCompositionLemma} we have that $k_{0}$ defines a homotopy from $f_{0}f_{2}^{-1}f_{1}$ to $f_{0}f_{2}^{-1}f_{1}f_{0}f_{0}^{-1}$ with respect to $\cylinder$. We also have that \[ f_{0}f_{2}^{-1}f_{1}f_{0}f_{0}^{-1} = f_{0}f_{2}^{-1}f_{2}f_{0}^{-1}. \] %
%
Let \ar{\cyl(a_{1}),a_{1},k_{1}} denote the arrow $f_{0} \circ h_{2} \circ \cyl(f_{0}^{-1})$ of $\cl{A}$. Appealing again to Lemma \ref{HomotopyPreAndPostCompositionLemma} we have that $k_{1}$ defines a homotopy from $f_{0}f_{2}^{-1}f_{2}f_{0}^{-1}$ to $f_{0}f_{0}^{-1}$ with respect to $\cylinder$. We deduce that the arrow \ar[7]{\cyl(a_{1}),a_{1},{(k_{0} + k_{1}) + h_{0}}} of $\cl{A}$ defines a homotopy from $f_{0}f_{2}^{-1}f_{1}$ to $id(a_{1})$ with respect to $\cylinder$. 

We also have that $h_{2}$ defines a homotopy from \[ f_{1}f_{0}f_{2}^{-1} = f_{2}f_{2}^{-1} \] to $id(a_{2})$ with respect to $\cylinder$. This completes the proof of the claim. \end{proof}

\begin{prpn} \label{TwoOutOfThreeHomotopyEquivalencesProposition} Let $\cylinder = \big( \cyl, i_0, i_1, v, \subdiv, r_0, r_1, s \big)$ be a cylinder in $\cl{A}$ equipped with an involution structure $v$ and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$. Then homotopy equivalences with respect to $\cylinder$ have the two-out-of-three property. \end{prpn}

\begin{proof} Follows immediately from Lemma \ref{TwoOutOfThreeHomotopyEquivalencesFirstLemma}, Lemma \ref{TwoOutOfThreeHomotopyEquivalencesSecondLemma}, and Proposition \ref{CompositionHomotopiesProposition}. \end{proof}

\begin{prpn} \label{RetractionHomotopyEquivalenceIsHomotopyEquivalenceProposition} Let $\cylinder = \big( \cyl, i_{0}, i_{1} \big)$ be a cylinder in $\cl{A}$. Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$ which is a homotopy equivalence with respect to $\cylinder$. Suppose that we have commutative diagrams \twosq{a_{2},a_{0},a_{3},a_{1},g_{0},f,f',g_{1},a_{0},a_{2},a_{1},a_{3},r_{0},f',f,r_{1}} in $\cl{A}$ such that $r_{0}$ is a retraction of $g_{0}$, and such that $r_{1}$ is a retraction of $g_{1}$. Then $f'$ is a homotopy equivalence with respect to $\cylinder$. \end{prpn} 

\begin{proof} Let \ar{a_{1},a_{0},f^{-1}} be a homotopy inverse of $f$. Let $h_{0}$ be a homotopy from $f^{-1}f$ to $id(a_{0})$ with respect to $\cylinder$, and let $h_{1}$ be a homotopy from $ff^{-1}$ to $id(a_{1})$ with respect to $\cylinder$. 

Let \ar{a_{3},a_{2},{(f')^{-1}}} denote the arrow $r_{0} \circ f^{-1} \circ g_{1}$ of $\cl{A}$. Let \ar{\cyl(a_{2}),a_{2},h_{0}'} denote the arrow $r_{0} \circ h_{0} \circ \cyl(g_{0})$ of $\cl{A}$. We claim that $h_{0}'$ defines a homotopy from $(f')^{-1}f'$ to $id(a_{2})$ with respect to $\cylinder$. 

Firstly the following diagram in $\cl{A}$ commutes. \trapeziumstwo{a_{2},\cyl(a_{2}),a_{3},a_{0},\cyl(a_{0}),a_{1},a_{0},i_{0}(a_{2}),\cyl(g_{0}),h_{0},f',g_{1},f^{-1},g_{0},i_{0}(a_{0}),f} %
%
Hence the following diagram in $\cl{A}$ commutes. \sq[{7,3}]{a_{2},\cyl(a_{2}),a_{3},a_{2},i_{0}(a_{2}),r_{0} \circ h_{0} \circ \cyl(g_{0}),f',r_{0} \circ f^{-1} \circ \cyl(g_{0})} %
%
Thus the following diagram in $\cl{A}$ commutes. \sq{a_{2},\cyl(a_{2}),a_{3},a_{2},i_{0}(a_{2}),h_{0}',f',{(f')^{-1}}} %
%
Secondly the following diagram in $\cl{A}$ commutes. \squareabovetriangle{a_{2},\cyl(a_{2}),a_{0},\cyl(a_{0}),a_{0},i_{1}(a_{2}),\cyl(g_{0}),g_{0},i_{1}(a_{0}),h_{0},id} %
%
Hence the following diagram in $\cl{A}$ commutes. \squarewithdiagonal{a_{2},\cyl(a_{2}),a_{2},a_{0},i_{1}(a_{2}),h_{0} \circ \cyl(g_{0}),id,r_{0},g_{0}} %
%
Thus the following diagram in $\cl{A}$ commutes. \tri{a_{2},\cyl(a_{2}),a_{2},i_{1}(a_{2}),h_{0}',id} %
This completes the proof of the claim.

Let \ar{\cyl(a_{3}),a_{3},h_{1}'} denote the arrow $r_{1} \circ h_{1} \circ \cyl(g_{1})$ of $\cl{A}$. We claim that $h_{1}'$ defines a homotopy from $(f')^{-1}f'$ to $id(a_{3})$ with respect to $\cylinder$. 

Firstly the following diagram in $\cl{A}$ commutes. \stackedsq{a_{1},\cyl(a_{1}),a_{0},a_{1},a_{2},a_{3},i_{0}(a_{1}),h_{1},f^{-1},f,r_{0},r_{1},f'} %
%
Hence appealing to the commutativity of the diagram \sq{a_{3},\cyl(a_{3}),a_{1},\cyl(a_{1}),i_{0}(a_{3}),\cyl(g_{1}),g_{1},i_{0}(a_{1})} in $\cl{A}$ we have that the following diagram in $\cl{A}$ commutes. \sq{a_{3},\cyl(a_{3}),a_{2},a_{3},i_{0}(a_{3}),r_{1} \circ h_{1} \circ \cyl(g_{1}),r_{0} \circ f^{-1} \circ g_{1},f'} %
%
Thus the following diagram in $\cl{A}$ commutes. \sq{a_{3},cyl(a_{3}),a_{2},a_{3},i_{0}(a_{3}),h_{1}',{(f')^{-1}},f'}%
%
Secondly the following diagram in $\cl{A}$ commutes. \squareabovetriangle{a_{3},\cyl(a_{3}),a_{1},\cyl(a_{1}),a_{1},i_{1}(a_{3}),\cyl(g_{1}),g_{1},i_{1}(a_{1}),h_{1},id} %
%
Hence the following diagram in $\cl{A}$ commutes. \squarewithdiagonal{a_{3},\cyl(a_{3}),a_{3},a_{1},i_{1}(a_{3}),h_{1} \circ \cyl(g_{1}),id,r_{1},g_{1}} %
%
Thus the following diagram in $\cl{A}$ commutes. \tri{a_{3},\cyl(a_{3}),a_{3},i_{1}(a_{3}),h_{1}',id} %
This completes the proof of the claim. 

\end{proof}

\begin{lem} \label{RightHomotopyInverseIsHomotopyInverseLemma} Let $\cylinder = \big( \cyl, i_0, i_1, v, \subdiv, r_0, r_1, s \big)$ be a cylinder in $\cl{A}$ equipped with an involution structure $v$ and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$. Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$ which is a homotopy equivalence with respect to $\cylinder$, and let \ar{a_{1},a_{0},g} be an arrow of $\cl{A}$ such that there is a homotopy \ar{\cyl(a_{1}),a_{1},h} from $fg$ to $id(a_{1})$ with respect to $\cylinder$. Then $g$ is a homotopy inverse of $f$ with respect to $\cylinder$. \end{lem} 

\begin{proof} Let \ar{a_{1},a_{0},f^{-1}} be a homotopy inverse of $f$ with respect to $\cylinder$, and let \ar{\cyl(a_{0}),a_{0},k} denote the corresponding homotopy from $f^{-1}f$ to $id(a_{0})$ with respect to $\cylinder$. By Lemma \ref{HomotopyPreAndPostCompositionLemma} we have that the arrow \ar[5]{\cyl(a_{1}),a_{0},k \circ \cyl(g)} of $\cl{A}$ defines a homotopy from $f^{-1}fg$ to $g$ with respect to $\cylinder$. 

By Lemma \ref{HomotopyPreAndPostCompositionLemma} once more we also have that the arrow \ar[5]{\cyl(a_{1}),a_{0}, {f^{-1} \circ h^{-1}}} of $\cl{A}$ defines a homotopy from $f^{-1}$ to $f^{-1}fg$ with respect to $\cylinder$. %
%
Hence the arrow \ar[12]{\cyl(a_{1}),a_{0},\big( k \circ \cyl(g) \big) + \big( f^{-1} \circ h^{-1} \big)} of $\cl{A}$ defines a homotopy from $f^{-1}$ to $g$ with respect to $\cylinder$. Let us denote it by $l$ for brevity. 

Appealing again to Lemma \ref{HomotopyPreAndPostCompositionLemma} we have that the arrow \ar[5]{\cyl(a_{0}),a_{0}, l \circ \cyl(f)} of $\cl{A}$ defines a homotopy from $f^{-1}f$ to $gf$ with respect to $\cylinder$. Then \ar[8]{\cyl(a_{0}),a_{0},{\big(l \circ \cyl(f)\big) + k^{-1}}} defines a homotopy from $id(a_{0})$ to $gf$ with respect to $\cylinder$. 

Thus \ar[10]{\cyl(a_{0}),a_{0},{\Big( \big(l \circ \cyl(f)\big) + k^{-1}\Big)^{-1}}} defines a homotopy from $gf$ to $id(a_{0})$ with respect to $\cylinder$. \end{proof}

\begin{rmk} An analogous argument shows that if \ar{a_{0},a_{1},f} satisfies the hypotheses of Lemma \ref{RightHomotopyInverseIsHomotopyInverseLemma}, and if $g$ is an arrow of $\cl{A}$ such that there is a homotopy from $gf$ to $id(a_{0})$ with respect to $\cylinder$, then $g$ is a homotopy inverse of $f$ with respect to $\cylinder$. We will not need this. \end{rmk}

\begin{defn} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$. Let $a_{0}$ and $a_{1}$ be objects of $\cl{A}$. We refer to an arrow \ar{\cyl^{2}(a_{0}),a_{1},\sigma} of $\cl{A}$ as a {\em double homotopy} with respect to $\cylinder$. \end{defn}

\begin{defn} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$. Let \ar{\cyl^{2}(a_{0}),a_{1},\sigma} be a double homotopy with respect to $\cylinder$. We refer to the arrows $h_{0}, h_{1}, h_{2}$, and $h_{3}$ of $\cl{A}$ defined by the commutative diagrams below as the {\em  boundary homotopies} of $\sigma$ with respect to $\cylinder$. \twotriangles[{5,3,0}]{\cyl(a_{0}),\cyl^{2}(a_{0}),a_{1},i_{0}\big(\cyl(a_{0})\big),\sigma,h_{0},\cyl(a_{0}),\cyl^{2}(a_{0}),a_{1},\cyl\big(i_{1}(a_{0})\big),\sigma,h_{1}} \twotriangles[{5,3,0}]{\cyl(a_{0}),\cyl^{2}(a_{0}),a_{1},\cyl\big(i_{0}(a_{0})\big),\sigma,h_{2},\cyl(a_{0}),\cyl^{2}(a_{0}),a_{1},i_{1}\big(\cyl(a_{0})\big),\sigma,h_{3}} \end{defn}

\begin{rmk} \label{DoubleHomotopyPictorialNotationRemark} The following diagrams in $\cl{A}$ commute. 

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=3 em, row sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| a_{0}       \& |(1-0)| \cyl(a_{0}) \&[2em] |(2-0)| a_{0}       \& |(3-0)| \cyl(a_{0}) \\ 
|(0-1)| \cyl(a_{0}) \& |(1-1)| a_{1}       \&      |(2-1)| \cyl(a_{0}) \& |(3-1)| a_{1} \\[1em]
|(0-2)| a_{0}       \& |(1-2)| \cyl(a_{0}) \&[2em] |(2-2)| a_{0}       \& |(3-2)| \cyl(a_{0}) \\ 
|(0-3)| \cyl(a_{0}) \& |(1-3)| a_{1}       \&      |(2-3)| \cyl(a_{0}) \& |(3-3)| a_{1} \\
};
	
\draw[->] (0-0) to node[auto] {$i_{0}(a_{0})$} (1-0);
\draw[->] (1-0) to node[auto] {$h_{2}$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$i_{0}(a_{0})$} (0-1);
\draw[->] (0-1) to node[auto,swap] {$h_{0}$} (1-1);
\draw[->] (2-0) to node[auto] {$i_{0}(a_{0})$} (3-0);
\draw[->] (3-0) to node[auto] {$h_{1}$} (3-1);
\draw[->] (2-0) to node[auto,swap] {$i_{1}(a_{0})$} (2-1);
\draw[->] (2-1) to node[auto,swap] {$h_{0}$} (3-1);
\draw[->] (0-2) to node[auto] {$i_{1}(a_{0})$} (1-2);
\draw[->] (1-2) to node[auto] {$h_{1}$} (1-3);
\draw[->] (0-2) to node[auto,swap] {$i_{1}(a_{0})$} (0-3);
\draw[->] (0-3) to node[auto,swap] {$h_{3}$} (1-3);
\draw[->] (2-2) to node[auto] {$i_{1}(a_{0})$} (3-2);
\draw[->] (3-2) to node[auto] {$h_{2}$} (3-3);
\draw[->] (2-2) to node[auto,swap] {$i_{0}(a_{0})$} (2-3);
\draw[->] (2-3) to node[auto,swap] {$h_{3}$} (3-3);

\end{tikzpicture} 

\end{diagram} We denote by $f_{0}$, $f_{1}$, $f_{2}$, and $f_{3}$ the arrows of $\cl{A}$ obtained by taking either route through each of these four commutative diagrams, proceeding clockwise respectively from the top left diagram. Thus the following diagrams in $\cl{A}$ commute.

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=3 em, row sep=4 em, nodes={anchor=center}]
{ 
|(0-0)| a_{0})      \& |(1-0)| \cyl(a_{0}) \&[2em] |(2-0)| a_{0} \& |(3-0)| \cyl(a_{0}) \\ 
|(0-1)| \cyl(a_{0}) \& |(1-1)| a_{1}       \&      |(2-1)| \cyl(a_{0}) \& |(3-1)| a_{1} \\
|(0-2)| a_{0}      \& |(1-2)| \cyl(a_{0}) \& |(2-2)| a_{0} \& |(3-2)| \cyl(a_{0}) \\ 
|(0-3)| \cyl(a_{0}) \& |(1-3)| a_{1}      \& |(2-3)| \cyl(a_{0}) \& |(3-3)| a_{1} \\
};

\draw[->] (0-0) to node[auto] {$i_{0}(a_{0})$} (1-0);
\draw[->] (1-0) to node[auto] {$h_{2}$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$i_{0}(a_{0})$} (0-1);
\draw[->] (0-1) to node[auto,swap] {$h_{0}$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$f_{0}$} (1-1); 
\draw[->] (2-0) to node[auto] {$i_{0}(a_{0})$} (3-0);
\draw[->] (3-0) to node[auto] {$h_{1}$} (3-1);
\draw[->] (2-0) to node[auto,swap] {$i_{1}(a_{0})$} (2-1);
\draw[->] (2-1) to node[auto,swap] {$h_{0}$} (3-1);
\draw[->] (2-0) to node[auto,swap] {$f_{1}$} (3-1); 
\draw[->] (0-2) to node[auto] {$i_{1}(a_{0})$} (1-2);
\draw[->] (1-2) to node[auto] {$h_{1}$} (1-3);
\draw[->] (0-2) to node[auto,swap] {$i_{1}(a_{0})$} (0-3);
\draw[->] (0-3) to node[auto,swap] {$h_{3}$} (1-3);
\draw[->] (0-2) to node[auto,swap] {$f_{3}$} (1-3); 
\draw[->] (2-2) to node[auto] {$i_{1}(a_{0})$} (3-2);
\draw[->] (3-2) to node[auto] {$h_{2}$} (3-3);
\draw[->] (2-2) to node[auto,swap] {$i_{0}(a_{0})$} (2-3);
\draw[->] (2-3) to node[auto,swap] {$h_{3}$} (3-3);
\draw[->] (2-2) to node[auto,swap] {$f_{2}$} (3-3); 

\end{tikzpicture} 

\end{diagram} %
%
In summary:

\begin{itemize}[itemsep=1em, topsep=1em]

\item[(0)] $h_{0}$ defines a homotopy from $f_{0}$ to $f_{1}$ with respect to $\cylinder$,

\item[(1)] $h_{1}$ defines a homotopy from $f_{1}$ to $f_{3}$ with respect to $\cylinder$,

\item[(2)] $h_{2}$ defines a homotopy from $f_{0}$ to $f_{2}$ with respect to $\cylinder$, 

\item[(3)] $h_{3}$ defines a homotopy from $f_{2}$ to $f_{3}$ with respect to $\cylinder$. 

\end{itemize} %
%
To express (1)--(4) we will frequently depict $\sigma$ as follows. \doublehomotopy{h_{0},h_{1},h_{2},h_{3},\sigma,f_{0},f_{1},f_{2},f_{3}} \end{rmk}

\begin{prpn} Let $\cylinder = \big( \cyl, i_0, i_1, p, \Gamma_{ul} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$ and an upper left connection structure $\Gamma_{ul}$. Let \pair{a_{0},a_{1},f_{0},f_{1}} be arrows of $\cl{A}$, and let \ar{\cyl(a_{0}),a_{1},h} be a homotopy from $f_{0}$ to $f_{1}$ with respect to $\cylinder$. Let \ar{\cyl^{2}(a_{0}),a_{1},\sigma} denote the arrow $h \circ \Gamma_{ul}(a_{0})$ of $\cl{A}$. Then $\sigma$ defines a double homotopy with respect to $\cylinder$ with boundary homotopies as depicted below. \doublehomotopy{h,id,h,id,\sigma,f_{0},f_{1},f_{1},f_{1}} \end{prpn}

\begin{proof} Follows immediately from the fact that $\Gamma_{ul}$ defines an upper left connection structure with respect to $\cylinder$. \end{proof}

\begin{prpn} Let $\cylinder = \big( \cyl, i_0, i_1, p, \Gamma_{lr} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$ and a lower right connection structure $\Gamma_{lr}$. Let \pair{a_{0},a_{1},f_{0},f_{1}} be arrows of $\cl{A}$, and let \ar{\cyl(a_{0}),a_{1},h} be a homotopy from $f_{0}$ to $f_{1}$ with respect to $\cylinder$. Let \ar{\cyl^{2}(a_{0}),a_{1},\sigma} denote the arrow $h \circ \Gamma_{lr}(a_{0})$ of $\cl{A}$. Then $\sigma$ defines a double homotopy with respect to $\cylinder$ with boundary homotopies as depicted below. \doublehomotopy{id,h,id,h,\sigma,f_{0},f_{0},f_{0},f_{1}} \end{prpn}

\begin{proof} Follows immediately from the fact that $\Gamma_{lr}$ defines a lower right connection structure with respect to $\cylinder$. \end{proof} 

\begin{prpn} Let $\cylinder = \big( \cyl, i_0, i_1, p, \Gamma_{ur} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$ and an upper right connection structure $\Gamma_{ur}$. Let \pair{a_{0},a_{1},f_{0},f_{1}} be arrows of $\cl{A}$, and let \ar{\cyl(a_{0}),a_{1},h} be a homotopy from $f_{0}$ to $f_{1}$ with respect to $\cylinder$. Let \ar{\cyl^{2}(a_{0}),a_{1},\sigma} denote the arrow $h \circ \Gamma_{ur}(a_{0})$ of $\cl{A}$. Then $\sigma$ defines a double homotopy with respect to $\cylinder$ with the boundary homotopies depicted below. \doublehomotopy{h,h^{-1},id,id,\sigma,f_{0},f_{1},f_{0},f_{0}} \end{prpn} 

\begin{proof} Follows immediately from the fact that $\Gamma_{ur}$ defines an upper right connection structure with respect to $\cylinder$. \end{proof}

\begin{defn} Let $\cylinder = \big( \cyl, i_0, i_1 ,p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. Suppose that we have a pair of commutative diagrams in $\cl{A}$ as follows. \twotriangles{a,a_{0},a_{1},j_{0},f_{0},j_{1},a,a_{0},a_{1},j_{0},f_{1},j_{1}} A {\em homotopy under $a$} from $f_{0}$ to $f_{1}$ with respect to $\cylinder$ and $(j_{0},j_{1})$ is a homotopy \ar{\cyl(a_{0}),a_{1},h} from $f_{0}$ to $f_{1}$ with respect to $\cylinder$ such that the following diagram in $\cl{A}$ commutes. \sq{\cyl(a),a,\cyl(a_{0}),a_{1},p(a),j_{1},\cyl(j_{0}),h} \end{defn}

\begin{defn} Let $\cylinder = \big( \cyl, i_0, i_1 ,p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. Suppose that we have a pair of commutative diagrams in $\cl{A}$ as follows. \twoothertriangles{a_{0},a_{1},a,f_{0},j_{1},j_{0},a_{0},a_{1},a,f_{1},j_{0},j_{1}} A {\em homotopy over $a$} from $f_{0}$ to $f_{1}$ with respect to $\cylinder$ and $(j_{0},j_{1})$ is a homotopy \ar{\cyl(a_{0}),a_{1},h} from $f_{0}$ to $f_{1}$ with respect to $\cylinder$ such that the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{0}),a_{1},\cyl(a),a,h,j_{1},\cyl(j_{0}),p(a)} \end{defn}

\begin{defn} \label{UnderHomotopyCoCylinderDefinition} Let $\cocylinder = \big( \cocyl, e_0, e_1 , c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that we have a pair of commutative diagrams in $\cl{A}$ as follows. \twotriangles{a,a_{0},a_{1},j_{0},f_{0},j_{1},a,a_{0},a_{1},j_{0},f_{1},j_{1}} A {\em homotopy under $a$} from $f_{0}$ to $f_{1}$ with respect to $\cocylinder$ and $(j_{0},j_{1})$ is a homotopy \ar{a_{0},\cocyl(a_{1}),h} from $f_{0}$ to $f_{1}$ with respect to $\cocylinder$ such that the following diagram in $\cl{A}$ commutes. \sq{a,\cocyl(a),a_{0},\cocyl(a_{1}),c(a),\cocyl(j_{1}),j_{0},h} \end{defn}

\begin{rmk} Let $\cocylinder = \big( \cocyl, e_0, e_1 , c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that we have a pair of commutative diagrams in $\cl{A}$ as follows. \twotriangles{a,a_{0},a_{1},j_{0},f_{0},j_{1},a,a_{0},a_{1},j_{0},f_{1},j_{1}} Then an arrow \ar{a_{0},\cocyl(a_{1}),h} of $\cl{A}$ is a homotopy under $a$ from $f_{0}$ to $f_{1}$ with respect to $\cocylinder$ if and only if $h^{op}$ is a homotopy over $a$ from $f_{0}^{op}$ to $f_{1}^{op}$ with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$ equipped with the contraction structure $c^{op}$, and with respect to the arrows $(j_{1}^{op},j_{0}^{op})$ of $\cl{A}^{op}$. \end{rmk}

\begin{defn} \label{OverHomotopyCoCylinderDefinition} Let $\cocylinder = \big( \cocyl, e_0, e_1 ,c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that we have a pair of commutative diagrams in $\cl{A}$ as follows. \twoothertriangles{a_{0},a_{1},a,f_{0},j_{1},j_{0},a_{0},a_{1},a,f_{1},j_{1},j_{0}} A {\em homotopy over $a$} from $f_{0}$ to $f_{1}$ with respect to $\cocylinder$ and $(j_{0},j_{1})$ is a homotopy \ar{a_{0},\cocyl(a_{1}),h} from $f_{0}$ to $f_{1}$ with respect to $\cocylinder$ such that the following diagram in $\cl{A}$ commutes. \sq{a_{0},\cocyl(a_{1}),a,\cocyl(a),h,\cocyl(j_{1}),j_{0},c(a)} \end{defn}

\begin{rmk} Let $\cocylinder = \big( \cocyl, e_0, e_1 ,c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that we have a pair of commutative diagrams in $\cl{A}$ as follows. \twoothertriangles{a_{0},a_{1},a,f_{0},j_{1},j_{0},a_{0},a_{1},a,f_{1},j_{1},j_{0}} An arrow \ar{a_{0},\cocyl(a_{1}),h} of $\cl{A}$ is a homotopy over $a$ from $f_{0}$ to $f_{1}$ with respect to $\cocylinder$ and $(j_{0},j_{1})$ if and only if $h^{op}$ is a homotopy under $a$ from $f_{0}^{op}$ to $f_{1}^{op}$ with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$ equipped with the contraction structure $c^{op}$, and with respect to the arrows $(j_{1}^{op},j_{0}^{op})$ of $\cl{A}^{op}$. \end{rmk}

\begin{prpn} \label{OverHomotopyCylinderGivesOverHomotopyCoCylinderProposition} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, and let $\cocylinder = \big( \cocyl, e_0, e_1, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$. Let \ar{{\mathsf{Hom}_{\cl{A}}\big(\cyl(-),- \big)},{\mathsf{Hom}_{\cl{A}}\big(-,\cocyl(-) \big)},\mathsf{adj}} denote the corresponding natural isomorphism, adopting the shorthand of Recollection \ref{AdjunctionRecollection}. Suppose that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

Suppose that we have a pair of commutative diagrams in $\cl{A}$ as follows. \twoothertriangles{a_{0},a_{1},a,f_{0},j_{1},j_{0},a_{0},a_{1},a,f_{1},j_{1},j_{0}} If an arrow \ar{\cyl(a_{0}),a_{1},h} of $\cl{A}$ defines a homotopy over $a$ from $f_{0}$ to $f_{1}$ with respect to $\cylinder$ and $(j_{0},j_{1})$, then the arrow \ar[4]{a_{0},\cocyl(a_{1}),\mathsf{adj}(h)} of $\cl{A}$ defines a homotopy over $a$ from $f_{0}$ to $f_{1}$ with respect to $\cocylinder$ and $(j_{0},j_{1})$. \end{prpn}

\begin{proof} Firstly by Proposition \ref{HomotopyCylinderGivesHomotopyCoCylinderProposition} if $h$ is a homotopy from $f_{0}$ to $f_{1}$ with respect to $\cylinder$ then $\mathsf{adj}(h)$ is a homotopy from $f_{0}$ to $f_{1}$ with respect to $\cocylinder$. 

Secondly since $h$ is a homotopy over $a$ with respect to $\cylinder$ and $(j_{0},j_{1})$ the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{0}),a_{1},\cyl(a),a,h,j_{1},\cyl(j_{0}),p(a)} Moreover the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{\cyl(a_{0}),\cyl(a),a_{0},a,\cyl(j_{0}),p(a),j_{0},p(a_{0})} %
%
Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{0}),a_{1},a_{0},a,h,j_{1},p(a_{0}),j_{0}} Thus we have that \[ \mathsf{adj}(j_{1} \circ h) = \mathsf{adj}\big( j_{0} \circ p(a_{0}) \big). \] %
%
Moreover by the naturality of the isomorphism $\mathsf{adj}$ the following diagram in $\cl{A}$ commutes. \tri[{4,3}]{a_{0},\cocyl(a_{1}),\cocyl(a),{\mathsf{adj}(h)},\cocyl(j_{1}),{\mathsf{adj}(j_{1} \circ h)}} Hence the following diagram in $\cl{A}$ commutes. \tri[{4,3}]{a_{0},\cocyl(a_{1}),\cocyl(a),{\mathsf{adj}(h)},\cocyl(j_{1}),{\mathsf{adj} \big( j_{0} \circ p(a_{0}) \big) }} %
%
Since the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$ the following diagram in $\cl{A}$ also commutes. \triother{a_{0},a,\cocyl(a),j_{0},c(a),\mathsf{adj}(j_{0} \circ p(a_{0})\big)} Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{a_{0},\cocyl(a_{1}),a,\cocyl(a),\mathsf{adj}(h),\cocyl(j_{1}),j_{0},c(a)}
\end{proof}

\begin{prpn} \label{UnderHomotopyCylinderGivesUnderHomotopyCoCylinderProposition} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, and let $\cocylinder = \big( \cocyl, e_0, e_1, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$. Let \ar{{\mathsf{Hom}_{\cl{A}}\big(\cyl(-),- \big)},{\mathsf{Hom}_{\cl{A}}\big(-,\cocyl(-) \big)},\mathsf{adj}} denote the corresponding natural isomorphism, adopting the shorthand of Recollection \ref{AdjunctionRecollection}. Suppose that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

Suppose that we have a pair of commutative diagrams in $\cl{A}$ as follows. \twotriangles{a,a_{0},a_{1},j_{0},f_{0},j_{1},a,a_{0},a_{1},j_{0},f_{1},j_{1}}If an arrow \ar{\cyl(a_{0}),a_{1},h} of $\cl{A}$ defines a homotopy under $a$ from $f_{0}$ to $f_{1}$ with respect to $\cylinder$ and $(j_{0},j_{1})$, then the arrow \ar[4]{a_{0},\cocyl(a_{1}),\mathsf{adj}(h)} of $\cl{A}$ defines a homotopy under $a$ from $f_{0}$ to $f_{1}$ with respect to $\cocylinder$ and $(j_{0},j_{1})$. \end{prpn}

\begin{proof} Firstly by Proposition \ref{HomotopyCylinderGivesHomotopyCoCylinderProposition} if $h$ is a homotopy from $f_{0}$ to $f_{1}$ with respect to $\cylinder$ then $\mathsf{adj}(h)$ is a homotopy from $f_{0}$ to $f_{1}$ with respect to $\cocylinder$. 

Secondly since $h$ is a homotopy under $a$ with respect to $\cylinder$ and $(j_{0},j_{1})$ the following diagram in $\cl{A}$ commutes. \sq{\cyl(a),a,\cyl(a_{0}),a_{1},p(a),j_{1},\cyl(j_{0}),h} Thus we have that \[ \mathsf{adj}\big(h \circ \cyl(j_{0})\big) = \mathsf{adj}(j_{1} \circ p(a) \big). \] %
%
Moreover by the naturality of the isomorphism $\mathsf{adj}$ the following diagram in $\cl{A}$ commutes. \triother[{4,3}]{a,a_{0},\cocyl(a_{1}),j_{0},\mathsf{adj}(h),\mathsf{adj}\big(h \circ \cyl(j_{0})\big)} %
%
Hence the following diagram in $\cl{A}$ commutes. \triother[{4,3}]{a,a_{0},\cocyl(a_{1}),j_{0},\mathsf{adj}(h),\mathsf{adj}\big(j_{1} \circ p(a)\big)} Since the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$ the following diagram in $\cl{A}$ also commutes. \tri{a,\cocyl(a),\cocyl(a_{1}),c(a),\cocyl(j_{1}),\mathsf{adj}\big(j_{1} \circ p(a)\big)} Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{a,\cocyl(a),a_{0},\cocyl(a_{1}),c(a),\cocyl(j_{1}),j_{0},\mathsf{adj}(h)} \end{proof}

\begin{cor} \label{OverHomotopyCoCylinderGivesOverHomotopyCylinderCorollary} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, and let $\cocylinder = \big( \cocyl, e_0, e_1, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$. Let \ar{{\mathsf{Hom}_{\cl{A}}\big(\cyl(-),- \big)},{\mathsf{Hom}_{\cl{A}}\big(-,\cocyl(-) \big)},\mathsf{adj}} denote the corresponding natural isomorphism, adopting the shorthand of Recollection \ref{AdjunctionRecollection}. Suppose that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

Suppose that we have a pair of commutative diagrams in $\cl{A}$ as follows. \twoothertriangles{a_{0},a_{1},a,f_{0},j_{1},j_{0},a_{0},a_{1},a,f_{1},j_{1},j_{0}}If an arrow \ar{a_{0},\cocyl(a_{1}),h} of $\cl{A}$ defines a homotopy over $a$ from $f_{0}$ to $f_{1}$ with respect to $\cocylinder$ and $(j_{0},j_{1})$, then the arrow \ar{\cyl(a_{0}),a_{1},\mathsf{adj}^{-1}(h)} of $\cl{A}$ defines a homotopy over $a$ from $f_{0}$ to $f_{1}$ with respect to $\cylinder$ and $(j_{0},j_{1})$. \end{cor}

\begin{proof} Follows immediately from Proposition \ref{UnderHomotopyCylinderGivesUnderHomotopyCoCylinderProposition} by duality. \end{proof}

\begin{cor} \label{UnderHomotopyCoCylinderGivesUnderHomotopyCylinderCorollary} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, and let $\cocylinder = \big( \cocyl, e_0, e_1, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$. Let \ar{{\mathsf{Hom}_{\cl{A}}\big(\cyl(-),- \big)},{\mathsf{Hom}_{\cl{A}}\big(-,\cocyl(-) \big)},\mathsf{adj}} denote the corresponding natural isomorphism, adopting the shorthand of Recollection \ref{AdjunctionRecollection}. Suppose that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

Suppose that we have a pair of commutative diagrams in $\cl{A}$ as follows. \twotriangles{a,a_{0},a_{1},j_{0},f_{0},j_{1},a,a_{0},a_{1},j_{0},f_{1},j_{1}} If an arrow \ar{a_{0},\cocyl(a_{1}),h} of $\cl{A}$ defines a homotopy under $a$ from $f_{0}$ to $f_{1}$ with respect to $\cocylinder$ and $(j_{0},j_{1})$, then the arrow \ar[4]{\cyl(a_{0}),a_{1},\mathsf{adj}^{-1}(h)} of $\cl{A}$ defines a homotopy under $a$ from $f_{0}$ to $f_{1}$ with respect to $\cylinder$ and $(j_{0},j_{1})$. \end{cor}

\begin{proof} Follows immediately from Proposition \ref{OverHomotopyCylinderGivesOverHomotopyCoCylinderProposition} by duality. \end{proof}

\begin{prpn} \label{IdentityHomotopyIsOverHomotopyProposition} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure. Suppose that we have a commutative diagram in $\cl{A}$ as follows. \triother{a_{0},a_{1},a,f,j_{1},j_{0}} %
%
Then the identity homotopy from $f$ to itself with respect to $\cylinder$ is moreover a homotopy over $a$ with respect to $\cylinder$ and $(j_{0},j_{1})$. \end{prpn}

\begin{proof} Let $h$ denote the identity homotopy from $f$ to itself with respect to $\cylinder$. By definition of $h$ the following diagram in $\cl{A}$ commutes. \tri[{4,3}]{\cyl(a_{0}),a_{0},a_{1},p(a_{0}),f,h} %
%
Thus the following diagram in $\cl{A}$ commutes. \squareofthreetrianglesthree{\cyl(a_{0}),a_{1},a_{0},\cyl(a),a,h,j_{1},\cyl(j_{0}),p(a),p(a_{0}),f,j_{0}}
 
\end{proof}

\begin{prpn} \label{ReverseHomotopyIsOverHomotopyProposition} Let $\cylinder = \big( \cyl, i_0, i_1, p, v \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, and an involution structure $v$ compatible with $p$. 

Suppose that we have arrows \ar{a_{0},a,j_{0}} and \ar{a_{1},a,j_{1}} of $\cl{A}$ and a homotopy \ar{\cyl(a_{0}), a_{1}, h} with respect to $\cylinder$ such that the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{0}),a_{1},\cyl(a),a,h,j_{1},\cyl(j_{0}),p(a)} Then the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{0}),a_{1},\cyl(a),a,{h^{-1}},j_{1},\cyl(j_{0}),p(a)} \end{prpn}

\begin{proof} By definition of $h^{-1}$ the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{0}),\cyl(a_{0}),a_{1},v(a_{0}),h,h^{-1}} %
%
Thus we have that the following diagram in $\cl{A}$ commutes. \squareofthreetrianglesfour{\cyl(a_{0}),a_{1},\cyl(a),\cyl(a_{0}),\cyl(a),a,h^{-1},j_{1},\cyl(j_{0}),v(a),p(a),v(a_{0}),h,\cyl(j_{0})} %
% 
Since $v$ is compatible with $p$ we also have that the following diagram in $\cl{A}$ commutes. \tri{\cyl(a),\cyl(a),a,v(a),p(a),p(a)} %
%
Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes as required. \sq{\cyl(a_{0}),a_{1},\cyl(a),a,h^{-1},j_{1},\cyl(j_{0}),p(a)}
\end{proof}

\begin{cor} \label{ReverseHomotopyIsOverHomotopyCorollary} Let $\cylinder = \big( \cyl, i_0, i_1, p, v \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, and an involution structure $v$ compatible with $p$. 

Suppose that we have a pair of commutative diagrams in $\cl{A}$ as follows. \twoothertriangles{a_{0},a_{1},a,f_{0},j_{1},j_{0},a_{0},a_{1},a,f_{1},j_{1},j_{0}} Let \ar{\cyl(a_{0}), a_{1}, h} be a homotopy over $a$ from $f_{0}$ to $f_{1}$ with respect to $\cylinder$ and $(j_{0},j_{1})$. Then the reverse homotopy \ar{\cyl(a_{0}),a_{1},h^{-1}} from $f_{1}$ to $f_{0}$ with respect to $\cylinder$ is moreover a homotopy under $a$ with respect to $\cylinder$ and $(j_{0},j_{1})$ . \end{cor}

\begin{proof} Follows immediately from Proposition \ref{ReverseHomotopyIsOverHomotopyProposition}.
\end{proof}

\begin{prpn} \label{CompositeHomotopyIsOverHomotopyProposition} Let $\cylinder = \big( \cyl, i_0, i_1, p, \subdiv, r_0, r_1, s \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ compatible with $p$. 

Suppose that we have arrows \ar{a_{0},a,j_{0}} and \ar{a_{1},a,j_{1}} of $\cl{A}$ and homotopies \ar{\cyl(a_{0}), a_{1}, h} and \ar{\cyl(a_{0}), a_{1}, k} with respect to $\cylinder$ such that the diagrams \twosq{\cyl(a_{0}),a_{1},\cyl(a),a,h,j_{1},\cyl(j_{0}),p(a),\cyl(a_{0}),a_{1},\cyl(a),a,k,j_{1},\cyl(j_{0}),p(a)} and \sq{a_{0},\cyl(a_{0}),\cyl(a_{0}),a_{1},i_{0}(a_{0}),k,i_{1}(a_{0}),h} in $\cl{A}$ commute. Then the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{\cyl(a_{0}),a_{1},\cyl(a),a,h+k,j_{1},\cyl(j_{0}),p(a)} \end{prpn}

\begin{proof} Appealing to the universal property of $\subdiv(a_{1})$ there is an arrow \ar{\subdiv(a_{0}),a_{1},r} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),\cyl(a_{0}),\subdiv(a_{0}),a_{1},{i_{0}(a_{0})},{r_{0}(a_{0})},{i_{1}(a_{0})},{r_{1}(a_{0})},h,k,r} %
%
The following diagram in $\cl{A}$ commutes. \squarewithdiagonal{\cyl(a_{0}),\subdiv(a_{0}),a,a_{1},r_{0}(a_{0}),r,p(a) \circ \cyl(j_{0}),j_{1},h} %
%
The following diagram in $\cl{A}$ also commutes. \squarewithdiagonal{\cyl(a_{0}),\subdiv(a_{0}),a,a_{1},r_{1}(a_{0}),r,p(a) \circ \cyl(j_{0}),j_{1},k} %
%
Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),\cyl(a_{0}),\subdiv(a_{0}),a,i_{0}(a_{0}),r_{0}(a_{0}),i_{1}(a_{0}),r_{1}(a_{0}),p(a_{0}) \circ \cyl(j_{0}),p(a_{0}) \circ \cyl(j_{0}), j_{1} \circ r} %
%
Let \ar{\subdiv,\id_{\cl{A}},\overline{p}} denote the canonical 2-arrow of $\cl{C}$ of Definition \ref{SubdivisionCompatibleWithContractionDefinition}. The following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),\cyl(a_{0}),\subdiv(a_{0}),a_{0},i_{0}(a_{0}),r_{0}(a_{0}),i_{1}(a_{0}),r_{1}(a_{0}),p(a_{0}),p(a_{0}),\overline{p}(a_{0})} %
%
Thus the following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),\cyl(a_{0}),\subdiv(a_{0}),a,i_{0}(a_{0}),r_{0}(a_{0}),i_{1}(a_{0}),r_{1}(a_{0}),j_{0} \circ p(a_{0}),j_{0} \circ p(a_{0}), j_{0} \circ \overline{p}(a_{0})} %
%
Moreover the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{0}),a_{0},\cyl(a),a,p(a_{0}),j_{0},\cyl(j_{0}),p(a)} %
%
Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),\cyl(a_{0}),\subdiv(a_{0}),a,i_{0}(a_{0}),r_{0}(a_{0}),i_{1}(a_{0}),r_{1}(a_{0}),p(a) \circ \cyl(j_{0}), p(a) \circ \cyl(j_{0}), j_{0} \circ \overline{p}(a_{0})} % 
% 
Appealing to the universal property of $\subdiv(a_{0})$ we deduce that the following diagram in $\cl{A}$ commutes. \sq{\subdiv(a_{0}),a_{1},a_{0},a,r,j_{1},\overline{p}(a_{0}),j_{0}} %
%
Since the subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ is compatible with $p$ we also have that that the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{0}),\subdiv(a_{0}),a_{0},s(a_{0}),\overline{p}(a_{0}),p(a_{0})} %
%
By definition of $h+k$ the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{0}),\subdiv(a_{0}),a_{1},s(a_{0}),r,{h+k}} %
%
Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \squareofthreetrianglestwo{\cyl(a_{0}),a_{1},\subdiv(a_{0}),a_{0},a,h+k,j_{1},p(a_{0}),j_{0},s(a_{0}),r,\overline{p}(a_{0})} %
%
Appealing once more to the commutativity of the diagram \sq{\cyl(a_{0}),a_{0},\cyl(a),a,p(a_{0}),j_{0},\cyl(j_{0}),p(a)} in $\cl{A}$ we conclude that the following diagram in $\cl{A}$ commutes. \sq[{3,3.5,0,0}]{\cyl(a_{0}),a_{1},\cyl(a),a,h+k,j_{1},\cyl(j_{0}),p(a)}
 
\end{proof} 

\begin{cor} \label{CompositeHomotopyIsOverHomotopyCorollary} Let $\cylinder = \big( \cyl, i_0, i_1, p, \subdiv, r_0, r_1, s \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$ and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ compatible with $p$. 

Suppose that we have three commutative diagrams in $\cl{A}$ as follows.

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=3 em, row sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| a_{0} \& |(1-0)|   \&[2em] |(2-0)| a_{0} \& |(3-0)|   \&[2em] |(4-0)| a_{0}  \& |(5-0)|   \\ 
|(0-1)| a_{1} \& |(1-1)| a \&      |(2-1)| a_{1} \& |(3-1)| a \&      |(4-1)| a_{1}  \& |(5-1)| a \\
};
	
\draw[->] (0-0) to node[auto] {$j_{0}$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$f_{0}$} (0-1);
\draw[->] (0-1) to node[auto,swap] {$j_{1}$} (1-1);
\draw[->] (2-0) to node[auto] {$j_{0}$} (3-1);
\draw[->] (2-0) to node[auto,swap] {$f_{1}$} (2-1);
\draw[->] (2-1) to node[auto,swap] {$j_{1}$} (3-1);
\draw[->] (4-0) to node[auto] {$j_{0}$} (5-1);
\draw[->] (4-0) to node[auto,swap] {$f_{2}$} (4-1);
\draw[->] (4-1) to node[auto,swap] {$j_{1}$} (5-1);

\end{tikzpicture} 

\end{diagram} %
%
Let \ar{\cyl(a_{0}), a_{1}, h} be a homotopy over $a$ from $f_{0}$ to $f_{1}$ with respect to $\cylinder$ and $(j_{0},j_{1})$. Let \ar{\cyl(a_{0}), a_{1}, k} be a homotopy under $a$ from $f_{1}$ to $f_{2}$ with respect to $\cylinder$ and $(j_{0},j_{1})$. Then the homotopy \ar[4]{\cyl(a_{0}),a_{1},h + k} from $f_{0}$ to $f_{2}$ with respect to $\cylinder$ is moreover a homotopy under $a$ with respect to $\cylinder$ and $(j_{0},j_{1})$. \end{cor}

\begin{proof} Follows immediately from Proposition \ref{CompositeHomotopyIsOverHomotopyProposition}. \end{proof} 

\begin{rmk} Analogues of Proposition \ref{IdentityHomotopyIsOverHomotopyProposition}, Corollary \ref{ReverseHomotopyIsOverHomotopyCorollary}, and Corollary \ref{CompositeHomotopyIsOverHomotopyCorollary} for homotopies under an object can all be proven. Moreover all of these results dualise to the setting of over and under homotopies with respect to a co-cylinder. We will not need any of this. \end{rmk}
 
\begin{defn} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, and suppose that we have a commutative diagram in $\cl{A}$ as follows. \tri{a,a_{0},a_{1},j_{0},f,j_{1}} A {\em homotopy inverse under $a$} of $f$ with respect to $\cylinder$ and $(j_{0},j_{1})$ is an arrow \ar{a_{1},a_{0},f^{-1}} of $\cl{A}$ such that the diagram \tri{a,a_{1},a_{0},j_{1},f^{-1},j_{0}} in $\cl{A}$ commutes together with a homotopy under $a$ from $f^{-1}f$ to $id(a_{0})$ with respect to $\cylinder$ and $(j_{0},j_{0})$ and a homotopy under $a$ from $ff^{-1}$ to $id(a_{1})$ with respect to $\cylinder$ and $(j_{1},j_{1})$. \end{defn}

\begin{defn} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. Suppose that we have a commutative diagram in $\cl{A}$ as follows. \tri{a,a_{0},a_{1},j_{0},f,j_{1}} Then $f$ is a {\em homotopy equivalence under $a$} with respect to $\cylinder$ and $(j_{0},j_{1})$ if it admits a homotopy inverse under $a$ with respect to $\cylinder$ and $(j_{0},j_{1})$. \end{defn} 

\begin{defn} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. Suppose that we have a commutative diagram in $\cl{A}$ as follows. \triother{a_{0},a_{1},a,f,j_{1},j_{0}} A {\em homotopy inverse over $a$} of $f$ with respect to $\cylinder$ and $(j_{0},j_{1})$ is an arrow \ar{a_{1},a_{0},f^{-1}} of $\cl{A}$ such that the diagram \triother{a_{1},a_{0},a,f^{-1},j_{0},j_{1}} in $\cl{A}$ commutes together with a homotopy over $a$ from $f^{-1}f$ to $id(a_{0})$ with respect to $\cylinder$ and $(j_{0},j_{0})$ and a homotopy over $a$ from $ff^{-1}$ to $id(a_{1})$ with respect to $\cylinder$ and $(j_{1},j_{1})$. \end{defn}

\begin{defn} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, and suppose that we have a commutative diagram in $\cl{A}$ as follows. \triother{a_{0},a_{1},a,f,j_{1},j_{0}} Then $f$ is a {\em homotopy equivalence over $a$} with respect to $\cylinder$ and $(j_{0},j_{1})$ if it admits a homotopy inverse over $a$ with respect to $\cylinder$ and $(j_{0},j_{1})$. \end{defn} 

\begin{defn} Let $\cocylinder = \big( \cocyl, e_0, e_1, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that we have a commutative diagram in $\cl{A}$ as follows. \tri{a,a_{0},a_{1},j_{0},f,j_{1}} Then $f$ is a {\em homotopy equivalence under $a$} with respect to $\cocylinder$ if $f^{op}$ is a homotopy equivalence over $a$ with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$ equipped with the contraction structure $c^{op}$, and $(j_{1}^{op},j_{0}^{op})$. \end{defn} 

\begin{defn} Let $\cocylinder = \big( \cocyl, e_0, e_1, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that we have a commutative diagram in $\cl{A}$ as follows. \triother{a_{0},a_{1},a,f,j_{1},j_{0}} Then $f$ is a {\em homotopy equivalence over $a$} with respect to $\cocylinder$ if $f^{op}$ is a homotopy equivalence under $a$ with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$ equipped with the contraction structure $c^{op}$, and $(j_{1}^{op},j_{0}^{op})$. \end{defn} 

\begin{prpn} \label{UnderHomotopyEquivalenceCylinderIffUnderHomotopyEquivalenceCoCylinderProposition} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, and let $\cocylinder = \big( \cocyl, e_0, e_1, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$ and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

Suppose that we have a commutative diagram in $\cl{A}$ as follows. \tri{a, a_{0}, a_{1}, j_{0}, f, j_{1}} Then $f$ is a homotopy equivalence under $a$ with respect to $\cylinder$ if and only if it is a homotopy equivalence under $a$ with respect to $\cocylinder$. \end{prpn}

\begin{proof} Follows immediately from Proposition \ref{UnderHomotopyCylinderGivesUnderHomotopyCoCylinderProposition} and Corollary \ref{UnderHomotopyCoCylinderGivesUnderHomotopyCylinderCorollary}. \end{proof} 

\begin{prpn}  \label{OverHomotopyEquivalenceCylinderIffOverHomotopyEquivalenceCoCylinderProposition} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, and let $\cocylinder = \big( \cocyl, e_0, e_1, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

Suppose that we have a commutative diagram in $\cl{A}$ as follows. \triother{a,a_{0},a_{1},f,j_{0},j_{1}} Then $f$ is a homotopy equivalence over $a$ with respect to $\cylinder$ if and only if it is a homotopy equivalence over $a$ with respect to $\cocylinder$. \end{prpn}

\begin{proof} Follows immediately from Proposition \ref{OverHomotopyCylinderGivesOverHomotopyCoCylinderProposition} and Corollary \ref{UnderHomotopyCoCylinderGivesUnderHomotopyCylinderCorollary}. \end{proof} 

\begin{defn} Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$. An arrow \ar{a_{0},a_{1},f} of $\cl{A}$ is a {\em retraction} of $j$ if the diagram \tri{a_{0},a_{1},a_{0},j,f,id} in $\cl{A}$ commutes. \end{defn} 
 
\begin{defn} \label{StrongDeformationRetractionCylinderDefinition} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$. 

An arrow \ar{a_{1},a_{0},f} of $\cl{A}$ is a {\em strong deformation retraction} of $j$ with respect to $\cylinder$ if $f$ is a retraction of $j$, and if there is a homotopy under $a_{0}$ from $jf$ to $id(a_{1})$ with respect to $\cylinder$ and $(j,j)$. \end{defn}

\begin{defn}  Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. An arrow \ar{a_{0},a_{1},j} of $\cl{A}$ {\em admits a strong deformation retraction} with respect to $\cylinder$ if there is an arrow \ar{a_{1},a_{0},f} of $\cl{A}$ which defines a strong deformation retraction of $j$ with respect to $\cylinder$. \end{defn}

\begin{defn} \label{StrongDeformationRetractionCoCylinderDefinition} Let $\cocylinder = \big( \cocyl, e_0, e_1, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$. 

An arrow \ar{a_{1},a_{0},f} of $\cl{A}$ is a {\em strong deformation retraction} of $j$ with respect to $\cocylinder$ if $f$ is a retraction of $j$, and if there is a homotopy over $a_{0}$ from $jf$ to $id(a_{1})$ with respect to $\cocylinder$ and $(f,f)$. \end{defn} 

\begin{rmk} Let $\cocylinder = \big( \cocyl, e_0, e_1, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$. 

An arrow \ar{a_{1},a_{0},f} of $\cl{A}$ is a {\em strong deformation retraction} of $j$ with respect to $\cocylinder$ if and only if $j^{op}$ defines a strong deformation retraction of $f^{op}$ with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}$, equipped with the contraction structure defined by $c^{op}$.  \end{rmk}

\begin{defn} Let $\cocylinder = \big( \cocyl, e_0, e_1, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. An arrow \ar{a_{0},a_{1},j} of $\cl{A}$ {\em admits a strong deformation retraction} with respect to $\cocylinder$ if there is an arrow \ar{a_{1},a_{0},f} of $\cl{A}$ which defines a strong deformation retraction of $j$ with respect to $\cocylinder$. \end{defn} 

\begin{prpn} \label{StrongDeformationRetractionCoCylinderCharacterisationProposition} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, and let $\cocylinder = \big( \cocyl, e_0, e_1, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$. Then an arrow \ar{a_{1},a_{0},f} of $\cl{A}$ is a strong deformation retraction of $j$ with respect to $\cocylinder$ if and only if $f$ is a retraction of $j$, and if there exists a homotopy over $a_{0}$ from $jf$ to $id(a_{1})$ with respect to $\cylinder$ and $(f,f)$. \end{prpn}

\begin{proof} Follows immediately from Corollary \ref{OverHomotopyCoCylinderGivesOverHomotopyCylinderCorollary}. \end{proof} 

\begin{lem} \label{HomotopyPreAndPostCompositionIsOverHomotopyLemma} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure. Suppose that we have four commutative diagrams in $\cl{A}$ as follows. \twoothertriangles{a_{0},a_{1},a,g_{0},j_{1},j_{0},a_{1},a_{2},a,f_{0},j_{2},j_{1}} \twoothertriangles{a_{1},a_{2},a,f_{1},j_{2},j_{1},a_{2},a_{3},a,g_{1},j_{3},j_{2}} %
%
Suppose also that we have a homotopy \ar{\cyl(a_{1}),a_{2},h} over $a$ from $f_{0}$ to $f_{1}$ with respect to $\cylinder$ and $(j_{1},j_{2})$. Then the arrow \ar[7]{\cyl(a_{0}),a_{3},g_{1} \circ h \circ \cyl(g_{0})} of $\cl{A}$ defines a homotopy over $a$ from $g_{1}f_{0}g_{0}$ to $g_{1}f_{1}g_{0}$ with respect to $\cylinder$ and $(j_{0},j_{3})$. \end{lem}

\begin{proof} That the arrow \ar[7]{\cyl(a_{0}),a_{3},g_{1} \circ h \circ \cyl(g_{0})} of $\cl{A}$  defines a homotopy from $g_{1}f_{0}g_{0}$ to $g_{1}f_{1}g_{0}$ with respect to $\cylinder$ is Lemma \ref{HomotopyPreAndPostCompositionLemma}. 

In addition since $h$ defines a homotopy over $a$ from $f_{0}$ to $f_{1}$ with respect to $\cylinder$ and $(j_{1},j_{2})$ the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{1}),a_{2},\cyl(a),a,h,j_{2},\cyl(j_{1}),p(a)} %
%
Since the diagram \triother{a_{0},a_{1},a,g_{0},j_{1},j_{0}} in $\cl{A}$ commutes we also have that the following diagram in $\cl{A}$ commutes. \tri[{4,3}]{\cyl(a_{0}),\cyl(a_{1}),\cyl(a),\cyl(g_{0}),\cyl(j_{1}),\cyl(j_{0})} %
%
Moreover we have that the following diagram in $\cl{A}$ commutes. \tri{a_{2},a_{3},a,g_{1},j_{3},j_{2}} %
%
Putting the last three observations together we have that the following diagram in $\cl{A}$ commutes. \sq[{7,3}]{\cyl(a_{0}),a_{3},\cyl(a),a,g_{1} \circ h \circ \cyl(g_{0}),j_{3},\cyl(j_{0}),p(a)}%

\end{proof} 

\begin{rmk} An analogous result holds for homotopies under an object with respect to $\cylinder$. We will not need this. \end{rmk}

\end{section}

introduction.tex
\begin{section}{Introduction} \label{IntroductionSection}

Model structures are highly cherished --- powerful but hard to find. In this work we build a model structure from the simple point of departure of a structured interval in a monoidal category --- more generally, a structured cylinder and a structured co-cylinder in a category.

\subsection*{Abstract homotopy theory} \label{PlaceWithinAbstractHomotopyTheorySubsection} \label{PlaceWithinAbstractHomotopySubsection} The first steps towards an abstract homotopy theory were for us taken in the early 1950s by Kan --- there is a considerable prehistory, in the work of Whitehead for example. Kan isolated in \cite{KanAbstractHomotopyTheoryII} the notion of a cylinder in a category and of homotopy with respect to a cylinder. 

Kan's immediate interest lay in the categories of cubical and simplicial sets. Whilst both categories admit a cylinder, the corresponding homotopies cannot be composed or reversed. What are now known as Kan complexes were introduced as a remedy. 

Abstract homotopy theory has subsequently evolved in two branches which have been explored rather independently. The first is the theory of model categories and the many variants and weakenings. The second is much less known. Its origins lie in the observation that the cylinder in topological spaces admits a much richer structure than the cylinder of the categories of simplicial or cubical sets. Two directions have been followed in capturing this richer structure of the topological cylinder in an abstract setting. 

\begin{figure}[h] \label{FigureAbstractHomotopyTheory}

\centering

\begin{tikzpicture} [>=stealth]

\node[align=center,text width=2cm,anchor=north] (kan) at (4,0) {Kan \\[0.5em]  1956 \\[0.5em] Cylinder in a category};

\node[align=center,text width=3cm,anchor=north,inner ysep=1em] (quillen) at (0,-5) {Quillen \\[0.5em]  1967 \\[0.5em] Model categories};

\node[align=center,text width=2cm,anchor=north,inner ysep=1em] (grandis) at (5,-5) {Grandis \\[0.5em]  1990s \\[0.5em] Structured cylinders};

\node[align=center,text width=2.5cm,anchor=north,inner ysep=1em] (kamps) at (9,-5) {Kamps \\[0.5em]  Late 1960s \\[0.5em] Kan conditions};

\draw[bend right=45] (kan.west) to (quillen.north);
\draw[bend left=45] (kan.east) to (7,-3);
\draw[bend right=45] (7,-3) to (grandis.north);
\draw[bend left=45] (7,-3) to (kamps.north);
\draw[<-,dashed] (quillen.east) to node[auto] {our work} ++ (2.1,0);
\end{tikzpicture}

\caption{Approaches to abstract homotopy theory}

\end{figure}

The first, begun by Kamps in the late 1960s in works such as \cite{KampsKanBedingungenUndAbstrakteHomotopietheorie}, explores the homotopy theory coming from a cylinder whose associated cubical set satisfies properties similar to those of a Kan complex. This is a global approach --- the axioms require consideration of all arrows in one's category. 

The second emerges out of work such as \cite{BrownHigginsOnTheAlgebraOfCubes} of Brown, Higgins, and others on cubical sets equipped with connections. It is of a structural and categorical nature involving a rainbow of natural transformations intertwining a cylinder and its corresponding double cylinder. This approach has been explored by Grandis in works such as \cite{GrandisCategoricallyAlgebraicFoundationsForHomotopicalAlgebra}. 

The book \cite{KampsPorterAbstractHomotopyAndSimpleHomotopyTheory} of Kamps and Porter gives a nice overview of both directions, with an emphasis on the first.

\subsection*{Outline}

The present work builds a bridge to model categories from the catgegorical approach to homotopy theory via structured intervals, cylinders, and co-cylinders. This is depicted by the dashed arrow in Figure 1. 

In a nutshell we proceed along the following lines. Each section begins with an introduction to its contents --- here we provide only a fil d'Ariane. 

The theorems towards which all of our work leads are to be found in \ref{ModelStructureSection}. Given a richly structured cylinder and co-cylinder in a category satisfying a certain strictness hypothesis we prove that homotopy equivalences, cofibrations, and fibrations --- defined from an abstract point of view in the same way as for topological spaces --- equip this category with a model structure. 

More precisely our theory typically gives rise to not one but two model structures. We introduce in \ref{CofibrationsAndFibrationsSection} a notion of a normally cloven cofibration and a normally cloven fibration, in addition to the afore-mentioned notions of cofibration and fibration. One of our model structures is defined by homotopy equivalences, cofibrations, and normally cloven fibrations. The other is defined by homotopy equivalences, normally cloven cofibrations, and fibrations. 

The structures on a cylinder and a co-cylinder with which we work, and our strictness hypothesis, are defined in \ref{StructuresUponACylinderOrACoCylinderSection}. Often in practise we construct a structured cylinder and co-cylinder by means of a structured interval in a monoidal category. This is discussed in \ref{StructuresUponAnIntervalSection}. 

We work throughout in a 2-categorical setting introduced in \ref{FormalCategoryTheoryPreliminariesSection}, which allows us to express a duality between homotopy theory with respect to a cylinder and homotopy theory with respect to a co-cylinder. This duality manifests itself throughout. In \ref{CylindricalAdjunctionsSection} we discuss a notion of adjunction between a cylinder and a co-cylinder, in the presence of which their respective homotopy theories intertwine.

For the remainder of this outline we will not indicate the particular structures involved at different points, or whether we are working with a cylinder, a co-cylinder, or both. These matters are carefully treated in the rest of the work.

In \ref{HomotopyAndRelativeHomotopySection} we define homotopies and relative homotopies with respect to a cylinder or co-cylinder. We demonstrate that we can compose, reverse, and construct identity homotopies and relative homotopies. 

We show in \ref{MappingCylindersAndMappingCoCylindersSection} that if our strictness hypothesis holds we obtain a mapping cylinder factorisation into a normally cloven cofibration followed by a strong deformation retraction, and that we obtain a mapping co-cylinder factorisation into a section of strong deformation retraction followed by a normally cloven fibration.

In \ref{DoldTheoremSection} we characterise trivial fibrations as strong deformation retractions and characterise trivial cofibrations as sections of strong deformation retractions. This is by means of an abstraction of Dold's theorem for topological spaces on homotopy equivalences under or over an object.

We moreover prove in \ref{FactorisationAxiomsSection} that if our strictness hypothesis holds, a trivial cofibration is exactly a section of a strong deformation retraction and dually a trivial fibration is exactly a strong deformation retraction. With our mapping cylinder and mapping co-cylinder factorisations to hand, we deduce that the factorisation axioms for our model structures hold.

In \ref{LiftingAxiomsSection} we prove that the canonical map from the mapping cylinder of a map to the cylinder at its target admits a strong deformation retraction. We prove that normally cloven fibrations have the right lifting property with respect to sections of strong deformation retractions. We deduce that normally cloven fibrations have the covering homotopy extension property --- introduced in \ref{CoveringHomotopyExtensionPropertySection} --- with respect to cofibrations. 

This allows us to prove that cofibrations have the left lifting property with respect to trivial normally cloven fibrations. We conclude that the lifting axioms hold for one of our model structures. The lifting axioms for the other model structure follow by duality.  

\subsection*{Folk model structure} \label{ExamplesFutureDirectionsSubsection} The author initially thought this work as a step towards the construction of a theory of $n$-groupoids satisfying a strong interpretation of the homotopy hypothesis. The original aim was to find a construction of the folk model structure on categories and groupoids which could generalise to a folk model structure on our proposed category of $n$-groupoids. 

These ideas on $n$-groupoids are currently being explored in joint work with Marius Thaule. We demonstrate in \ref{ExampleSection} that our work indeed gives a new construction of the folk model structure on categories and groupoids. 

\subsection*{Further examples} Our work gives rise to many other model structures. We discuss three. 

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(1)] Let $\mathsf{Ch}(\cl{A})$ denote the category of chain complexes in an additive category $\cl{A}$ with finite limits and colimits. The cylinder and co-cylinder functors \ar{\mathsf{Ch}(\cl{A}),\mathsf{Ch}(\cl{A})} of homological algebra can be equipped with all the structures of \ref{StructuresUponACylinderOrACoCylinderSection}. We refer the reader for example to \S{4.4.2} of the book \cite{GrandisDirectedAlgebraicTopology} of Grandis. Our strictness hypothesis is satisfied. 

Our work thus gives a model structure on $\mathsf{Ch}(\cl{A})$ whose weak equivalences are chain homotopy equivalences. This model structure was constructed by Golasi{\'n}ski and Gromadzki in \cite{GolasinskiGromadzkiTheHomotopyCategoryOfChainComplexesIsAChainComplex}. Appealing to a characterisation due to Kamps in \cite{KampsNoteOnNormalSequencesOfChainComplexes} of the fibrations and cofibrations, this construction is quite different to ours.

\item[(2)] Let $\mathsf{Kan}_{\Delta}$ denote the category of algebraic Kan complexes introduced by Nikolaus in \cite{NikolausAlgebraicModelsForHigherCategories}. The objects of $\mathsf{Kan}_{\Delta}$ are Kan complexes with a chosen filling for every horn. The arrows of $\mathsf{Kan}_{\Delta}$ are morphisms of simplicial sets which respect the chosen horn fillings.

Our work gives a model structure on $\mathsf{Kan}_{\Delta}$ which we think of as akin to the model structure on topological spaces constructed by Str{\o}m in \cite{StromTheHomotopyCategoryIsAHomotopyCategory}. A different model structure on $\mathsf{Kan}_{\Delta}$ was constructed by Nikolaus in \cite{NikolausAlgebraicModelsForHigherCategories}, which we think of as akin to the Serre model structure on topological spaces.

The author conjectures that the identity functor defines a Quillen equivalence between the two model structures on $\mathsf{Kan}_{\Delta}$.

\item[(3)] Let $\mathsf{Top}$ denote the category of all topological spaces. The unit interval is exponentiable with respect to the cartesian monoidal structure on $\mathsf{Top}$ and can be equipped with all of the structures of \ref{StructuresUponAnIntervalSection}. 

Our strictness hypothesis does not however hold.  Thus our work does not immediately give rise to a model structure on $\mathsf{Top}$. 

Nevertheless homotopy equivalences in $\mathsf{Top}$ can be understood as homotopy equivalences with respect to the Moore co-cylinder, which to a topological space $X$ associates the set of pairs $(t,f)$ of a real number $t \in [0,\infty)$ and a map \ar{{[0,\infty)},X,f} such that $f(x) =t$ for all $x \geq t$, where this set is equipped with the subspace topology with respect to $[0,\infty) \times X^{[0,\infty)}$. Our strictness hypothesis does hold for the Moore co-cylinder.

The Moore co-cylinder does not however admit connection structures. There are two ways to get around this. Firstly it is possible to generalise our work slightly to allow for double cylinders and double co-cylinders which are not necessarily obtained by applying the cylinder or co-cylinder functor twice. We can then take our Moore double co-cylinder to consist of Moore rectangles as considered in the paper \cite{BrownMooreHyperrectanglesOnASpaceFormAStrictCubicalOmegaCategory} of Brown. Secondly it should be possible to replace the connection structures in our work by the `strengths' of the paper \cite{VanDenBergGarnerTopologicalAndSimplicialModelsOfIdentityTypes} of van den Berg and Garner.

Following either route, our work with respect to both the Moore co-cylinder and the usual cylinder and co-cylinder in $\mathsf{Top}$ gives rise to a model structure on $\mathsf{Top}$. This is the model structure constructed by Str{\o}m in \cite{StromTheHomotopyCategoryIsAHomotopyCategory}.

Around Easter 2012 Tobias Barthel and Bill Richter suggested a construction of the Moore co-cylinder to the author which could be carried out in a quite general setting. Thus our side-stepping of the failure of the strictness hypothesis to hold in $\mathsf{Top}$ may be able to be carried out more widely.   

The significance of the Moore co-cylinder for the construction of the Str{\o}m model structure on $\mathsf{Top}$ is also explored in the paper \cite{BarthelRiehlOnTheConstructionOfFunctorialFactorizationsForModelCategories} of Barthel and Riehl.

\end{itemize}

\subsection*{A further direction} The model structures to which our work gives rise have very strong properties. Every object is both fibrant and cofibrant, yet the raison d'{\^e}tre of the theory of model categories is to work with abstract homotopy theories for which this is not the case! We might therefore ask how our work fits into the world of model categories as a whole.

We expect that our work give a new approach to the construction of model structures quite broadly. In particular the weak equivalences need not all be homotopy equivalences with respect to a cylinder or a co-cylinder. 

The three fundamental examples of the model structure of simplicial sets, the model structures on chain complexes with quasi-isomorphisms as weak equivalences, and the Serre model structure on topological spaces should all, we anticipate, be able to be constructed in the manner we have in mind. The folk model structure on 2-categories would be another example.   

Let us sketch the idea. Roughly speaking we turn the usual approach upside down.

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(1)] We begin with what will afterwards become a subcategory of fibrant and cofibrant objects. In the example of simplicial sets we work with the category of algebraic Kan complexes. 

We require that weak equivalences between the objects of this subcategory can be viewed as homotopy equivalences with respect to a cylinder or co-cylinder. We equip this subcategory with a model structure. 

\item[(2)] We explicitly construct what will afterwards become a fibrant or cofibrant replacement functor. If this fibrant and cofibrant replacement functor has strong properties, we can construct a model structure on our original category by means of the model structure of (1). 

\end{itemize}

This is being explored jointly with Dimitri Ara. 

\subsection*{Acknowledgements} I thank Kari, the wonderful funny bee to whom this work is dedicated, for everything. It is impossible to express how much your lifting of my spirit --- cakes of course playing a crucial role! --- has contributed to this work.

I thank Mum and Dad for everything over the years. I cannot express my gratitude. I would specifically like to thank you for your unwavering encouragement and belief throughout the writing of this work --- I am very lucky to have been able to phone up babbling incomprehensibly about mathematics!

I have the grandest pair of brothers in the world, Chris and Andrew. I thank you both for all the many laughter filled times --- for always being there for me.

I thank Babcia, Dziadziu, Grandma, and, beyond the grave, Grandad for everything they have done for me. I feel deeply that this work owes a great deal to your hard work throughout your lives.

To Alex, Tom, Richy, Shaun, Sian, Phil, and Katherine --- I count myself incredibly lucky to have got to know you all, and that we have such a close knit friendship. You all bring great joy to my life!

To Merav, Daniel, Nimrod, Henning, and Anne -- wonderful dinners, black and white films, and the fun brought into our lives by little Nimrod gave Kari and me great happiness in our time in Oxford, and I will always look back with an ineffable fondness. 

I thank Rapha{\"e}l Rouquier greatly for his supervision over my four years in Oxford. That every turn in my path was greeted with enthusiasm, wisdom, and sincerity was a great encouragement to me.

I thank Tobias Barthel for many enjoyable discussions. I treasure our correspondence, and our sharing of mathematical dreams!   

I thank Nicholas Cooney for many pleasant conversations over coffee and in the pub, and for the great help with the submission of this work as my thesis! 

I thank Ulrike Tillmann for her advice and understanding at a crucial point in my time as a graduate student, and for her continuing encouragement.

I thank Michael Collins for his support and encouragement in the months before and during my time at Oxford, and for his insight in suggesting Rapha{\"e}l as a supervisor.  

To all at Oxenford Cricket Club --- especially my 2nd XI teammates --- for three years of immensely enjoyable summer Saturdays.

I thank Tim Porter and Kobi Kremnizer for their examination of my thesis, for their encouragement, and for their very helpful comments and suggestions. 

I thank Nils Baas greatly for his kindness in inviting us and helping us to settle down in Trondheim. I thank Nils and Andrew Stacey for their support in my finding a position at NTNU.

I thank Bj{\o}rn Ian Dundas for arranging the funding of my employment at NTNU as a guest researcher from January to March 2012. 

Finally to all our friends and family in Sandefjord, T{\o}nsberg, Larvik, and Trondheim for their help, well wishes, and understanding since we moved to Norway in August 2011 --- and of course for the dinners, waffles, and cakes\ldots a recurring theme!

\end{section}

lifting_axioms.tex
\begin{section}{Lifting axioms} \label{LiftingAxiomsSection}

Let $\cylinder$ be a cylinder in a formal category $\cl{A}$ and let $\cocylinder$ be a co-cylinder in $\cl{A}$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$. We prove that if an arrow $j$ of $\cl{A}$ is a cofibration then the canonical arrow $m^{\cylinder}_{j}$ of $\cl{A}$ defined in \ref{MappingCylindersAndMappingCoCylindersSection} admits a strong deformation retraction. For this we assume for the first time that our cylinder is equipped with upper and lower right connection structures, which we require to be compatible with subdivision. 

For topological spaces the fact that $m^{\cylinder}_{j}$ admits a strong deformation retraction if $j$ is a cofibration is due to Str{\o}m. It is proven in \S{2} of the paper \cite{StromNoteOnCofibrations}. Str{\o}m's proof is however quite different to ours. It relies on the fact that the homotopy theory of topological spaces is defined with respect to a cartesian monoidal structure.

Next we prove that a normally cloven fibration has the right lifting property with respect to arrows admitting a strong deformation retraction. We deduce that normally cloven fibrations have the covering homotopy extension property with respect to cofibrations. 

In a similar way we prove that trivial normally cloven fibrations have the right lifting property with respect to cofibrations. Dualising we deduce that fibrations have the right lifting property with respect to trivial normally cloven cofibrations and that trivial fibrations have the right lifting property with respect to normally cloven cofibrations. 

\begin{assum} Let $\cl{C}$ be a 2-category with a final object such that pushouts and pullbacks of 2-arrows of $\cl{C}$ give rise to pushouts and pullbacks in formal categories in the sense of Definition \ref{PushoutsPullbacks2ArrowsArePushoutsPullbacksInFormalCategoriesTerminology}. Let $\cl{A}$ be an object of $\cl{C}$. As before we view $\cl{A}$ as a formal category, writing of objects and arrows of $\cl{A}$. \end{assum} 

\begin{prpn} \label{RetractionIsStrongDeformationRetractionProposition} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{lr}, \Gamma_{ur} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $v$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ compatible with $p$, a lower right connection structure $\Gamma_{lr}$, and an upper right connection structure $\Gamma_{ur}$. Suppose that $\Gamma_{lr}$ and $\Gamma_{ur}$ are compatible with the subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$, and that $\cyl$ preserves mapping cylinders with respect to $\cylinder$. 

Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$ which is a cofibration with respect to $\cylinder$, and suppose that $\big( a^{\cylinder}_{j}, d^{0}_{j}, d^{1}_{j} \big)$ defines a mapping cylinder of $j$ with respect to $\cylinder$. Let \ar{a^{\cylinder}_{j},\cyl(a_{1}),m^{\cylinder}_{j}} denote the corresponding canonical arrow of $\cl{A}$ of Notation \ref{CanonicalArrowFromMappingCylinderToCylinderDefinition}. Let \ar{\cyl(a_{1}),a^{\cylinder}_{j},r^{\cylinder}_{j}} be the arrow of $\cl{A}$ of Proposition \ref{RetractionToMappingCylinderProposition}. %
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Then $r^{\cylinder}_{j}$ is a strong deformation retraction of $m^{\cylinder}_{j}$ with respect to $\cylinder$. \end{prpn}

\begin{proof} By Proposition \ref{RetractionToMappingCylinderProposition} we have that $r^{\cylinder}_{j}$ is a retraction of $m^{\cylinder}_{j}$. It remains to prove that there is a homotopy \ar{\cyl^{2}(a_{1}),\cyl(a_{1}),\sigma} from $m^{\cylinder}_{j} \circ r^{\cylinder}_{j}$ to $id\big( \cyl(a_{1}) \big)$ such that the following diagram in $\cl{A}$ commutes. \sq{\cyl\big(a^{\cylinder}_{j}\big),a^{\cylinder}_{j},\cyl^{2}(a_{1}),\cyl(a_{1}),p\big(a^{\cylinder}_{j}\big),m^{\cylinder}_{j},\cyl\big(m^{\cylinder}_{j}\big),\sigma} 

Let us first construct $\sigma$. The following diagram in $\cl{A}$ commutes. \squarewithdiagonal{a_{1},\cyl(a_{1}),\cyl(a_{1}),a^{\cylinder}_{j},i_{0}(a_{1}),r^{\cylinder}_{j},i_{0}(a_{1}),m^{\cylinder}_{j},d^{1}_{j}} %
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By definition of $\Gamma_{ur}$ as an upper right connection structure, we also have that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),a_{1},\cyl(a_{1}),i_{1}\big(\cyl(a_{1})\big),\Gamma_{ur}(a_{1}),p(a_{1}),i_{0}(a_{1})} %
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Putting the last two observations together, we have that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),a_{1},\cyl(a_{1}),i_{1}\big(\cyl(a_{1})\big),m^{\cylinder}_{j} \circ r^{\cylinder}_{j} \circ \Gamma_{ur}(a_{1}),p(a_{1}),i_{0}(a_{1})} %
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By definition of $\Gamma_{lr}$ as a lower right connection structure the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),a_{1},\cyl(a_{1}),i_{0}\big(\cyl(a_{1})\big),\Gamma_{ur}(a_{1}),p(a_{1}),i_{0}(a_{1})} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \sq[{9,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),\cyl^{2}(a_{1}),\cyl(a_{1}),i_{0}\big(\cyl(a_{1})\big),\Gamma_{lr}(a_{1}),i_{1}\big(\cyl(a_{1})\big),m^{\cylinder}_{j} \circ r^{\cylinder}_{j} \circ \Gamma_{ur}(a_{1})} %
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We define $\sigma$ to be the arrow \ar[15]{\cyl^{2}(a_{1}),\cyl(a_{1}),\big( m^{\cylinder}_{j} \circ r^{\cylinder}_{j} \circ \Gamma_{ur}(a_{1}) \big) + \Gamma_{lr}(a_{1})} of $\cl{A}$. 

Let us first prove that the following diagram in $\cl{A}$ commutes. \tri[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}), \cyl(a_{1}),i_{0}\big(\cyl(a_{1})\big),\sigma,m^{\cylinder}_{j} \circ r^{\cylinder}_{j}} %
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By definition of $\Gamma_{ur}$ as an upper right connection structure the following diagram in $\cl{A}$ commutes. \tri[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),\cyl(a_{1}),i_{0}\big(\cyl(a_{1})\big),\Gamma_{ur}(a_{1}),id} %
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Thus the following diagram in $\cl{A}$ commutes. \tri[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),\cyl(a_{1}),i_{0}\big(\cyl(a_{1})\big),m^{\cylinder}_{j} \circ r^{\cylinder}_{j} \circ \Gamma_{ur}(a_{1}),m^{\cylinder}_{j} \circ r^{\cylinder}_{j}} %
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We also have that the following diagram in $\cl{A}$ commutes. \sq[{9,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),\cyl^{2}(a_{1}),\cyl(a_{1}),i_{0}\big(\cyl(a_{1})\big),\sigma,i_{0}\big(\cyl(a_{1})\big),m^{\cylinder}_{j} \circ r^{\cylinder}_{j} \circ \Gamma_{ur}(a_{1})} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes, as required. \tri[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}), \cyl(a_{1}),i_{0}\big(\cyl(a_{1})\big),\sigma,m^{\cylinder}_{j} \circ r^{\cylinder}_{j}} 

Next let us prove that the following diagram in $\cl{A}$ commutes. \tri[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),\cyl(a_{1}),i_{1}\big(\cyl(a_{1})\big),\sigma,id} %
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By definition of $\Gamma_{lr}$ as a lower right connection structure we have that the following diagram in $\cl{A}$ commutes. \tri[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),\cyl(a_{1}),i_{1}\big(\cyl(a_{1})\big),\Gamma_{lr}(a_{1}),id} %
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We also have that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),\cyl^{2}(a_{1}),\cyl(a_{1}),i_{1}\big(\cyl(a_{1})\big),\sigma,i_{1}\big(\cyl(a_{1})\big),\Gamma_{lr}(a_{1})} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes, as required. \tri[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),\cyl(a_{1}),i_{1}\big(\cyl(a_{1})\big),\Gamma_{lr}(a_{1}),id} 

Let us now prove that the following diagram in $\cl{A}$ commutes. \sq{\cyl\big(a^{\cylinder}_{j}\big),a^{\cylinder}_{j},\cyl^{2}(a_{1}),\cyl(a_{1}),p\big(a^{\cylinder}_{j}\big),m^{\cylinder}_{j},\cyl\big(m^{\cylinder}_{j}\big),\sigma} %
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Let \ar{\subdiv\big(\cyl(a_{1})\big),\cyl(a_{1}),u} denote the canonical arrow of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout[{5,3,0}]{\cyl(a_{1}),\cyl^{2}(a_{1}),\cyl^{2}(a_{1}),\subdiv\big(\cyl(a_{1})\big),\cyl(a_{1}),i_{0}\big(\cyl(a_{1})\big),r_{0}\big(\cyl(a_{1})\big),i_{1}\big(\cyl(a_{1})\big),r_{1}\big(\cyl(a_{1})\big),\Gamma_{lr}(a_{1}),m^{\cylinder}_{j} \circ r^{\cylinder}_{j} \circ \Gamma_{ur}(a_{1}),u} %
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Let \ar{\subdiv,\id,\overline{p}} denote the canonical 2-arrow of $\cl{C}$ of Definition \ref{SubdivisionCompatibleWithContractionDefinition}. We claim that the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{1}),\subdiv(a_{1}),a_{1},\cyl(a_{1}),s(a_{1}),u \circ \subdiv\big(i_{0}(a_{1})\big),p(a_{1}),i_{0}(a_{1})} 

By definition of $\Gamma_{lr}(a_{1})$ as a lower right connection structure the following diagram in $\cl{A}$ commutes. \sq[{5,3}] {\cyl(a_{1}),\cyl^{2}(a_{1}),a_{1},\cyl(a_{1}),\cyl\big(i_{0}(a_{1})\big),\Gamma_{ur}(a_{1}),p(a_{1}),i_{0}(a_{1})} %
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We also have that the following diagram in $\cl{A}$ commutes. \squareabovetriangle[{5,3,0}]{\cyl(a_{1}),\subdiv(a_{1}),\cyl^{2}(a_{1}),\subdiv\big(\cyl(a_{1})\big),\cyl(a_{1}),r_{0}(a_{1}),\subdiv\big(i_{0}(a_{1})\big),\cyl\big(i_{0}(a_{1})\big),r_{0}\big(\cyl(a_{1})\big),u,\Gamma_{lr}(a_{1})} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{1}),\subdiv(a_{1}),a_{1},\cyl(a_{1}),r_{0}(a_{1}),u \circ \subdiv\big(i_{0}(a_{1})\big),p(a_{1}),i_{0}(a_{1})}
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By definition of $\Gamma_{ur}(a_{1})$ as an upper right connection structure the following diagram in $\cl{A}$ commutes. \sq[{5,3}] {\cyl(a_{1}),\cyl^{2}(a_{1}),a_{1},\cyl(a_{1}),\cyl\big(i_{0}(a_{1})\big),\Gamma_{ur}(a_{1}),p(a_{1}),i_{0}(a_{1})} %
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Thus we have that the following diagram in $\cl{A}$ commutes. \trapeziumstwo[{3,3,2,0}]{\cyl(a_{1}),\subdiv(a_{1}),a_{1},\cyl^{2}(a_{1}),\subdiv\big(\cyl(a_{1})\big),\cyl(a_{1}),\cyl(a_{1}),r_{1}(a_{1}),\subdiv\big(i_{0}(a_{1})\big),u,p(a_{1}),i_{0}(a_{1}),m^{\cylinder}_{j} \circ r^{\cylinder}_{j},\cyl\big(i_{0}(a_{1})\big),r_{1}\big(\cyl(a_{1})\big),\Gamma_{ur}(a_{1})} %
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Earlier in the proof we observed that the following diagram in $\cl{A}$ commutes. \tri{a_{1},\cyl(a_{1}),\cyl(a_{1}),i_{0}(a_{1}),m^{\cylinder}_{j} \circ r^{\cylinder}_{j},i_{0}(a_{1})} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{1}),\subdiv(a_{1}),a_{1},\cyl(a_{1}),r_{1}(a_{1}),u \circ \subdiv\big(i_{0}(a_{1})\big),p(a_{1}),i_{0}(a_{1})} %
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We have now shown that the following diagram in $\cl{A}$ commutes. \pushout{a_{1},\cyl(a_{1}),\cyl(a_{1}),\subdiv(a_{1}),\cyl(a_{1}),i_{0}(a_{1}),r_{0}(a_{1}),i_{1}(a_{1}),r_{1}(a_{1}),i_{0}(a_{1}) \circ p(a_{1}),i_{0}(a_{1}) \circ p(a_{1}), u \circ \subdiv\big(i_{0}(a_{1})\big)} %
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By definition of $\overline{p}$ the following diagram in $\cl{A}$ commutes. \pushout{a_{1},\cyl(a_{1}),\cyl(a_{1}),\subdiv(a_{1}),a_{1},i_{0}(a_{1}),r_{0}(a_{1}),i_{1}(a_{1}),r_{1}(a_{1}),p(a_{1}), p(a_{1}), \overline{p}(a_{1})} %
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Thus the following diagram in $\cl{A}$ commutes. \pushout{a_{1},\cyl(a_{1}),\cyl(a_{1}),\subdiv(a_{1}),\cyl(a_{1}),i_{0}(a_{1}),r_{0}(a_{1}),i_{1}(a_{1}),r_{1}(a_{1}),i_{0}(a_{1}) \circ p(a_{1}),i_{0}(a_{1}) \circ p(a_{1}), i_{0}(a_{1}) \circ \overline{p}(a_{1})} %
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Appealing to the universal property of $\subdiv(a_{1})$ we deduce that the following diagram in $\cl{A}$ commutes.\sq[{5,3}]{\subdiv(a_{1}),\subdiv\big(\cyl(a_{1})\big),a_{1},\cyl(a_{1}),\subdiv\big(i_{0}(a_{1})\big),u,\overline{p}(a_{1}),i_{0}(a_{1})} %
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Since the subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ is compatible with $p$ we also have that the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{1}),\subdiv(a_{1}),a_{1},s(a_{1}),\overline{p}(a_{1}),p(a_{1})} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes, completing the proof of the claim. \squarewithdiagonaltwo[{4,3.5}]{\cyl(a_{1}),\subdiv(a_{1}),a_{1},\cyl(a_{1}),s(a_{1}),u \circ \subdiv\big(i_{0}(a_{1})\big),p(a_{1}),i_{0}(a_{1}),\overline{p}(a_{1})} 

We also claim that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl^{2}(a_{0}),\subdiv\big(\cyl(a_{0})\big),\cyl(a_{0}),\cyl(a_{1}),s\big(\cyl(a_{0})\big),u \circ \subdiv\big(\cyl(j)\big),p\big(\cyl(a_{0})\big),\cyl(j)} %
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We have that the following diagram in $\cl{A}$ commutes. \trapeziumstwo[{3,3,2,0}]{\cyl^{2}(a_{0}),\subdiv\big(\cyl(a_{0})\big),\cyl(a_{0}),\cyl^{2}(a_{1}),\subdiv\big(\cyl(a_{1})\big),\cyl(a_{1}),\cyl(a_{1}),r_{1}\big(\cyl(a_{0})\big),\subdiv\big(\cyl(j)\big),u,\Gamma_{ur}(a_{0}),\cyl(j),m^{\cylinder}_{j} \circ r^{\cylinder}_{j},\cyl^{2}(j),r_{1}\big(\cyl(a_{1})\big),\Gamma_{ur}(a_{1})} %
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The following diagram in $\cl{A}$ also commutes. \squarewithdiagonal{\cyl(a_{0}),\cyl(a_{1}),\cyl(a_{1}),a^{\cylinder}_{j},\cyl(j),r^{\cylinder}_{j},\cyl(j),m^{\cylinder}_{j},d^{0}_{j}} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl^{2}(a_{0}),\subdiv\big(\cyl(a_{0})\big),\cyl(a_{0}),\cyl(a_{1}),r_{1}\big(\cyl(a_{0})\big),u \circ \subdiv\big(\cyl(j)\big),\Gamma_{ur}(a_{0}),\cyl(j)} %
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The following diagram in $\cl{A}$ commutes. \squareabovetriangle[{5,3,0}]{\cyl^{2}(a_{0}),\subdiv\big(\cyl(a_{0})\big),\cyl^{2}(a_{1}),\subdiv\big(\cyl(a_{1})\big),\cyl(a_{1}),r_{0}\big(\cyl(a_{0})\big),\subdiv\big(\cyl(j)\big),\cyl^{2}(j),r_{0}\big(\cyl(a_{1})\big),u,\Gamma_{lr}(a_{1})} %
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The following diagram in $\cl{A}$ also commutes. \sq{\cyl^{2}(a_{0}),\cyl^{2}(a_{1}),\cyl(a_{0}),\cyl(a_{1}),\cyl^{2}(j),\Gamma_{lr}(a_{1}),\Gamma_{lr}(a_{0}),\cyl(j)} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl^{2}(a_{0}),\subdiv\big(\cyl(a_{0})\big),\cyl(a_{0}),\cyl(a_{1}),r_{0}\big(\cyl(a_{0})\big),u \circ \subdiv\big(\cyl(j)\big),\Gamma_{lr}(a_{0}),\cyl(j)} %
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We have now shown that the following diagram in $\cl{A}$ commutes. \pushout[{5,3,0}]{\cyl(a_{0}),\cyl^{2}(a_{0}),\cyl^{2}(a_{0}),\subdiv\big(\cyl(a_{0})\big),\cyl(a_{1}),i_{0}\big(\cyl(a_{0})\big),r_{0}\big(\cyl(a_{0})\big),i_{1}\big(\cyl(a_{0})\big),r_{1}\big(\cyl(a_{0})\big),\cyl(j) \circ \Gamma_{lr}(a_{0}),\cyl(j) \circ \Gamma_{ur}(a_{0}), u \circ \subdiv\big(\cyl(j)\big)} %
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Let \ar{\subdiv \circ \cyl,\cyl,x} denote the canonical 2-arrow of $\cl{C}$ of Definition \ref{RightConnectionsCylinderCompatibilityDefinition}. By definition of $x$ the following diagram in $\cl{A}$ commutes. \pushout[{5,3,0}]{\cyl(a_{0}),\cyl^{2}(a_{0}),\cyl^{2}(a_{0}),\subdiv\big(\cyl(a_{0})\big),\cyl(a_{0}),i_{0}\big(\cyl(a_{0})\big),r_{0}\big(\cyl(a_{0})\big),i_{1}\big(\cyl(a_{0})\big),r_{1}\big(\cyl(a_{0})\big),\Gamma_{lr}(a_{0}),\Gamma_{ur}(a_{0}),x(a_{0})} %
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Thus the following diagram in $\cl{A}$ commutes. \pushout[{5,3,0}]{\cyl(a_{0}),\cyl^{2}(a_{0}),\cyl^{2}(a_{0}),\subdiv\big(\cyl(a_{0})\big),\cyl(a_{1}),i_{0}\big(\cyl(a_{0})\big),r_{0}\big(\cyl(a_{0})\big),i_{1}\big(\cyl(a_{0})\big),r_{1}\big(\cyl(a_{0})\big),\cyl(j) \circ \Gamma_{lr}(a_{0}),\cyl(j) \circ \Gamma_{ur}(a_{0}),\cyl(j) \circ x(a_{0})} %
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Appealing to the universal property of $\subdiv\big(\cyl(a_{0})\big)$ we deduce that the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{\subdiv\big(\cyl(a_{0})\big),\subdiv\big(\cyl(a_{1})\big),\cyl(a_{0}),\cyl(a_{1}),\subdiv\big(\cyl(j)\big),u,x(a_{0}),\cyl(j)} % 
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Since $\Gamma_{lr}$ and $\Gamma_{ur}$ are compatible with the subdivision structure $\big( \subdiv,r_{0}, r_{1}, s \big)$ we also have that the following diagram in $\cl{A}$ commutes. \tri[{5,4}]{\cyl^{2}(a_{0}),\subdiv\big(\cyl(a_{0})\big),\cyl(a_{0}),s\big(\cyl(a_{0})\big),x(a_{0}),p\big(\cyl(a_{0})\big)} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes, completing the proof of the claim. \squarewithdiagonaltwo[{5,3}]{\cyl^{2}(a_{0}),\subdiv\big(\cyl(a_{0})\big),\cyl(a_{0}),\cyl(a_{1}),s\big(\cyl(a_{0})\big),u \circ \subdiv(\cyl(j)\big),p\big(\cyl(a_{0})\big), \cyl(j), x(a_{0})} 

Next we claim that the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{\cyl^{2}(a_{0}),\cyl\big(a^{\cylinder}_{j}\big),\cyl(a_{0}),\cyl(a_{1}),\cyl(d^{0}_{j}),\sigma \circ \cyl\big(m^{\cylinder}_{j}\big),p\big(\cyl(a_{0})\big),\cyl(j)} %
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We have that the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{0}),a^{\cylinder}_{j},\cyl(a_{1}),d^{0}_{j},m^{\cylinder}_{j},\cyl(j)} %
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Thus the following diagram in $\cl{A}$ commutes. \tri[{4,3}]{\cyl^{2}(a_{0}),\cyl\big(a^{\cylinder}_{j}\big),\cyl^{2}(a_{1}),\cyl(d^{0}_{j}),\cyl\big(m^{\cylinder}_{j}\big),\cyl^{2}(j)} %
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The following diagram in $\cl{A}$ also commutes. \sq[{5,3}]{\cyl^{2}(a_{0}),\cyl^{2}(a_{1}),\subdiv\big(\cyl(a_{0})\big),\subdiv\big(\cyl(a_{1})\big),\cyl^{2}(j),s\big(\cyl(a_{1})\big),s\big(\cyl(a_{0})\big),\subdiv\big(\cyl(j)\big)} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl^{2}(a_{0}),\cyl\big(a^{\cylinder}_{j}\big),\subdiv\big(\cyl(a_{0})\big),\subdiv\big(\cyl(a_{1})\big),\cyl(d^{0}_{j}),s\big(\cyl(a_{1})\big) \circ \cyl\big(m^{\cylinder}_{j}\big),s\big(\cyl(a_{0})\big),\subdiv\big(\cyl(j)\big)} %
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Earlier in the proof we established that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl^{2}(a_{0}),\subdiv\big(\cyl(a_{0})\big),\cyl(a_{0}),\cyl(a_{1}),s\big(\cyl(a_{0})\big),u \circ \subdiv\big(\cyl(j)\big),p\big(\cyl(a_{0})\big),\cyl(j)} %
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In addition by definition of $\sigma$ the following diagram in $\cl{A}$ commutes. \tri[{5,3}]{\cyl^{2}(a_{1}),\subdiv\big(\cyl(a_{1})\big),\cyl(a_{1}),s\big(\cyl(a_{1})\big),u,\sigma} %
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Putting the last three observations together, we have that the following diagram in $\cl{A}$ commutes, completing our proof of the claim. \sq[{4,3}]{\cyl^{2}(a_{0}),\cyl\big(a^{\cylinder}_{j}\big),\cyl(a_{0}),\cyl(a_{1}),\cyl(d^{0}_{j}),\sigma \circ \cyl\big(m^{\cylinder}_{j}\big),p\big(\cyl(a_{0})\big),\cyl(j)} %
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Next we claim that the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{\cyl(a_{1}),\cyl\big(a^{\cylinder}_{j}\big),a_{1},\cyl(a_{1}),\cyl(d^{1}_{j}),\sigma \circ \cyl\big(m^{\cylinder}_{j}\big),p(a_{1}),i_{0}(a_{1})} %
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We have that the following diagram in $\cl{A}$ commutes. \tri{a_{1},a^{\cylinder}_{j},\cyl(a_{1}),d^{1}_{j},m^{\cylinder}_{j},i_{0}(a_{1})} %
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Thus the following diagram in $\cl{A}$ commutes. \tri[{4,3}]{\cyl(a_{1}),\cyl\big(a^{\cylinder}_{j}\big),\cyl^{2}(a_{1}),\cyl(d^{1}_{j}),\cyl\big(m^{\cylinder}_{j}\big),\cyl\big(i_{0}(a_{1})\big)} %
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The following diagram in $\cl{A}$ also commutes. \sq[{5,3}]{\cyl(a_{1}),\cyl^{2}(a_{1}),\subdiv(a_{1}),\subdiv\big(\cyl(a_{1})\big),\cyl\big(i_{0}(a_{1})\big),s\big(\cyl(a_{1})\big),s(a_{1}),\subdiv\big(i_{0}(a_{1})\big)} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{\cyl(a_{1}),\cyl\big(a^{\cylinder}_{j}\big),a_{1},\cyl(a_{1}),\cyl(d^{1}_{j}),s\big(\cyl(a_{1})\big) \circ \cyl\big(m^{\cylinder}_{j}\big),s(a_{1}),\subdiv\big(i_{0}(a_{1})\big)} %
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Earlier in the proof we established that the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{1}),\subdiv(a_{1}),a_{1},\cyl(a_{1}),s(a_{1}),u \circ \subdiv\big(i_{0}(a_{1})\big),p(a_{1}),i_{0}(a_{1})} %
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In addition by definition of $\sigma$ the following diagram in $\cl{A}$ commutes. \tri[{5,3}]{\cyl^{2}(a_{1}),\subdiv\big(\cyl(a_{1})\big),\cyl(a_{1}),s\big(\cyl(a_{1})\big),u,\sigma} %
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Putting the last three observations together we have that the following diagram in $\cl{A}$ commutes, completing the proof of the claim. \sq[{4,3}]{\cyl(a_{1}),\cyl\big(a^{\cylinder}_{j}\big),a_{1},\cyl(a_{1}),\cyl(d^{1}_{j}),\sigma \circ \cyl\big(m^{\cylinder}_{j}\big),p(a_{1}),i_{0}(a_{1})} %
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We have now shown that the following diagram in $\cl{A}$ commutes. \pushout[{5,3,-2}]{\cyl(a_{0}),\cyl^{2}(a_{0}),\cyl(a_{1}),\cyl(a^{\cylinder}_{j}),\cyl(a_{1}),\cyl\big(i_{0}(a_{0})\big),\cyl(d^{0}_{j}),\cyl(j),\cyl(d^{1}_{j}),\cyl(j) \circ p\big(\cyl(a_{0})\big),i_{0}(a_{1}) \circ p(a_{1}), \sigma \circ \cyl\big(m^{\cylinder}_{j}\big)} %
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The following diagram in $\cl{A}$ commutes. \squareabovetriangle{\cyl^{2}(a_{0}),\cyl\big(a^{\cylinder}_{j}\big),\cyl(a_{0}),a^{\cylinder}_{j},\cyl(a_{1}),\cyl(d^{0}_{j}),p\big(a^{\cylinder}_{j}\big),p\big(\cyl(a_{0})\big),d^{0}_{j},m^{\cylinder}_{j},\cyl(j)} %
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The following diagram in $\cl{A}$ also commutes. \squareabovetriangle{\cyl(a_{1}),\cyl\big(a^{\cylinder}_{j}\big),a_{1},a^{\cylinder}_{j},\cyl(a_{1}),\cyl(d^{1}_{j}),p\big(a^{\cylinder}_{j}\big),p(a_{1}),d^{1}_{j},m^{\cylinder}_{j},i_{0}(a_{1})} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \pushout[{5,3,-2}]{\cyl(a_{0}),\cyl^{2}(a_{0}),\cyl(a_{1}),\cyl(a^{\cylinder}_{j}),\cyl(a_{1}),\cyl\big(i_{0}(a_{0})\big),\cyl(d^{0}_{j}),\cyl(j),\cyl(d^{1}_{j}),\cyl(j) \circ p\big(\cyl(a_{0})\big),i_{0}(a_{1}) \circ p(a_{1}), m^{\cylinder}_{j} \circ p(a^{\cylinder}_{j})} %
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Since $\cyl$ preserves mapping cylinders with respect to $\cylinder$ the following diagram in $\cl{A}$ is co-cartesian. \sq[{5,3}]{\cyl(a_{0}),\cyl^{2}(a_{0}),\cyl(a_{1}),\cyl(a^{\cylinder}_{j}),\cyl\big(i_{0}(a_{0})\big),\cyl(d^{0}_{j}),\cyl(j),\cyl(d^{1}_{j})} %
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Appealing to the universal property of $\cyl\big(a^{\cylinder}_{j}\big)$ we deduce that the following diagram in $\cl{A}$ commutes, as required. \sq{\cyl(a^{\cylinder}_{j}),a^{\cylinder}_{j},\cyl^{2}(a_{1}),\cyl(a_{1}),p(a^{\cylinder}_{j}),m^{\cylinder}_{j},\cyl(m^{\cylinder}_{j}),\sigma} 

\end{proof}

\begin{prpn} \label{NormallyClovenFibrationsHaveRLPWithRespectToArrowsAdmittingAStrongDeformationRetractionProposition} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$ which admits a strong deformation retraction \ar{a_{1},a_{0},r} with respect to $\cylinder$, and let \ar{a_{2},a_{3},f} be an arrow of $\cl{A}$ which is a normally cloven fibration with respect to $\cylinder$. 

For any arrows \ar{a_{0},a_{2},g_{0}} and \ar{a_{1},a_{3},g_{1}} of $\cl{A}$ such that the diagram \sq{a_{0},a_{2},a_{1},a_{3},g_{0},f,j,g_{1}} in $\cl{A}$ commutes there is an arrow \ar{a_{1},a_{2},l} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \liftingsquare{a_{0},a_{2},a_{1},a_{3},g_{0},f,j,g_{1},l}  \end{prpn}

\begin{proof} Since $r$ is a strong deformation retraction of $j$ with respect to $\cylinder$ there is a homotopy \ar{\cyl(a_{1}),a_{1},h} under $a_{0}$ from $jr$ to $id(a_{1})$ with respect to $\cylinder$. In particular the following diagram in $\cl{A}$ commutes. \tri{a_{1},\cyl(a_{1}),a_{1},i_{0}(a_{1}),h, j \circ r} %
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By assumption we have that the following diagram in $\cl{A}$ commutes. \sq{a_{0},a_{2},a_{1},a_{3},g_{0},f,j,g_{1}} %
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Together the commutativity of these two diagrams implies that the following diagram in $\cl{A}$ commutes. \sq{a_{1},a_{2},\cyl(a_{1}),a_{3},g_{0} \circ r,f, i_{0}(a_{1}), g_{1} \circ h} %
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For any object $a$ of $\cl{A}$ let \ar{\Delta^{\cylinder}_{f,a},\Omega^{\cylinder}_{f,a},k_{a}} denote the map of the cleavage with which $f$ is equipped. %
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Let \ar{\cyl(a_{1}),a_{2},k} denote the arrow $k_{a_{1}}(g_{0} \circ r, g_{1} \circ h)$ of $\cl{A}$. We have that the following diagram in $\cl{A}$ commutes. \liftingsquare{a_{1},a_{2},\cyl(a_{1}),a_{3},g_{0}\circ r,f,i_{0}(a_{1}),g_{1} \circ h,k} 

Let \ar{a_{1},a_{2},l} denote the arrow $k \circ i_{1}(a_{1})$ of $\cl{A}$. We claim that the following diagram in $\cl{A}$ commutes. \liftingsquare{a_{0},a_{2},a_{1},a_{3},g_{0},f,j,g_{1},l} %
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Firstly we have that the following diagram in $\cl{A}$ commutes. \squareofthreetrianglestwo{a_{1},a_{2},\cyl(a_{1}),a_{1},a_{3},l,f,id,g_{1},i_{1}(a_{1}),k,h} Thus the triangle \tri{a_{1},a_{2},a_{3},l,f,g_{1}} in $\cl{A}$ commutes. It remains to prove the commutativity of the triangle \tri{a_{0},a_{1},a_{2},j,l,g_{0}} in $\cl{A}$. 

Let \ar{\cyl(a_{0}),a_{2},k'} denote the arrow $k_{a_{0}}\big(g_{0} \circ r \circ j, g_{1} \circ h \circ \cyl(j) \big)$ of $\cl{A}$. Since $f$ is a normally cloven fibration with respect to $\cylinder$ its cleavage satisfies property (ii) of Definition \ref{NormallyClovenFibrationCylinderDefinition}. Thus the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{0}),\cyl(a_{1}),a_{3},\cyl(j),k,k'} %
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By definition of $k'$ we have that the following diagram in $\cl{A}$ commutes. \liftingsquare[{7,3}]{a_{0},a_{2},\cyl(a_{0}),a_{3},g_{0} \circ r \circ j,f,i_{0}(a_{1}),g_{1} \circ h \circ \cyl(j),k'} %
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By the commutativity of the diagram \tri{a_{0},a_{1},a_{0},j,r,id} in $\cl{A}$ we thus have that the following diagram in $\cl{A}$ commutes. \liftingsquare[{7,3}]{a_{0},a_{2},\cyl(a_{0}),a_{3},g_{0}, f, i_{0}(a_{0}), g_{1} \circ h \circ \cyl(j),k'} %
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The following diagram in $\cl{A}$ also commutes by definition of $h$. \sq{\cyl(a_{0}),a_{0},\cyl(a_{1}),a_{1},p(a_{0}),j,\cyl(j),h} %
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Hence the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{0}),a_{0},a_{3},p(a_{0}),g_{1} \circ j,g_{1} \circ h \circ \cyl(j)} %
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We now have that the following diagram in $\cl{A}$ commutes. \liftingsquare[{7,3}]{a_{0},a_{2},\cyl(a_{0}),a_{3},g_{0}, f, i_{0}(a_{0}), g_{1} \circ j \circ p(a_{0}),k'} %
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Since $f$ is a normally cloven fibration with respect to $\cylinder$ its cleavage satisfies property (i) of Definition \ref{NormallyClovenFibrationCylinderDefinition}. Thus the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{0}),a_{0},a_{2},p(a_{0}),g_{0},k'} %
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Putting everything together, we have that the following diagram in $\cl{A}$ commutes. \squarewithdiagonalthree{\cyl(a_{0}),\cyl(a_{1}),a_{0},a_{2},\cyl(j),k,p(a_{0}),g_{0},k'} %
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Thus the following diagram in $\cl{A}$ commutes, as required. \trapeziumsfive[{3,3.5,0,0,0}]{a_{0},a_{1},\cyl(a_{0}),\cyl(a_{1}),a_{0},a_{2},j,l,id,g_{0},i_{1}(a_{0}),i_{1}(a_{1}),\cyl(j),p(a_{0}),k} 
\end{proof} 

\begin{cor} \label{NormallyClovenFibrationRLPTrivialCofibrationCorollary} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{lr}, \Gamma_{ur} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $v$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ compatible with $p$, a lower right connection structure $\Gamma_{lr}$, and an upper right connection structure $\Gamma_{ur}$. Suppose that $\Gamma_{lr}$ and $\Gamma_{ur}$ are compatible with the subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$, and that $\cyl$ preserves mapping cylinders with respect to $\cylinder$. 

Let $\cocylinder = \big( \cocyl, e_0, e_1, c, v', \subdiv', r_{0}', r_{1}', s', \Gamma_{ul}'  \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$, an involution structure $v'$ compatible with $c$, a subdivision structure $\big( \subdiv', r_{0}', r_{1}', s' \big)$ compatible with $c$, and an upper left connection structure $\Gamma_{ul}'$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$ which is a trivial cofibration with respect to $\cylinder$, and let \ar{a_{2},a_{3},f} be an arrow of $\cl{A}$ which is a normally cloven fibration with respect to $\cocylinder$. 

For any commutative diagram \sq{a_{0},a_{1},a_{2},a_{3},g_{0},f,j,g_{1}} in $\cl{A}$ there is an arrow \ar{a_{1},a_{2},l} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \liftingsquare{a_{0},a_{1},a_{2},a_{3},g_{0},f,j,g_{1},l} \end{cor}

\begin{proof} By Corollary \ref{TrivialCofibrationAdmitsStrongDeformationRetractionCorollary} we have that $j$ admits a strong deformation retraction with respect to $\cylinder$. By Proposition \ref{NormallyClovenFibrationCoCylinderIffNormallyClovenFibrationCylinderProposition} we have that $f$ is a normally cloven fibration with respect to $\cylinder$. Thus we may appeal to Proposition \ref{NormallyClovenFibrationsHaveRLPWithRespectToArrowsAdmittingAStrongDeformationRetractionProposition} for a construction of $l$. \end{proof} 

\begin{cor} \label{TrivialFibrationRLPNormallyClovenCofibrationCorollary} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{ul} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $v$ compatible with $p$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ compatible with $p$, and an upper left connection structure $\Gamma_{ul}$. 

Let $\cocylinder = \big( \cocyl, e_0, e_1, c, v', \subdiv', r_{0}', r_{1}', s', \Gamma_{lr}', \Gamma_{ur}' \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$, an involution structure $v'$ compatible with $c$, a subdivision structure $\big( \subdiv', r_{0}', r_{1}', s' \big)$ compatible with $c$, a lower right connection structure $\Gamma_{lr}'$, and an upper right connection structure $\Gamma_{ur}'$. 

Suppose that $\Gamma_{lr}'$ and $\Gamma_{ur}'$ are compatible with the subdivision structure $\big( \subdiv', r_{0}', r_{1}', s' \big)$, and that $\cocyl$ preserves mapping co-cylinders with respect to $\cocylinder$. %
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Suppose that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$ which is a normally cloven cofibration with respect to $\cylinder$, and let \ar{a_{2},a_{3},f} be an arrow of $\cl{A}$ which is a trivial fibration with respect to $\cocylinder$. 

For any commutative diagram \sq{a_{0},a_{1},a_{2},a_{3},g_{0},f,j,g_{1}} in $\cl{A}$, there is an arrow \ar{a_{1},a_{2},l} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \liftingsquare{a_{0},a_{1},a_{2},a_{3},g_{0},f,j,g_{1},l} \end{cor}

\begin{proof} Follows immediately from Corollary \ref{NormallyClovenFibrationRLPTrivialCofibrationCorollary} by duality. \end{proof} 

\begin{cor} \label{CHEPHoldsCorollary} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{lr}, \Gamma_{ur} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $v$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ compatible with $p$, a lower right connection structure $\Gamma_{lr}$, and an upper right connection structure $\Gamma_{ur}$. %
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Suppose that $\Gamma_{lr}$ and $\Gamma_{ur}$ are compatible with the subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$, and that $\cyl$ preserves mapping cylinders with respect to $\cylinder$. 

Let $f$ be an arrow of $\cl{A}$ which is a normally cloven fibration with respect to $\cylinder$, and let $j$ be an arrow of $\cl{A}$ which is a cofibration with respect to $\cylinder$. Then $f$ has the covering homotopy extension property with respect to $j$ and $\cylinder$. \end{cor} 

\begin{proof} Follows immediately from Proposition \ref{RetractionIsStrongDeformationRetractionProposition} and Proposition \ref{NormallyClovenFibrationsHaveRLPWithRespectToArrowsAdmittingAStrongDeformationRetractionProposition}. \end{proof}

\begin{cor} \label{TrivialNormallyClovenFibrationRLPCofibrationCorollary} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{ul}, \Gamma_{lr}, \Gamma_{ur} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $v$ compatible with $p$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ compatible with $p$, an upper left connection structure $\Gamma_{ul}$, a lower right connection structure $\Gamma_{lr}$, and an upper right connection structure $\Gamma_{ur}$. %
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Suppose that $\Gamma_{lr}$ and $\Gamma_{ur}$ are compatible with the subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$, and that $\cyl$ preserves mapping cylinders with respect to $\cylinder$. 

Let $\cocylinder = \big( \cocyl, e_{0}, e_{1}, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. %

Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$ which is a cofibration with respect to $\cylinder$, and let \ar{a_{2},a_{3},f} be an arrow of $\cl{A}$ which is a trivial normally cloven fibration with respect to $\cocylinder$. 

For any commutative diagram \sq{a_{0},a_{2},a_{1},a_{3},g_{0},f,j,g_{1}} in $\cl{A}$, there is an arrow \ar{a_{1},a_{2},l} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \liftingsquare{a_{0},a_{2},a_{1},a_{3},g_{0},f,j,g_{1},l} \end{cor}

\begin{proof} By Proposition \ref{NormallyClovenFibrationCoCylinderIffNormallyClovenFibrationCylinderProposition} we have that $f$ is a normally cloven fibration with respect to $\cylinder$. Thus by Corollary \ref{CHEPHoldsCorollary} $f$ has the covering homotopy extension property with respect to $j$ and $\cylinder$. 

Moreover by Corollary \ref{TrivialFibrationIsStrongDeformationRetractionCorollary} there is an arrow \ar{a_{3},a_{2},j'} of $\cl{A}$ such that $f$ is a strong deformation retraction of $j'$ with respect to $\cocylinder$. Thus we may appeal to Proposition \ref{CHEPImpliesLiftingAxiomProposition} for a construction of $l$. \end{proof}

\begin{cor} \label{FibrationRLPTrivialNormallyClovenCofibrationCorollary} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. Let $\cocylinder = \big( \cocyl, e_{0}, e_{1}, c, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{ul}, \Gamma_{lr}, \Gamma_{ur} \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$, an involution structure $v$ compatible with $c$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ compatible with $c$, an upper left connection structure $\Gamma_{ul}$, a lower right connection structure $\Gamma_{lr}$, and an upper right connection structure $\Gamma_{ur}$. 

Suppose that $\Gamma_{lr}$ and $\Gamma_{ur}$ are compatible with the subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$, and that $\cocyl$ preserves mapping co-cylinders with respect to $\cocylinder$. %
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Suppose that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. %

Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$ which is a trivial normally cloven cofibration with respect to $\cylinder$, and let \ar{a_{2},a_{3},f} be an arrow of $\cl{A}$ which is a fibration with respect to $\cocylinder$. 

For any commutative diagram \sq{a_{0},a_{2},a_{1},a_{3},g_{0},f,j,g_{1}} in $\cl{A}$, there is an arrow \ar{a_{1},a_{2},l} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \liftingsquare{a_{0},a_{2},a_{1},a_{3},g_{0},f,j,g_{1},l} \end{cor}

\begin{proof} Follows immediately from Corollary \ref{TrivialNormallyClovenFibrationRLPCofibrationCorollary} by duality. \end{proof} 


\end{section}

mapping_cylinders_and_mapping_co-cylinders.tex
\begin{section}{Mapping cylinders and mapping co-cylinders} \label{MappingCylindersAndMappingCoCylindersSection}

We introduce a notion of a mapping cylinder $a^{\cylinder}_{j}$ of an arrow \ar{a_{0},a_{1},f} of a formal category $\cl{A}$ with respect to a cylinder $\cylinder$ in $\cl{A}$. There is a canonical arrow $m^{\cylinder}_{f}$ from $a^{\cylinder}_{f}$ to $\cyl(a_{1})$. This arrow admits a retraction if $f$ is a cofibration with respect to $\cylinder$.

 We introduce a dual notion of a mapping co-cylinder $a^{\cocylinder}_{f}$ of $f$ with respect to a co-cylinder $\cocylinder$ in $\cl{A}$. There is an analogous canonical arrow from $\cocyl(a_{0})$ to $a^{\cocylinder}_{f}$. This arrow admits a section if $f$ is a fibration with respect to $\cocylinder$.

In \ref{LiftingAxiomsSection} we will prove that if $\cylinder$ admits sufficient structure the retraction of $m^{\cylinder}_{f}$ for $f$ a cofibration is moreover a strong deformation retraction with respect to $\cylinder$. This will be a vital step towards establishing the lifting axioms for a model structure.

We present a proof due to van den Berg and Garner that if $\cylinder$ has strictness of right identities then every arrow $f$ of $\cl{A}$ factors through $a^{\cylinder}_{f}$ into a normally cloven cofibration followed by a strong deformation retraction. The proof is based on \S{6} of \cite{VanDenBergGarnerTopologicalAndSimplicialModelsOfIdentityTypes}. If $\cocylinder$ has strictness of right identities we deduce dually that $f$ factors through $a^{\cocylinder}_{f}$ into an arrow admitting a strong deformation retraction followed by a normally cloven fibration. In \ref{FactorisationAxiomsSection} we will build upon these observations to obtain a proof that the factorisation axioms for a model structure hold.

That $\cylinder$ has strictness of right identities is an indispensable assumption. We discuss this a little more in the introduction to \ref{FactorisationAxiomsSection}. To carry through the ideas of van den Berg and Garner we assume that $\cylinder$ admits a lower right connection structure. Instead van den Berg and Garner work with a `strength', which is a structure of a slightly different nature to those which we considered in \ref{StructuresUponACylinderOrACoCylinderSection}. 

The Moore co-cylinder in topological spaces is a key motivating example for van den Berg and Garner. It does not admit a lower connection structure, but does admit a strength. We refer the reader to \S{4.2} of \cite{VanDenBergGarnerTopologicalAndSimplicialModelsOfIdentityTypes} for more on this. Everywhere that we make use of a connection in this work it should be possible to instead make use of a strength. 

In abstract homotopy theory the observation that the mapping cylinder of an arrow \ar{a_{0},a_{1},f} of $\cl{A}$ gives rise to a factorisation into a cofibration followed by a strong deformation retraction goes back to Kamps. It can be found in \S{4} of \cite{KampsKanBedingungenUndAbstrakteHomotopietheorie}, and is also Theorem 5.11 in the book \cite{KampsPorterAbstractHomotopyAndSimpleHomotopyTheory} of Kamps and Porter. 

The cleavage of the arrow \ar{a_{0},a^{\cylinder}_{f},j} constructed in these works does not however satisfy our conditions for $j$ to be a normally cloven cofibration. The cleavage we construct after van den Berg and Garner does demonstrate $j$ to be a normally cloven cofibration, which is vital for us.

This aside, a quite different proof that the mapping cylinder of $f$ gives rise to a factorisation into a cofibration followed by a strong deformation retraction was given by Grandis under the additional assumption that $\cylinder$ admits an `extended acceleration structure' in \S{4.6.5} -- \S{4.6.7} of the book \cite{GrandisDirectedAlgebraicTopology}. 

The key step in this proof concerning double mapping cylinders is Theorem 1.8 in an earlier paper \cite{GrandisDirectedHomotopyTheoryII} of Grandis, the hypotheses of which are weaker than those of \cite{GrandisDirectedAlgebraicTopology}. The proof is erroneous, but can be repaired under the assumption that $\cylinder$ admits a `zero collapse structure' in the sense of \cite{GrandisCategoricallyAlgebraicFoundationsForHomotopicalAlgebra}, or alternatively if $\cylinder$ has strictness of right identities. We also bring the reader's attention to Theorem 3.8 of the still earlier paper \cite{GrandisHomotopicalAlgebraInHomotopicalCategories} of Grandis. 

\begin{assum} Let $\cl{C}$ be a 2-category with a final object such that pushouts and pullbacks of 2-arrows of $\cl{C}$ give rise to pushouts and pullbacks in formal categories in the sense of Definition \ref{PushoutsPullbacks2ArrowsArePushoutsPullbacksInFormalCategoriesTerminology}. Let $\cl{A}$ be an object of $\cl{C}$. As before we view $\cl{A}$ as a formal category, writing of objects and arrows of $\cl{A}$. \end{assum} 

\begin{defn} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$. A {\em mapping cylinder} with respect to $\cylinder$ of an arrow \ar{a_{0},a_{1},f} of $\cl{A}$ is an object $a^{\cylinder}_{f}$ of $\cl{A}$ together with an arrow \ar{\cyl(a_{0}),a^{\cylinder}_{f},d^{0}_{f}} of $\cl{A}$ and an arrow \ar{a_{1},a^{\cylinder}_{f},d^{1}_{f}} of $\cl{A}$ such that the following diagram in $\cl{A}$ is co-cartesian. \sq{a_{0},\cyl(a_{0}),a_{1},a^{\cylinder}_{f},i_{0}(a_{0}),d^{0}_{f},f,d^{1}_{f}} \end{defn}

\begin{defn} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$. Then $\cl{A}$ {\em has mapping cylinders} with respect to $\cylinder$ if for every arrow $f$ of $\cl{A}$ we have a mapping cylinder $\big(a^{\cylinder}_{f}, d^{0}_{f},d^{1}_{f} \big)$ of $f$ with respect to $\cylinder$. \end{defn}

\begin{defn} Let $\cocylinder = \big( \cocyl, e_0, e_1 \big)$ be a co-cylinder in $\cl{A}$. A {\em mapping co-cylinder} of $f$ with respect to $\cocylinder$ of an arrow \ar{a_{0},a_{1},f} of $\cl{A}$ is an object $a^{\cocylinder}_{f}$ of $\cl{A}$ together with an arrow \ar{a^{\cocylinder}_{f},a_{0},d^{0}_{f}} of $\cl{A}$ and an arrow \ar{a^{\cocylinder}_{f},\cocyl(a_{1}),d^{1}_{f}} of $\cl{A}$, such that the following diagram in $\cl{A}$ is cartesian. \sq[{4,3}]{a^{\cocylinder}_{f},a_{0},\cocyl(a_{1}),a_{1},d^{0}_{f},f,d^{1}_{f},e_{0}(a_{1})} \end{defn}

\begin{rmk} Let $\cocylinder = \big( \cocyl, e_0, e_1 \big)$ be a co-cylinder in $\cl{A}$. Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$. Then $\big(a^{\cocylinder}_{f},d^{0}_{f},d^{1}_{f} \big)$ defines a mapping co-cylinder of $f$ with respect to $\cocylinder$ if and only if $\big( a^{\cocylinder}_{f}, (d^{0}_{f})^{op}, (d^{1}_{f})^{op} \big)$ defines a mapping cylinder of $f$ with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$. \end{rmk}

\begin{defn} Let $\cocylinder = \big( \cocyl, e_0, e_1 \big)$ be a co-cylinder in $\cl{A}$. Then $\cl{A}$ {\em has mapping co-cylinders} with respect to $\cocylinder$ if for every arrow $f$ of $\cl{A}$ we have a mapping co-cylinder $\big( a^{\cocylinder}_{f},d^{0}_{f},d^{1}_{f} \big)$ of $f$ with respect to $\cocylinder$. \end{defn}

\begin{defn} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$. Then $\cyl$ {\em preserves mapping cylinders} with respect to $\cylinder$ if for every pair of an arrow \ar{a_{0},a_{1},f} of $\cl{A}$ and a mapping cylinder $\big( a^{\cylinder}_{f}, d^{0}_{f}, d^{1}_{f} \big)$ of $f$ with respect to $\cylinder$, we have that $\big( \cyl\big(a^{\cylinder}_{f}\big), \cyl(d^{0}_{f}), \cyl(d^{1}_{f}) \big)$ defines a mapping cylinder of $\cyl(f)$ with respect to $\cylinder$. \end{defn} 

\begin{defn} Let $\cocylinder = \big( \cocyl, e_0, e_1 \big)$ be a co-cylinder in $\cl{A}$. Then $\cocyl$ {\em preserves mapping co-cylinders} with respect to $\cocylinder$ if $\cocyl$ preserves mapping cylinders with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$. \end{defn} 

\begin{notn} \label{CanonicalArrowFromMappingCylinderToCylinderDefinition} Let $\cylinder = \big( \cyl, i_{0}, i_{1} \big)$ be a cylinder in $\cl{A}$. Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$, and suppose that $\big( a^{\cylinder}_{f}, d^{0}_{f}, d^{1}_{f} \big)$ defines a mapping cylinder of $f$ with respect to $\cylinder$. 

We denote by $m^{\cylinder}_{f}$ the canonical arrow of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),a_{1},a^{\cylinder}_{f},\cyl(a_{1}),i_{0}(a_{0}),d^{0}_{f},f,d^{1}_{f},\cyl(f),i_{0}(a_{1}),m^{\cylinder}_{f}} \end{notn}

\begin{notn} Let $\cocylinder = \big( \cocyl, e_{0}, e_{1} \big)$ be a co-cylinder in $\cl{A}$. Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$, and suppose that $\big( a^{\cocylinder}_{f}, d^{0}_{f}, d^{1}_{f} \big)$ defines a mapping co-cylinder of $f$ with respect to $\cocylinder$. 

We denote by $m^{\cocylinder}_{f}$ the canonical arrow of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pullback{a^{\cocylinder}_{f},a_{0},\cocyl(a_{1}),a_{1},\cocyl(a_{0}),d^{0}_{f},f,d^{1}_{f},e_{0}(a_{1}),e_{0}(a_{0}),\cocyl(f),m^{\cocylinder}_{f}} \end{notn}

\begin{prpn} \label{RetractionToMappingCylinderProposition} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$. Let \ar{a_{0},a_{1},j} be an arrow of $\cl{A}$, and suppose that $\big( a^{\cylinder}_{j}, d^{0}_{j}, d^{1}_{j} \big)$ defines a mapping cylinder of $j$ with respect to $\cylinder$. 

Suppose that $j$ is a cofibration with respect to $\cylinder$, and let \ar{\cyl(a_{1}),a^{\cylinder}_{j},r^{\cylinder}_{j}} denote the corresponding arrow of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),a_{1},\cyl(a_{1}),a^{\cylinder}_{j},i_{0}(a_{0}),\cyl(j),j,i_{0}(a_{1}),d^{0}_{j},d^{1}_{j},r^{\cylinder}_{j}} Then $r^{\cylinder}_{j}$ is a retraction of $m^{\cylinder}_{j}$. \end{prpn}

\begin{proof} The following diagram in $\cl{A}$ commutes.

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=11 em, row sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| \cyl(a_{0})        \& |(1-0)| a^{\cylinder}_{j} \\ 
|(0-1)| a^{\cylinder}_{j} \& |(1-1)| \cyl(a_{1}) \\
};
	
\draw[->] (0-0) to node[auto] {$d^{0}_{j}$} (1-0);
\draw[->] (1-0) to node[auto] {$m^{\cylinder}_{j}$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$\cyl(j)$} (1-1);
\draw[->] (1-1) to node[auto] {$r^{\cylinder}_{j}$} (0-1);
\draw[->] (0-0) to node[auto,swap] {$d^{0}_{j}$} (0-1);
\end{tikzpicture} 

\end{diagram} 

The following diagram in $\cl{A}$ also commutes.

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=11 em, row sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| a_{1}        \& |(1-0)| a^{\cylinder}_{j} \\ 
|(0-1)| a^{\cylinder}_{j} \& |(1-1)| \cyl(a_{1}) \\
};
	
\draw[->] (0-0) to node[auto] {$d^{1}_{j}$} (1-0);
\draw[->] (1-0) to node[auto] {$m^{\cylinder}_{j}$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$i_{0}(a_{1})$} (1-1);
\draw[->] (1-1) to node[auto] {$r^{\cylinder}_{j}$} (0-1);
\draw[->] (0-0) to node[auto,swap] {$d^{1}_{j}$} (0-1);
\end{tikzpicture} 

\end{diagram} %
%
Putting these observations together we have that the following diagram in $\cl{A}$ commutes. \pushout[{3,3,1}]{a_{0},\cyl(a_{0}),a_{1},a^{\cylinder}_{j},a^{\cylinder}_{j},i_{0}(a_{0}),d^{0}_{j},j,d^{1}_{j},d^{0}_{j},d^{1}_{j},r^{\cylinder}_{j} \circ m^{\cylinder}_{j}} %
%
Appealing to the universal property of $a^{\cylinder}_{j}$ we deduce that the following diagram in $\cl{A}$ commutes. \tri{a^{\cylinder}_{j},\cyl(a_{1}),a^{\cylinder}_{j},m^{\cylinder}_{j},r^{\cylinder}_{j},id} \end{proof} 

\begin{prpn} \label{MappingCylinderFactorisationProposition} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$, equipped with a contraction structure $p$. Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$, and suppose that $\big( a^{\cylinder}_{f}, d^{0}_{f}, d^{1}_{f} \big)$ defines a mapping cylinder of $f$ with respect to $\cylinder$. 

Let \ar{a^{\cylinder}_{f},a_{1},g} denote the canonical arrow of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),a_{1},a^{\cylinder}_{f},a_{1},i_{0}(a_{0}),d^{0}_{f},f,d^{1}_{f},f \circ p(a_{0}),id,g} 

Let \ar{a_{0},a^{\cylinder}_{f},j} denote the arrow $d^{0}_{f} \circ i_{1}(a_{0})$ of $\cl{A}$. 

The following diagram in $\cl{A}$ commutes. \tri{a_{0},a^{\cylinder}_{f},a_{1},j,g,f} \end{prpn}

\begin{proof} The following diagram in $\cl{A}$ commutes. 

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=4 em, row sep=4 em, nodes={anchor=center}]
{
|(0-0)| a_{0} \&                     \& |(2-0)| a^{\cylinder}_{f} \\ 
              \& |(1-1)| \cyl(a_{0}) \&                           \\ 
|(0-2)| a_{0} \&                     \& |(2-2)| a_{1}             \\
};
	
\draw[->] (0-0) to node[auto] {$j$} (2-0);
\draw[->] (0-0) to node[auto] {$i_{1}(a_{0})$} (1-1);
\draw[->] (1-1) to node[auto,swap] {$d^{0}_{f}$} (2-0);
\draw[->] (2-0) to node[auto] {$g$} (2-2);
\draw[->] (0-0) to node[auto,swap] {$id$} (0-2);
\draw[->] (1-1) to node[auto] {$p(a_{0})$} (0-2);
\draw[->] (0-2) to node[auto,swap] {$f$} (2-2);
\end{tikzpicture} 

\end{diagram} 

\end{proof}

\begin{defn} \label{MappingCylinderFactorisationDefinition} Let $\cylinder = \big( \cyl, i_0, i_1, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$, and suppose that $\big( a^{\cylinder}_{f}, d^{0}_{f}, d^{1}_{f} \big)$ defines a mapping cylinder of $f$ with respect to $\cylinder$. 

We refer to the factorisation of $f$ obtained via Proposition \ref{MappingCylinderFactorisationProposition} as the {\em mapping cylinder factorisation} of $f$ with respect to $\cylinder$. \end{defn} 

\begin{cor} \label{MappingCoCylinderFactorisationCorollary} Let $\cylinder = \big( \cocyl, e_0, e_1, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$, and suppose that $\big( a^{\cocylinder}_{f}, d^{0}_{f}, d^{1}_{f} \big)$ defines a mapping co-cylinder of $f$ with respect to $\cocylinder$. 

Let \ar{a_{0},a^{\cocylinder}_{f},j} denote the canonical arrow of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pullback[{3,3,-1}]{a^{\cocylinder}_{f},a_{0},\cocyl(a_{1}),a_{1},a_{0},d^{0}_{f},f,d^{1}_{f},e_{0}(a_{1}),id,c(a_{1}) \circ f,j} 

Let \ar{a^{\cocylinder}_{f},a_{1},g} denote the arrow $e_{1}(a_{1}) \circ d^{1}_{f}$ of $\cl{A}$. 

The following diagram in $\cl{A}$ commutes. \tri{a_{0},a^{\cocylinder}_{f},a_{1},j,g,f} \end{cor}

\begin{proof} Follows immediately from Proposition \ref{MappingCylinderFactorisationProposition} by duality. \end{proof}

\begin{defn} \label{MappingCoCylinderFactorisationDefinition} Let $\cocylinder = \big( \cocyl, e_0, e_1, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$, and suppose that $\big( a^{\cocylinder}_{f}, d^{0}_{f}, d^{1}_{f} \big)$ defines a mapping co-cylinder of $f$ with respect to $\cocylinder$. We refer to the factorisation of $f$ obtained via Proposition \ref{MappingCoCylinderFactorisationCorollary} as the {\em mapping co-cylinder factorisation} of $f$ with respect to $\cocylinder$. \end{defn}

\begin{prpn} \label{MappingCylinderFactorisationStrongDeformationRetractionWithRespectToCylinderProposition} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p, \Gamma_{lr} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$ and a lower right connection structure $\Gamma_{lr}$. Suppose that $\Gamma_{lr}$ is compatible with $p$, and that $\cyl$ preserves mapping cylinders with respect to $\cylinder$. 

Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$, and let $\big( a^{\cylinder}_{f}, d^{0}_{f}, d^{1}_{f} \big)$ be a mapping cylinder of $f$ with respect to $\cylinder$. 

Let \tri{a_{0},a^{\cylinder}_{f},a_{1},j,g,f} denote the corresponding mapping cylinder factorisation of $f$. 

There is a homotopy over $a_{1}$ from $d^{1}_{f} \circ g$ to $id\big(a^{\cylinder}_{f}\big)$ with respect to $\cylinder$ and $(g,g)$. \end{prpn}

\begin{proof} By construction $g$ is a retraction of $d^{1}_{f}$. The following diagram in $\cl{A}$ is co-cartesian since $\cyl$ preserves mapping cylinders with respect to $\cylinder$. \sq[{5,3}]{\cyl(a_{0}),\cyl^{2}(a_{0}),\cyl(a_{1}),\cyl\big(a^{\cylinder}_{f}\big),\cyl\big(i_{0}(a_{0})\big),\cyl(d^{0}_{f}),\cyl(j),\cyl(d^{1}_{f})} Moreover the following diagram in $\cl{A}$ commutes.  

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=3 em, row sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| \cyl(a_{0}) \&               \& |(2-0)| \cyl^{2}(a_{0}) \\
|(0-1)| \cyl(a_{1}) \& |(1-1)| a_{0} \& |(2-1)| \cyl(a_{0})     \\
|(0-2)| a_{1}       \&               \& |(2-2)| a^{\cylinder}_{f}    \\
};
	
\draw[->] (0-0) to node[auto] {$\cyl\big(i_{0}(a_{0})\big)$} (2-0);
\draw[->] (2-0) to node[auto] {$\Gamma_{lr}(a_{0})$} (2-1);
\draw[->] (0-0) to node[auto] {$p(a_{0})$} (1-1);
\draw[->] (1-1) to node[auto] {$i_{0}(a_{0})$} (2-1);
\draw[->] (1-1) to node[auto] {$f$} (0-2);
\draw[->] (0-2) to node[auto,swap] {$d^{1}_{f}$} (2-2);
\draw[->] (2-1) to node[auto] {$d^{0}_{f}$} (2-2);
\draw[->] (0-0) to node[auto,swap] {$\cyl(f)$} (0-1);
\draw[->] (0-1) to node[auto,swap] {$p(a_{1})$} (0-2);
\end{tikzpicture} 

\end{diagram} %
%
Thus there is a canonical arrow \ar{\cyl\big(a^{\cylinder}_{f}\big),a^{\cylinder}_{f},h} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout[{5,3,-1}]{\cyl(a_{0}),\cyl^{2}(a_{0}),\cyl(a_{1}),\cyl\big(a^{\cylinder}_{f}\big),a^{\cylinder}_{f},\cyl\big(i_{0}(a_{0})\big),\cyl(d^{0}_{f}),\cyl(f),\cyl(d^{1}_{f}),d^{0}_{f} \circ \Gamma_{lr}(a_{0}),d^{1}_{f} \circ p(a_{1}),h} We claim that $h$ defines a homotopy over $a_{1}$ from $d^{1}_{f} \circ g$ to $id(a^{\cylinder}_{f})$ with respect to $\cylinder$ and $(d^{1}_{f},d^{1}_{f})$. 

Let us first prove that the following diagram in $\cl{A}$ commutes. \tri[{5,3}]{a^{\cylinder}_{f},\cyl\big(a^{\cylinder}_{f}\big),a^{\cylinder}_{f},i_{0}\big(a^{\cylinder}_{f}\big),h,d^{1}_{f} \circ g} We have that the following diagram in $\cl{A}$ commutes. \trapeziums[{3,3,1,0}]{a_{1},a^{\cylinder}_{f},\cyl(a_{1}),\cyl\big(a^{\cylinder}_{f}\big),a_{1},a^{\cylinder}_{f},d^{1}_{f},i_{0}\big(a^{\cylinder}_{f}\big),h,id,d^{1}_{f},i_{0}(a_{1}),\cyl(d^{1}_{f}),p(a_{1})} %
%
The following diagram in $\cl{A}$ also commutes.  

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=3 em, row sep=4 em, nodes={anchor=center}]
{ 
|(0-0)| \cyl(a_{0}) \&                         \&[1em] |(2-0)| a^{\cylinder}_{f} \\
|(0-1)| a_{0}       \& |(1-1)| \cyl^{2}(a_{0}) \& |(2-1)| \cyl\big(a^{\cylinder}_{f}\big)     \\
|(0-2)| \cyl(a_{0}) \&                         \& |(2-2)| a^{\cylinder}_{f}    \\
};
	
\draw[->] (0-0) to node[auto] {$d^{0}_{f}$} (2-0);
\draw[->] (2-0) to node[auto] {$i_{0}\big(a^{\cylinder}_{f}\big)$} (2-1);
\draw[->] (0-0) to node[auto] {$i_{0}\big(\cyl(a_{0})\big)$} (1-1);
\draw[->] (1-1) to node[auto] {$\cyl(d^{0}_{f})$} (2-1);
\draw[->] (1-1) to node[auto] {$\Gamma_{lr}(a_{0})$} (0-2);
\draw[->] (0-2) to node[auto,swap] {$d^{0}_{f}$} (2-2);
\draw[->] (2-1) to node[auto] {$h$} (2-2);
\draw[->] (0-0) to node[auto,swap] {$p(a_{0})$} (0-1);
\draw[->] (0-1) to node[auto,swap] {$i_{0}(a_{0})$} (0-2);
\end{tikzpicture} 

\end{diagram} %
%
Putting these two observations together we have that the following diagram in $\cl{A}$ commutes. \pushout[{4,3,-1}]{a_{0},\cyl(a_{0}),a_{1},a^{\cylinder}_{f},a^{\cylinder}_{f},i_{0}(a_{0}),d^{0}_{f},f,d^{1}_{f},d^{0}_{f} \circ i_{0}(a_{0}) \circ p(a_{0}),d^{1}_{f},h \circ i_{0}\big(a^{\cylinder}_{f}\big)} We also have that the diagram \tri{a_{1},a^{\cylinder}_{f},a^{\cylinder}_{f},d^{1}_{f},d^{1}_{f} \circ g,d^{1}_{f}} in $\cl{A}$ commutes since $g$ is a retraction of $d^{1}_{f}$. Moreover the following diagram in $\cl{A}$ commutes.

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=3 em, row sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| \cyl(a_{0}) \& |(1-0)| a^{\cylinder}_{f} \\
|(0-1)| a_{0}       \& |(1-1)| a_{1}        \\
|(0-2)| \cyl(a_{0}) \& |(1-2)| a^{\cylinder}_{f}    \\
};
	
\draw[->] (0-0) to node[auto] {$d^{0}_{f}$} (1-0);
\draw[->] (1-0) to node[auto] {$g$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$p(a_{0})$} (0-1);
\draw[->] (1-1) to node[auto] {$d^{1}_{f}$} (1-2);
\draw[->] (0-2) to node[auto,swap] {$d^{0}_{f}$} (1-2);
\draw[->] (0-1) to node[auto,swap] {$i_{0}(a_{0})$} (0-2);
\draw[->] (0-1) to node[auto,swap] {$f$} (1-1);
\end{tikzpicture} 

\end{diagram} %
%
It follows from the commutativity of the last two diagrams that the following diagram in $\cl{A}$ commutes. \pushout[{4,3,-1}]{a_{0},\cyl(a_{0}),a_{1},a^{\cylinder}_{f},a^{\cylinder}_{f},i_{0}(a_{0}),d^{0}_{f},f,d^{1}_{f},d^{0}_{f} \circ i_{0}(a_{0}) \circ p(a_{0}),d^{1}_{f},d^{1}_{f} \circ g} %
%
Appealing to the universal property of $a^{\cylinder}_{f}$ we deduce that the following diagram in $\cl{A}$ commutes. \tri[{5,3}]{a^{\cylinder}_{f},\cyl\big(a^{\cylinder}_{f}\big),a^{\cylinder}_{f},i_{0}\big(a^{\cylinder}_{f}\big),h,d^{1}_{f} \circ g} 

Let us now prove that the following diagram in $\cl{A}$ commutes. \tri[{5,3}]{a^{\cylinder}_{f},\cyl\big(a^{\cylinder}_{f}\big),a^{\cylinder}_{f},i_{1}\big(a^{\cylinder}_{f}\big),h,id} We have that the following diagram in $\cl{A}$ commutes. 

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=3 em, row sep=4 em, nodes={anchor=center}]
{ 
|(0-0)| a_{1} \&                     \&[1em] |(2-0)| a^{\cylinder}_{f} \\
              \& |(1-1)| \cyl(a_{1}) \& |(2-1)| \cyl(a^{\cylinder}_{f})     \\
|(0-2)| a_{1} \&                     \& |(2-2)| a^{\cylinder}_{f}    \\
};
	
\draw[->] (0-0) to node[auto] {$d^{1}_{f}$} (2-0);
\draw[->] (2-0) to node[auto] {$i_{1}(a^{\cylinder}_{f})$} (2-1);
\draw[->] (0-0) to node[auto,swap] {$id$} (0-2);
\draw[->] (1-1) to node[auto] {$\cyl(d^{1}_{f})$} (2-1);
\draw[->] (1-1) to node[auto] {$p(a_{1})$} (0-2);
\draw[->] (0-2) to node[auto,swap] {$d^{1}_{f}$} (2-2);
\draw[->] (2-1) to node[auto] {$h$} (2-2);
\draw[->] (0-0) to node[auto] {$i_{1}(a_{1})$} (1-1);
\end{tikzpicture} 

\end{diagram} %
%
The following diagram in $\cl{A}$ also commutes.  

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=3 em, row sep=4 em, nodes={anchor=center}]
{ 
|(0-0)| \cyl(a_{0}) \&                         \&[1em] |(2-0)| a^{\cylinder}_{f} \\
                    \& |(1-1)| \cyl^{2}(a_{0}) \& |(2-1)| \cyl(a^{\cylinder}_{f})     \\
|(0-2)| \cyl(a_{0}) \&                         \& |(2-2)| a^{\cylinder}_{f}    \\
};
	
\draw[->] (0-0) to node[auto] {$d^{0}_{f}$} (2-0);
\draw[->] (2-0) to node[auto] {$i_{1}(a^{\cylinder}_{f})$} (2-1);
\draw[->] (0-0) to node[auto] {$i_{1}\big(\cyl(a_{0})\big)$} (1-1);
\draw[->] (1-1) to node[auto] {$\cyl(d^{0}_{f})$} (2-1);
\draw[->] (1-1) to node[auto] {$\Gamma_{lr}(a_{0})$} (0-2);
\draw[->] (0-2) to node[auto,swap] {$d^{0}_{f}$} (2-2);
\draw[->] (2-1) to node[auto] {$h$} (2-2);
\draw[->] (0-0) to node[auto,swap] {$id$} (0-2);
\end{tikzpicture} 

\end{diagram} Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \pushout[{4,3,0}]{a_{0},\cyl(a_{0}),a_{1},a^{\cylinder}_{f},a^{\cylinder}_{f},i_{0}(a_{0}),d^{0}_{f},f,d^{1}_{f},d^{0}_{f},d^{1}_{f},h \circ i_{1}(a^{\cylinder}_{f})} %
%
Appealing to the universal property of $a^{\cylinder}_{f}$ we deduce that the following diagram in $\cl{A}$ commutes, as we were aiming to show. \tri[{4,3}]{a^{\cylinder}_{f},\cyl(a^{\cylinder}_{f}),a^{\cylinder}_{f},i_{1}(a^{\cylinder}_{f}),h,id} 

We have now proven that $h$ defines a homotopy from $d^{1}_{f} \circ g$ to $id\big(a^{\cylinder}_{f}\big)$ with respect to $\cylinder$. It remains to demonstrate that the following diagram in $\cl{A}$ commutes. \sq{\cyl\big(a^{\cylinder}_{f}\big),a^{\cylinder}_{f},\cyl(a_{1}),a_{1},h,g,\cyl(g),p(a_{1})} %
%
Since $\Gamma_{lr}$ is compatible with $p$ the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{\cyl^{2}(a_{0}),\cyl(a_{0}),\cyl(a_{0}),a_{0},\Gamma_{lr}(a_{0}),p(a_{0}),p\big(\cyl(a_{0})\big),p(a_{0})} By definition of $h$ and $g$ we thus have that the following diagram in $\cl{A}$ commutes.  

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=3 em, row sep=4 em, nodes={anchor=center}]
{ 
|(0-0)| \cyl^{2}(a_{0}) \&                         \& |(2-0)| \cyl\big(a^{\cylinder}_{f}\big) \\
|(0-1)| \cyl(a_{0})     \& |(1-1)| \cyl^{2}(a_{0}) \& |(2-1)| a^{\cylinder}_{f}     \\
|(0-2)| a_{0}           \&                         \& |(2-2)| a_{1}    \\
};
	
\draw[->] (0-0) to node[auto] {$\cyl(d^{0}_{f})$} (2-0);
\draw[->] (2-0) to node[auto] {$h$} (2-1);
\draw[->] (0-0) to node[auto] {$\Gamma_{lr}(a_{0})$} (1-1);
\draw[->] (1-1) to node[auto] {$d^{0}_{f}$} (2-1);
\draw[->] (1-1) to node[auto] {$p(a_{0})$} (0-2);
\draw[->] (0-2) to node[auto,swap] {$f$} (2-2);
\draw[->] (2-1) to node[auto] {$g$} (2-2);
\draw[->] (0-0) to node[auto,swap] {$p\big(\cyl(a_{0})\big)$} (0-1);
\draw[->] (0-1) to node[auto,swap] {$p(a_{0})$} (0-2);
\end{tikzpicture} 

\end{diagram} By definition of $h$ and $g$ we also have that the following diagram in $\cl{A}$ commutes.  

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=4 em, row sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| \cyl(a_{1}) \& |(1-0)| \cyl\big(a^{\cylinder}_{f}\big) \\
|(0-1)| a_{1}       \& |(1-1)| a^{\cylinder}_{f}       \\
                    \& |(1-2)| a_{1}              \\
};
	
\draw[->] (0-0) to node[auto] {$\cyl(d^{1}_{f})$} (1-0);
\draw[->] (1-0) to node[auto] {$h$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$p(a_{1})$} (0-1);
\draw[->] (1-1) to node[auto] {$g$} (1-2);
\draw[->] (0-1) to node[auto] {$d^{1}_{f}$} (1-1);
\draw[->] (0-1) to node[auto,swap] {$id$} (1-2);
\end{tikzpicture} 

\end{diagram} Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \pushout[{5,3,-1}]{\cyl(a_{0}),\cyl^{2}(a_{0}),\cyl(a_{1}),\cyl\big(a^{\cylinder}_{f}\big),a_{1},\cyl\big(i_{0}(a_{0})\big),\cyl(d^{0}_{f}),\cyl(j),\cyl(d^{1}_{f}),f \circ p(a_{0}) \circ p\big(\cyl(a_{0})\big),p(a_{1}),g \circ h} %
%
By definition of $g$ the following diagram in $\cl{A}$ also commutes. \tri[{4,3}]{\cyl(a_{1}),\cyl(a^{\cylinder}_{f}),\cyl(a_{1}),\cyl(d^{1}_{f}),\cyl(g),id} Hence the following diagram in $\cl{A}$ commutes. \tri[{4,3}]{\cyl(a_{1}),\cyl(a^{\cylinder}_{f}),a_{1},\cyl(d^{1}_{f}),p(a_{1}) \circ \cyl(g),p(a_{1})} %
%
Moreover appealing to the definition of $g$ the following diagram in $\cl{A}$ commutes.   

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=3 em, row sep=4 em, nodes={anchor=center}]
{ 
|(0-0)| \cyl^{2}(a_{0}) \&                         \& |(2-0)| \cyl\big(a^{\cylinder}_{f}\big) \\
|(0-1)| \cyl(a_{0})     \& |(1-1)| \cyl(a_{0}) \& |(2-1)| \cyl(a_{1})     \\
|(0-2)| a_{0}           \&                         \& |(2-2)| a_{1}    \\
};
	
\draw[->] (0-0) to node[auto] {$\cyl(d^{0}_{f})$} (2-0);
\draw[->] (2-0) to node[auto] {$\cyl(g)$} (2-1);
\draw[->] (0-0) to node[auto] {$\cyl\big(p(a_{0})\big)$} (1-1);
\draw[->] (1-1) to node[auto] {$\cyl(f)$} (2-1);
\draw[->] (1-1) to node[auto] {$p(a_{0})$} (0-2);
\draw[->] (0-2) to node[auto,swap] {$f$} (2-2);
\draw[->] (2-1) to node[auto] {$p(a_{1})$} (2-2);
\draw[->] (0-0) to node[auto,swap] {$p\big(\cyl(a_{0})\big)$} (0-1);
\draw[->] (0-1) to node[auto,swap] {$p(a_{0})$} (0-2);
\end{tikzpicture} 

\end{diagram} Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \pushout[{5,3,-1}]{\cyl(a_{0}),\cyl^{2}(a_{0}),\cyl(a_{1}),\cyl\big(a^{\cylinder}_{f}\big),a_{1},\cyl\big(i_{0}(a_{0})\big),\cyl(d^{0}_{f}),\cyl(j),\cyl(d^{1}_{f}),f \circ p(a_{0}) \circ p\big(\cyl(a_{0})\big),p(a_{1}),p(a_{1}) \circ \cyl(g)} %
%
Appealing to the universal property of $\cyl\big(a^{\cylinder}_{f}\big)$ we deduce that the following diagram in $\cl{A}$ commutes. \sq{\cyl\big(a^{\cylinder}_{f}\big),a^{\cylinder}_{f},\cyl(a_{1}),a_{1},h,g,\cyl(g),p(a_{1})} This completes the proof of the claim.

\end{proof}

\begin{cor} \label{MappingCylinderFactorisationStrongDeformationRetractionWithRespectToCoCylinderCorollary} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p, \Gamma_{lr} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$ and a lower right connection structure $\Gamma_{lr}$. Suppose that $\Gamma_{lr}$ is compatible with $p$. Let $\cocylinder = \big( \cocyl, e_{0}, e_{1}, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. Suppose that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$, and let $\big( a^{\cylinder}_{f}, d^{0}_{f}, d^{1}_{f} \big)$ be a mapping cylinder of $f$ with respect to $\cylinder$. Let \tri{a_{0},a^{\cylinder}_{f},a_{1},j,g,f} denote the corresponding mapping cylinder factorisation of $f$. Then $g$ is a strong deformation retraction of $d^{1}_{f}$ with respect to $\cocylinder$. \end{cor}

\begin{proof} Since $\cyl$ is left adjoint to $\cocyl$ we have that $\cyl$ preserves mapping cylinders with respect to $\cylinder$. Thus the result follows immediately from Proposition \ref{MappingCylinderFactorisationStrongDeformationRetractionWithRespectToCylinderProposition} and Proposition \ref{StrongDeformationRetractionCoCylinderCharacterisationProposition}. \end{proof}

\begin{lem} \label{NormallyClovenCofibrationCriterionLemma} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p, \Gamma_{lr} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$ and a lower right connection structure $\Gamma_{lr}$. Suppose that $\Gamma_{lr}$ is compatible with $p$, and that $\cyl$ preserves mapping cylinders with respect to $\cylinder$. 

Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$, and let $\big( a^{\cylinder}_{f},d^{0}_{f},d^{1}_{f} \big)$ be a mapping cylinder of $f$ with respect to $\cylinder$. Let \tri{a_{0},a^{\cylinder}_{f},a_{1},j,g,f} denote the corresponding mapping cylinder factorisation of $f$. 

Suppose that there is an arrow \ar{a_{1},a^{\cylinder}_{f},l} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \liftingsquare{a_{0},a^{\cylinder}_{f},a_{1},a_{1},j,g,f,id,l} Then $f$ is a normally cloven cofibration with respect to $\cylinder$. \end{lem} 

\begin{proof} We first prove that $f$ is a cofibration with respect to $\cylinder$. Suppose that we have a commutative diagram in $\cl{A}$ as follows. \sq{a_{0},\cyl(a_{0}),a_{1},a_{2},i_{0}(a_{0}),k,f,f'} Let \ar{\cyl\big(a^{\cylinder}_{f}\big),a^{\cylinder}_{f},h} denote the canonical arrow of $\cl{A}$ constructed as in the proof of Proposition \ref{MappingCylinderFactorisationStrongDeformationRetractionWithRespectToCylinderProposition}. We have that the following diagram in $\cl{A}$ commutes. \pushout[{5,3,-1}]{\cyl(a_{0}),\cyl^{2}(a_{0}),\cyl(a_{1}),\cyl\big(a^{\cylinder}_{f}\big),a^{\cylinder}_{f},\cyl\big(i_{0}(a_{0})\big),\cyl(d^{0}_{f}),\cyl(f),\cyl(d^{1}_{f}),d^{0}_{f} \circ \Gamma_{lr}(a_{0}),d^{1}_{f} \circ p(a_{1}),h} %
%
Let \ar{a^{\cylinder}_{f},a_{1},u} denote the canonical arrow of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),a_{1},a^{\cylinder}_{f},a_{2},i_{0}(a_{0}),d^{0}_{f},f,d^{1}_{f},k,f',u} By definition of $l$ we have that the following diagram in $\cl{A}$ commutes. \triother{a_{0},a_{1},a^{\cylinder}_{f},f,l,j} Hence the following diagram in $\cl{A}$ commutes. \triother[{4,3}]{\cyl(a_{0}),\cyl(a_{1}),\cyl\big(a^{\cylinder}_{f}\big),\cyl(f),\cyl(l),\cyl(j)} %
%
Moreover by definition of $j$ we have that the following diagram in $\cl{A}$ commutes. \tri[{5,3}]{\cyl(a_{0}),\cyl^{2}(a_{0}),\cyl\big(a^{\cylinder}_{f}\big),\cyl\big(i_{1}(a_{0})\big),\cyl(d^{0}_{f}),\cyl(j)} Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{\cyl(a_{0}),\cyl^{2}(a_{0}),\cyl(a_{1}),\cyl\big(a^{\cylinder}_{f}\big),\cyl\big(i_{1}(a_{0})\big),\cyl(d^{0}_{f}),\cyl(f),\cyl(l)} %
%
By definition of $h$ and $u$ we also have that the following diagram in $\cl{A}$ commutes. \squareabovetriangle[{4,3,0}]{\cyl^{2}(a_{0}),\cyl\big(a^{\cylinder}_{f}\big),\cyl(a_{0}),a^{\cylinder}_{f},a_{2},\cyl(d^{0}_{f}),h,\Gamma_{lr}(a_{0}),d^{0}_{f},u,k} Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \trapeziums[{3,{3.5},1,0}]{\cyl(a_{0}),\cyl(a_{1}),\cyl^{2}(a_{0}),\cyl\big(a^{\cylinder}_{f}\big),\cyl(a_{0}),a_{2},\cyl(f),\cyl(l),u \circ h,id,k,\cyl\big(i_{1}(a_{0})\big),\cyl(d^{0}_{f}),\Gamma_{lr}(a_{0})} %
%
As in the proof of Proposition \ref{MappingCylinderFactorisationStrongDeformationRetractionWithRespectToCylinderProposition} we also have that the following diagram in $\cl{A}$ commutes. \sq[{5,3}]{a^{\cylinder}_{f},\cyl\big(a^{\cylinder}_{f}\big),a_{1},a^{\cylinder}_{f},i_{0}\big(a^{\cylinder}_{f}\big),h,g,d^{1}_{f}} %
%
Thus appealing to the definition of $u$ we have that the following diagram in $\cl{A}$ commutes. \squareabovetriangle[{5,3,0}]{a^{\cylinder}_{f},\cyl\big(a^{\cylinder}_{f}\big),a_{1},a^{\cylinder}_{f},a_{2},i_{0}\big(a^{\cylinder}_{f}\big),h,g,d^{1}_{f},u,f'} %
%
Moreover by definition of $l$ we have that the following diagram in $\cl{A}$ commutes. \tri{a_{1},a^{\cylinder}_{f},a_{1},l,g,id} Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \trapeziums[{3,{3.5},1,0}]{a_{1},\cyl(a_{1}),a^{\cylinder}_{f},\cyl\big(a^{\cylinder}_{f}\big),a_{1},a_{2},i_{0}(a_{1}),\cyl(l),u \circ h,id,f',l,i_{0}\big(a^{\cylinder}_{f}\big),g} %
%
We have now shown that the following diagram in $\cl{A}$ commutes. \pushout[{4,3,2}]{a_{0},\cyl(a_{0}),a_{1},\cyl(a_{1}),a_{2},i_{0}(a_{0}),\cyl(f),f,i_{0}(a_{1}),k,f',u \circ h \circ \cyl(l)} This completes our proof that $f$ is a cofibration with respect to $\cylinder$. 

We claim moreover that the cleavage which associates to an object $a_{2}$ and a pair of arrows $(f',g)$ of $\cl{A}$ as above the arrow \ar[7]{\cyl(a_{1}),a_{2},f' \circ h \circ \cyl(l)} of $\cl{A}$ equips $f$ with the structure of a normally cloven cofibration with respect to $\cylinder$. 

Let us prove that this cleavage has property (i) of Definition \ref{NormallyClovenCofibrationCylinderDefinition}. Suppose that we have a commutative diagram in $\cl{A}$ as follows. \tri{\cyl(a_{0}),a_{0},a_{2},p(a_{0}),f'',k} By definition of $u$ we then have that the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{0}),a^{\cylinder}_{f},a_{0},a_{2},d^{0}_{f},u,p(a_{0}),f''} %
%
Since $\Gamma_{lr}$ is compatible with $p$ the following diagram in $\cl{A}$ also commutes. \sq[{4,3}]{\cyl^{2}(a_{0}),\cyl(a_{0}),\cyl(a_{0}),a_{0},\Gamma_{lr}(a_{0}),p(a_{0}),p\big(\cyl(a_{0})\big),p(a_{0})} Putting the last two observations together and appealing to the definition of $h$ we have that the following diagram in $\cl{A}$ commutes. \trapeziumstwo[{3,3,0,1}]{\cyl^{2}(a_{0}),\cyl\big(a^{\cylinder}_{f}\big),\cyl(a_{0}),\cyl(a_{0}),a^{\cylinder}_{f},a_{0},a_{2},\cyl(d^{0}_{f}),h,u,p\big(\cyl(a_{0})\big),p(a_{0}),f'',\Gamma_{lr}(a_{0}),d^{0}_{f},p(a_{0})} %
%
By definition of $h$ and $u$ the following diagram in $\cl{A}$ also commutes. \squareabovetriangle[{4,3,0}]{\cyl(a_{1}),\cyl\big(a^{\cylinder}_{f}\big),a_{1},a^{\cylinder}_{f},a_{2},\cyl(d^{1}_{f}),h,p(a_{1}),d^{1}_{f},u,f'} %
%
Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \pushout[{5,3,-1}]{\cyl(a_{0}),\cyl^{2}(a_{0}),\cyl(a_{1}),\cyl\big(a^{\cyl}_{f}\big),a_{2},\cyl\big(i_{0}(a_{0})\big),\cyl(d^{0}_{f}),\cyl(f),\cyl(d^{1}_{f}),f'' \circ p(a_{0}) \circ p\big(\cyl(a_{0})\big),f' \circ p(a_{1}),u \circ h} %
%
We also have that the following diagram in $\cl{A}$ commutes. \squareofthreetriangles[{3,4,0,0}]{a_{0},a_{1},\cyl(a_{0}),a_{0},a_{2},f,f',id,f'',i_{0}(a_{0}),p(a_{0}),k} Hence the following diagram in $\cl{A}$ commutes. \squareabovetriangle{\cyl(a_{0}),\cyl(a_{1}),a_{0},a_{1},a_{2},\cyl(f),p(a_{1}),p(a_{0}),f,f',f''} %
%
By definition of $g$ the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{\cyl^{2}(a_{0}),\cyl\big(a^{\cylinder}_{f}\big),\cyl(a_{0}),\cyl(a_{1}),\cyl(d^{0}_{f}),\cyl(g),\cyl\big(p(a_{0})\big),\cyl(f)} Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \trapeziumstwo[{3,3,0,1}]{\cyl^{2}(a_{0}),\cyl\big(a^{\cylinder}_{f}\big),\cyl(a_{0}),\cyl(a_{0}),\cyl(a_{1}),a_{0},a_{2},\cyl(d^{0}_{f}),\cyl(g),f' \circ p(a_{1}),p\big(\cyl(a_{0})\big),p(a_{0}),f'',\cyl\big(p(a_{0})\big),\cyl(f),p(a_{0})} %
%
Moreover by definition of $g$ the following diagram in $\cl{A}$ commutes. \tri[{4,3}]{\cyl(a_{1}),\cyl\big(a^{\cylinder}_{f}\big),\cyl(a_{1}),\cyl(d^{1}_{f}),\cyl(g),id} Thus the following diagram in $\cl{A}$ commutes. \tri[{4,3}]{\cyl(a_{1}),\cyl\big(a^{\cylinder}_{f}\big),a_{2},\cyl(d^{1}_{f}),f' \circ p(a_{1}) \circ \cyl(g),f' \circ p(a_{1})} %
%
We have now shown that the following diagram in $\cl{A}$ commutes.  \pushout[{5,3,-1}]{\cyl(a_{0}),\cyl^{2}(a_{0}),\cyl(a_{1}),\cyl\big(a^{\cyl}_{f}\big),a_{2},\cyl\big(i_{0}(a_{0})\big),\cyl(d^{0}_{f}),\cyl(f),\cyl(d^{1}_{f}),f'' \circ p(a_{0}) \circ p\big(\cyl(a_{0})\big),f' \circ p(a_{1}),f' \circ p(a_{1}) \circ \cyl(g)} %
%
Appealing to the universal property which $\cyl\big(a^{\cylinder}_{f}\big)$ possesses due to our assumption that $\cyl$ preserves mapping cylinders we deduce that the following diagram in $\cl{A}$ commutes. \tri{\cyl\big(a^{\cylinder}_{f}\big),a^{\cylinder}_{f},a_{2},h,u,f' \circ p(a_{1}) \circ \cyl(g)} %
%
Moreover by definition of $l$ the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{1}),\cyl\big(a^{\cylinder}_{f}\big),\cyl(a_{1}),\cyl(l),\cyl(g),id} Putting the last two observations together, we have that the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{1}),\cyl\big(a^{\cylinder}_{f}\big),a_{2},\cyl(l),u \circ h,f' \circ p(a_{1})} 

This completes the proof that the cleavage satisfies property (i) of Definition \ref{NormallyClovenCofibrationCylinderDefinition}. That it moreover satisfies property (ii) of Definition \ref{NormallyClovenCofibrationCylinderDefinition} is clear. 
\end{proof}

\begin{lem} \label{MappingCylinderFactorisationGivesArrowSatisfyingNormallyClovenCofibrationCriterionLemma} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p, \subdiv, r_{0}, r_{1}, s \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$ and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$. Suppose that $\cylinder$ has strictness of right identities. 

Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$, and let $\big( a^{\cylinder}_{f},d^{0}_{f},d^{1}_{f} \big)$ be a mapping cylinder of $f$ with respect to $\cylinder$. Let \tri{a_{0},a^{\cylinder}_{f},a_{1},j,g,f} denote the corresponding mapping cylinder factorisation of $f$. 

Let $\big( a^{\cylinder}_{j},d^{0}_{j},d^{1}_{j} \big)$ be a mapping cylinder of $j$ with respect to $\cylinder$, and let \tri{a_{0},a^{\cylinder}_{j},a^{\cylinder}_{f},j',g',j} denote the corresponding mapping cylinder factorisation of $j$. Then there is an arrow \ar{a^{\cylinder}_{f},a^{\cylinder}_{j},l} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \liftingsquare{a_{0},a^{\cylinder}_{j},a^{\cylinder}_{f},a^{\cylinder}_{f},j',g',j,id,l} \end{lem}

\begin{proof} Since $\big( a^{\cylinder}_{j}, d^{0}_{j}, d^{1}_{j} \big)$ defines a mapping cylinder of $j$ with respect to $\cylinder$ the following diagram in $\cl{A}$ commutes. \sq{a_{0},\cyl(a_{0}),a^{\cylinder}_{f},a^{\cylinder}_{j},i_{0}(a_{0}),d^{0}_{j},j,d^{1}_{j}} %
%
By definition of $j$ we also have that the following diagram in $\cl{A}$ commutes. \tri{a_{0},\cyl(a_{0}),a^{\cylinder}_{f},i_{1}(a_{0}),d^{0}_{f},j} Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{a_{0},\cyl(a_{0}),\cyl(a_{0}),a^{\cylinder}_{j},i_{0}(a_{0}),d^{0}_{j},i_{1}(a_{0}),d^{1}_{j} \circ d^{0}_{f}} %
%
Thus there is a canonical arrow \ar{\subdiv(a_{0}),a^{\cylinder}_{j},u} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),\cyl(a_{0}),\subdiv(a_{0}),a^{\cylinder}_{j},i_{0}(a_{0}),r_{0}(a_{0}),i_{1}(a_{0}),r_{1}(a_{0}),d^{0}_{j},d^{1}_{j} \circ d^{0}_{f},u}  By definition of $g'$ the following diagram in $\cl{A}$ commutes. \sq{\cyl(a_{0}),a^{\cylinder}_{j},a_{0},a^{\cylinder}_{f},d^{0}_{j},g',p(a_{0}),j} %
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Appealing again to the commutativity by definition of $j$ of the diagram \triother{a_{0},\cyl(a_{0}),a^{\cylinder}_{f},i_{1}(a_{0}),d^{0}_{f},j} in $\cl{A}$ we thus have that the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{0}),a^{\cylinder}_{j},a^{\cylinder}_{f},d^{0}_{j},g',d^{0}_{f} \circ i_{1}(a_{0}) \circ p(a_{0})} %
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Appealing to the commutativity by definition of $u$ of the diagram \tri[{4,3}]{\cyl(a_{0}),\subdiv(a_{0}),a^{\cylinder}_{j},r_{0}(a_{0}),u,d^{0}_{j}} in $\cl{A}$ we deduce that the following diagram in $\cl{A}$ commutes. \tri[{4,3}]{\cyl(a_{0}),\subdiv(a_{0}),a^{\cylinder}_{f},r_{0}(a_{0}),g' \circ u, d^{1}_{f} \circ i_{1}(a_{0}) \circ p(a_{0})} %
%
We also have that the following diagram in $\cl{A}$ commutes by definition of $u$ and $g'$. \squareabovetriangle[{4,3,0,0}]{\cyl(a_{0}),\subdiv(a_{0}),a^{\cylinder}_{f},a^{\cylinder}_{j},a^{\cylinder}_{f},r_{1}(a_{0}),u,d^{0}_{f},d^{1}_{j},g',id} %
%
Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),\cyl(a_{0}),\subdiv(a_{0}),a^{\cylinder}_{f},i_{0}(a_{0}),r_{0}(a_{0}),i_{1}(a_{0}),r_{1}(a_{0}),d^{0}_{f} \circ i_{1}(a_{0}) \circ p(a_{0}), d^{0}_{f}, g' \circ u} %
%
Let \ar{\subdiv,\cyl,q_{r}} denote the canonical 2-arrow of $\cl{C}$ constructed as in Definition \ref{StrictnessLeftIdentitiesCylinderDefinition}. We have that the following diagram in $\cl{A}$  commutes. \pushout{a_{0},\cyl(a_{0}),\cyl(a_{0}),\subdiv(a_{0}),\cyl(a_{0}),i_{0}(a_{0}),r_{0}(a_{0}),i_{1}(a_{0}),r_{1}(a_{0}),i_{1}(a_{0}) \circ p(a_{0}),id,q_{r}(a_{0})} %
%
Then the following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),\cyl(a_{0}),\subdiv(a_{0}),a^{\cylinder}_{f},i_{0}(a_{0}),r_{0}(a_{0}),i_{1}(a_{0}),r_{1}(a_{0}),d^{0}_{f} \circ i_{1}(a_{0}) \circ p(a_{0}),d^{0}_{f},d^{0}_{f} \circ q_{r}(a_{0})} %
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Appealing to the universal property of $\subdiv(a_{0})$ we deduce that the following diagram in $\cl{A}$ commutes. \sq{\subdiv(a_{0}),a^{\cylinder}_{j},\cyl(a_{0}),a^{\cylinder}_{f},u,g',q_{r}(a_{0}),d^{0}_{f}} By definition of the homotopy $(d^{1}_{j} \circ d^{0}_{f}) + d^{0}_{j}$ with respect to $\cylinder$, we have that the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{0}),\subdiv(a_{0}),a^{\cylinder}_{j},s(a_{0}),u,{(d^{1}_{j} \circ d^{0}_{f}) + d^{0}_{j}}} %
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The following diagram in $\cl{A}$ also commutes since $\cylinder$ has strictness of right identities. \tri{\cyl(a_{0}),\subdiv(a_{0}),\cyl(a_{0}),s(a_{0}),q_{r}(a_{0}),id} Putting the last three observations together we have that the following diagram in $\cl{A}$ commutes. \tri[{7,3}]{\cyl(a_{0}),a^{\cylinder}_{j},a^{\cylinder}_{f},{(d^{1}_{j} \circ d^{0}_{f}) + d^{0}_{j}},g',d^{0}_{f}} %
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Since $\big( a^{\cylinder}_{f}, d^{0}_{f}, d^{1}_{f} \big)$ defines a mapping cylinder of $f$ with respect to $\cylinder$ the following diagram in $\cl{A}$ commutes. \sq{a_{0},\cyl(a_{0}),a_{1},a^{\cylinder}_{f},i_{0}(a_{0}),d^{0}_{f},f,d^{1}_{f}} %
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Hence the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{a_{0},\cyl(a_{0}),a_{1},a^{\cylinder}_{j},i_{0}(a_{0}),d^{1}_{j} \circ d^{0}_{f},f,d^{1}_{j} \circ d^{1}_{f}} We deduce that the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{a_{0},\cyl(a_{0}),a_{1},a^{\cylinder}_{j},i_{0}(a_{0}),{(d^{1}_{j} \circ d^{0}_{f}) + d^{0}_{j}},f,d^{1}_{j} \circ d^{1}_{f}} %
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Thus there is a canonical arrow \ar{a^{\cylinder}_{f},a^{\cylinder}_{j},l} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),a_{1},a^{\cylinder}_{f},a^{\cylinder}_{j},i_{0}(a_{0}),d^{0}_{f},f,d^{1}_{f},{(d^{1}_{j} \circ d^{0}_{f}) + d^{0}_{j}},d^{1}_{j} \circ d^{1}_{f},l} 

We claim that $l$ fits into a commutative diagram as in the statement of the proposition. Firstly it follows from the commutativity of the diagrams \tri{\cyl(a_{0}),a^{\cylinder}_{f},a^{\cylinder}_{j},d^{0}_{f},l,{(d^{1}_{j} \circ d^{0}_{f}) + d^{0}_{j}}} and \tri[{7,3}]{\cyl(a_{0}),a^{\cylinder}_{j},a^{\cylinder}_{f},{(d^{1}_{j} \circ d^{0}_{f}) + d^{0}_{j}},g',d^{0}_{f}} in $\cl{A}$ that the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{0}),a^{\cylinder}_{f},a^{\cylinder}_{f},d^{0}_{f},g' \circ l,d^{0}_{f}} %
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The following diagram in $\cl{A}$ also commutes by definition of $l$ and $g'$. \squareabovetriangle{a_{1},a^{\cylinder}_{f},a^{\cylinder}_{f},a^{\cylinder}_{j},a^{\cylinder}_{j},d^{1}_{f},l,d^{1}_{f},d^{1}_{j},g',id} %
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Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \pushout{a_{0},\cyl(a_{0}),a_{1},a^{\cylinder}_{f},a^{\cylinder}_{f},i_{0}(a_{0}),d^{0}_{f},f,d^{1}_{f},d^{0}_{f},d^{1}_{f},g' \circ l} Appealing to the universal property of $a^{\cylinder}_{f}$ we deduce that the following diagram in $\cl{A}$ commutes. \tri{a^{\cylinder}_{f},a^{\cylinder}_{j},a^{\cylinder}_{f},l,g',id} 

Secondly the following diagram in $\cl{A}$ commutes by definition of $j'$. \tri{a_{0},\cyl(a_{0}),a^{\cylinder}_{f},i_{1}(a_{0}),d^{0}_{j},j'} Hence the following diagram in $\cl{A}$ commutes. \tri{\cyl(a_{0}),\cyl(a_{0}),a^{\cylinder}_{j},i_{1}(a_{0}),{(d^{1}_{j} \circ d^{0}_{f}) + d^{0}_{j}},j'} %
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By definition of $j$ and $l$ we also have that the following diagram in $\cl{A}$ commutes. \threetriangles{a_{0},\cyl(a_{0}),\cyl(a_{0}),a^{\cylinder}_{j},j,l,i_{1}(a_{0}),d^{0}_{f},{(d^{1}_{j} \circ d^{0}_{f}) + d^{0}_{j}}} Putting the last two observations together we have that the following diagram in $\cl{A}$ commutes. \tri{a_{0},\cyl(a_{0}),a^{\cylinder}_{j},j,l,j'} This completes the proof of the claim.    
\end{proof}

\begin{prpn} \label{MappingCylinderFactorisationGivesNormallyClovenCofibrationProposition} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p, \subdiv, r_{0}, r_{1}, s, \Gamma_{lr} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, a subdivision structure $\big(\subdiv,r_{0},r_{1},s\big)$, and a lower right connection structure $\Gamma_{lr}$. Suppose that $\Gamma_{lr}$ is compatible with $p$, and that $\cylinder$ has strictness of right identities. Suppose moreover that $\cyl$ preserves mapping cylinders with respect to $\cylinder$. 

Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$, and let $\big( a^{\cylinder}_{f},d^{0}_{f},d^{1}_{f} \big)$ be a mapping cylinder of $f$ with respect to $\cylinder$. 

Let \tri{a_{0},a^{\cylinder}_{f},a_{1},j,g,f} denote the corresponding mapping cylinder factorisation of $f$. Then $j$ is a normally cloven cofibration with respect to $\cylinder$. \end{prpn} 

\begin{proof} Follows immediately from Lemma \ref{NormallyClovenCofibrationCriterionLemma} and Lemma \ref{MappingCylinderFactorisationGivesArrowSatisfyingNormallyClovenCofibrationCriterionLemma}. \end{proof}

\begin{cor} Let $\cocylinder = \big( \cocyl, e_{0}, e_{1}, c, \subdiv, r_{0}, r_{1},s,\Gamma_{lr} \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$, a subdivision structure $\big(\subdiv, r_{0}, r_{1}, s\big)$, and a lower right connection structure $\Gamma_{lr}$. Suppose that $\Gamma_{lr}$ is compatible with $c$, and that $\cocylinder$ has strictness of right identities. Suppose moreover that $\cocyl$ preserves mapping co-cylinders with respect to $\cocylinder$. 

Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$, and let $\big( a^{\cocylinder}_{f},d^{0}_{f},d^{1}_{f} \big)$ be a mapping co-cylinder of $f$ with respect to $\cocylinder$. 

Let \tri{a_{0},a^{\cocylinder}_{f},a_{1},j,g,f} denote the corresponding mapping co-cylinder factorisation of $f$. Then $g$ is a normally cloven fibration with respect to $\cocylinder$. \end{cor}

\begin{proof} Follows immediately from Proposition \ref{MappingCylinderFactorisationGivesNormallyClovenCofibrationProposition} by duality. \end{proof}

\end{section}

model_category_recollections.tex
\begin{section}{Model category recollections} \label{ModelCategoryRecollectionsSection} 

In \ref{ModelStructureSection} we shall bring together all the theory we have developed so far. In order to do so we now present a few recollections on model categories. 

The notion of a model category was introduced by Quillen in \cite{QuillenHomotopicalAlgebra}. We recall two definitions and prove that they are equivalent. Our arguments comprise part of the proof of Proposition 2 of \S{5} of \cite{QuillenHomotopicalAlgebra}. Our definitions are equivalent to the definition of a closed model category given in \S{5} of \cite{QuillenHomotopicalAlgebra}.

\begin{defn} \label{ModelStructureDefinition} Let $\cl{A}$ be a category with finite limits and colimits.  A {\em model structure} upon $\cl{A}$ consists of three sets $W$, $F$, and $C$ of arrows of $\cl{A}$ such that the following conditions are satisfied. 

\begin{itemize}[topsep=1em,itemsep=1em]

\item[(i)] If any two of the arrows in a commutative diagram \tri{a_{0},a_{1},a_{2},g_{0},g_{1},g_{2}} in $\cl{A}$ belong to $W$ so does the third. 

\item[(ii)] An arrow \ar{a_{2},a_{3},f} of $\cl{A}$ belongs to $F$ if and only if for every commutative diagram \sq{a_{0},a_{1},a_{2},a_{3},g_{0},f,j,g_{1}} in $\cl{A}$ such that $j$ belongs to both $W$ and $C$ there is an arrow \ar{a_{2},a_{1},l} of $\cl{A}$ such that the diagram \liftingsquare{a_{0},a_{2},a_{1},a_{3},g_{0},f,j,g_{1},l} in $\cl{A}$ commutes.

\item[(iii)] An arrow \ar{a_{2},a_{3},f} of $\cl{A}$ belongs to both $F$ and $W$ if and only if for every commutative diagram \sq{a_{0},a_{1},a_{2},a_{3},g_{0},f,j,g_{1}} in $\cl{A}$ such that $j$ belongs to $C$ there is an arrow \ar{a_{2},a_{1},l} of $\cl{A}$ such that the diagram \liftingsquare{a_{0},a_{2},a_{1},a_{3},g_{0},f,j,g_{1},l} in $\cl{A}$ commutes.

\item[(iv)] An arrow \ar{a_{0},a_{1},j} of $\cl{A}$ belongs to $C$ if and only if for every commutative diagram \sq{a_{0},a_{1},a_{2},a_{3},g_{0},f,j,g_{1}} in $\cl{A}$ such that $f$ belongs to both $F$ and $W$ there is an arrow \ar{a_{2},a_{1},l} of $\cl{A}$ such that the diagram \liftingsquare{a_{0},a_{2},a_{1},a_{3},g_{0},f,j,g_{1},l} in $\cl{A}$ commutes.

\item[(v)] An arrow \ar{a_{0},a_{1},j} belongs to both $C$ and $W$ if and only if for every commutative diagram \sq{a_{0},a_{1},a_{2},a_{3},g_{0},f,j,g_{1}} in $\cl{A}$ such that $f$ belongs to $F$ there is an arrow \ar{a_{2},a_{1},l} of $\cl{A}$ such that the diagram \liftingsquare{a_{0},a_{2},a_{1},a_{3},g_{0},f,j,g_{1},l} in $\cl{A}$ commutes.

\item[(vi)] For every arrow \ar{a_{0},a_{1},f} of $\cl{A}$ there is an arrow \ar{a_{0},a_{2},j} of $\cl{A}$ which belongs to $C$ and an arrow \ar{a_{2},a_{1},g} of $\cl{A}$ which belongs to $W$ and $F$ such that the following diagram in $\cl{A}$ commutes. \tri{a_{0},a_{2},a_{1},j,g,f}

\item[(vii)] For every arrow \ar{a_{0},a_{1},f} of $\cl{A}$ there is an arrow \ar{a_{0},a_{2},j} of $\cl{A}$ which belongs to $W$ and $C$ and an arrow \ar{a_{2},a_{1},g} of $\cl{A}$ which belongs to $F$ such that the following diagram in $\cl{A}$ commutes. \tri{a_{0},a_{2},a_{1},j,g,f}

\end{itemize}

\end{defn}

\begin{defn} Let $\cl{A}$ be a category with finite limits and colimits. Let $W$, $F$, and $C$ be sets of arrows of $\cl{A}$ which equip $\cl{A}$ with a model structure. We refer to an arrow of $\cl{A}$ which belongs to $W$ as a {\em weak equivalence}, to an arrow of $\cl{A}$ which belongs to $F$ as a {\em fibration}, and to an arrow of $\cl{A}$ which belongs to $C$ as a {\em cofibration}. We refer to an arrow of $\cl{A}$ which belongs to both $W$ and $C$ as a {\em trivial cofibration}, and to an arrow of $\cl{A}$ which belongs to both $W$ and $F$ as a {\em trivial fibration}. \end{defn}

\begin{defn} A {\em model category} is a category $\cl{A}$ which has finite limits and colimits together with a model structure upon $\cl{A}$. \end{defn}

\begin{prpn} \label{EquivalentDefinitionsOfModelStructureProposition} Let $\cl{A}$ be a category with finite limits and colimits. Let $W$, $F$, $C$ be sets of arrows of $\cl{A}$. Then $(W,F,C)$ equips $\cl{A}$ with a model structure if and only if the following conditions are satisfied. 

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] If any two of the arrows in a commutative diagram \tri{a_{0},a_{1},a_{2},g_{0},g_{1},g_{2}} in $\cl{A}$ belong to $W$ so does the third. 

\item[(ii)] Suppose that we have commutative diagrams \twosq{a_{2},a_{0},a_{3},a_{1},g_{0},j,j',g_{1},a_{0},a_{2},a_{1},a_{3},r_{0},j',j,r_{1}} in $\cl{A}$ such that $r_{0}$ is a retraction of $g_{0}$, and such that $r_{1}$ is a retraction of $g_{1}$. If $j$ belongs to $C$, then $j'$ belongs to $C$. If $j$ belongs to both $C$ and $W$, then $j'$ belongs to both $C$ and $W$.

\item[(iii)] Suppose that we have commutative diagrams \twosq{a_{2},a_{0},a_{3},a_{1},g_{0},f,f',g_{1},a_{0},a_{2},a_{1},a_{3},r_{0},f',f,r_{1}} in $\cl{A}$ such that $r_{0}$ is a retraction of $g_{0}$, and such that $r_{1}$ is a retraction of $g_{1}$. If $f$ belongs to $F$ then $f'$ belongs to $F$. If $f$ belongs to both $F$ and $W$ then $f'$ belongs to both $F$ and $W$.

\item[(iv)] For every diagram \sq{a_{0},a_{1},a_{2},a_{3},g_{0},f,j,g_{1}} in $\cl{A}$ such that $j$ belongs to both $W$ and $C$ and $f$ belongs to $F$ there is an arrow \ar{a_{2},a_{1},l} of $\cl{A}$ such that the diagram \liftingsquare{a_{0},a_{1},a_{2},a_{3},g_{0},f,j,g_{1},l} in $\cl{A}$ commutes.

\item[(v)] For every diagram \sq{a_{0},a_{1},a_{2},a_{3},g_{0},f,j,g_{1}} in $\cl{A}$ such that $j$ belongs to $C$ and $f$ belongs to $W$ and $F$ there is an arrow \ar{a_{2},a_{1},l} of $\cl{A}$ such that the diagram \liftingsquare{a_{0},a_{1},a_{2},a_{3},g_{0},f,j,g_{1},l} in $\cl{A}$ commutes.

\item[(vi)] For every arrow \ar{a_{0},a_{1},f} of $\cl{A}$ there is an arrow \ar{a_{0},a_{2},j} of $\cl{A}$ which belongs to $C$ and an arrow \ar{a_{2},a_{1},g} of $\cl{A}$ which belongs to $W$ and $F$ such that the following diagram in $\cl{A}$ commutes. \tri{a_{0},a_{2},a_{1},j,g,f}

\item[(vii)] For every arrow \ar{a_{0},a_{1},f} of $\cl{A}$ there is an arrow \ar{a_{0},a_{2},j} of $\cl{A}$ which belongs to $W$ and $C$ and an arrow \ar{a_{2},a_{1},g} of $\cl{A}$ which belongs to $F$ such that the following diagram in $\cl{A}$ commutes. \tri{a_{0},a_{2},a_{1},j,g,f}

\end{itemize}

\end{prpn}

\begin{proof} We first prove that if the conditions of Proposition \ref{EquivalentDefinitionsOfModelStructureProposition} are satisfied, then $(W,F,C)$ equips $\cl{A}$ with a model structure. Let us demonstrate that condition (ii) of Definition \ref{ModelStructureDefinition} holds. 

Given that condition (iv) of Proposition \ref{EquivalentDefinitionsOfModelStructureProposition} holds it suffices to show that if \ar{a_{2},a_{3},f} is an arrow of $\cl{A}$ with the property that for every commutative diagram \sq{a_{0},a_{1},a_{2},a_{3},g_{0},f,j,g_{1}} in $\cl{A}$ such that $j$ belongs to both $W$ and $C$ there is an arrow \ar{a_{2},a_{1},l} of $\cl{A}$ such that the diagram \liftingsquare{a_{0},a_{2},a_{1},a_{3},g_{0},f,j,g_{1},l} in $\cl{A}$ commutes then $f$ belongs to $F$.

By condition (vii) of Proposition \ref{EquivalentDefinitionsOfModelStructureProposition} there is an arrow \ar{a_{2},a_{4},j'} of $\cl{A}$ which belongs to both $W$ and $C$ and an arrow \ar{a_{4},a_{3},f'} of $\cl{A}$ which belongs to $F$ such that the following diagram in $\cl{A}$ commutes. \tri{a_{2},a_{4},a_{3},j',f',f} %
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By assumption there is an arrow \ar{a_{4},a_{2},l'} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \liftingsquare{a_{2},a_{2},a_{4},a_{3},id,f,j',f',l'} %
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In other words we have a pair of commutative diagrams in $\cl{A}$ as follows such that $l'$ is a retraction of $j'$. \twosq{a_{2},a_{4},a_{3},a_{3},j',f',f,id,a_{4},a_{2},a_{3},a_{3},l',f,f',id} %
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Appealing to condition (iii) of Proposition \ref{EquivalentDefinitionsOfModelStructureProposition} we deduce that $f$ belongs to $F$. 

Next let us demonstrate that condition (iii) of Definition \ref{ModelStructureDefinition} holds. Given that condition (v) of Proposition \ref{EquivalentDefinitionsOfModelStructureProposition} holds it suffices to show that if \ar{a_{2},a_{3},f} is now an arrow of $\cl{A}$ with the property that for every commutative diagram \sq{a_{0},a_{1},a_{2},a_{3},g_{0},f,j,g_{1}} in $\cl{A}$ such that $j$ belongs to $C$, there is an arrow \ar{a_{2},a_{1},l} of $\cl{A}$ such that the diagram \liftingsquare{a_{0},a_{2},a_{1},a_{3},g_{0},f,j,g_{1},l} in $\cl{A}$ commutes then $f$ belongs to both $W$ and $F$.

By condition (vi) of Proposition \ref{EquivalentDefinitionsOfModelStructureProposition} there is an arrow \ar{a_{2},a_{4},j'} of $\cl{A}$ which belongs to $C$ and an arrow \ar{a_{4},a_{3},f'} of $\cl{A}$ which belongs to both $F$ and $W$ such that the following diagram in $\cl{A}$ commutes. \tri{a_{2},a_{4},a_{3},j',f',f} %
By assumption there is an arrow \ar{a_{4},a_{2},l'} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \liftingsquare{a_{2},a_{2},a_{4},a_{3},id,f,j',f',l'} %
%
In other words we have a pair of commutative diagrams in $\cl{A}$ as follows such that $l'$ is a retraction of $j'$. \twosq{a_{2},a_{4},a_{3},a_{3},j',f',f,id,a_{4},a_{2},a_{3},a_{3},l',f,f',id} %
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Appealing to condition (iii) of Proposition \ref{EquivalentDefinitionsOfModelStructureProposition} we deduce that $f$ belongs to both $F$ and $W$. 

That conditions (iv) and (v) of Definition \ref{ModelStructureDefinition} hold given that conditions (ii), (iv), (v), (vi), and (vii) of Proposition \ref{EquivalentDefinitionsOfModelStructureProposition} hold follows formally by duality from the two arguments we have already given in this proof. 

Conversely suppose that $(W,F,C)$ equips $\cl{A}$ with a model structure. We must demonstrate that conditions (ii) and (iii) of Proposition \ref{EquivalentDefinitionsOfModelStructureProposition} are satisfied. 

Suppose that we have commutative diagrams \twosq{a_{2},a_{0},a_{3},a_{1},g_{0}',f,f',g_{1}',a_{0},a_{2},a_{1},a_{3},r_{0},f',f,r_{1}} in $\cl{A}$ such that $r_{0}$ is a retraction of $g_{0}$, such that $r_{1}$ is a retraction of $g_{1}$, and such that $f$ belongs to $F$. %
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Suppose that we have a commutative diagram in $\cl{A}$ as follows in which $j$ belongs to both $C$ and $W$. \sq{a_{0}',a_{2},a_{1}',a_{3}',g_{0},f',j,g_{1}} %
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Then the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{a_{0}',a_{0},a_{1}',a_{1},g_{0} \circ g_{0}',f,j,g_{1} \circ g_{1}'} %
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Since $f$ belongs to $F$ by condition (ii) of Definition \ref{ModelStructureDefinition} there is an arrow \ar{a_{1}',a_{0},l} of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \liftingsquare[{4,3}]{a_{0}',a_{0},a_{1}',a_{1},g_{0} \circ g_{0}',f,j,g_{1} \circ g_{1}',l} %
%
Thus the following diagram in $\cl{A}$ commutes. \liftingsquare[{6,3}]{a_{0}',a_{2},a_{1}',a_{3},r_{0} \circ g_{0} \circ g_{0}',f,j,r_{1} \circ g_{1} \circ g_{1}',r_{0} \circ l} %
%
Since $r_{0}$ is a retraction of $g_{0}$ and since $r_{1}$ is a retraction of $g_{1}$ we thus have that the following diagram in $\cl{A}$ commutes. \liftingsquare[{6,3}]{a_{0}',a_{2},a_{1}',a_{3},g_{0}',f,j,g_{1}',r_{0} \circ l} %
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An entirely similar argument appealing to condition (iii) rather than condition (ii) of Definition \ref{ModelStructureDefinition} proves that if $f$ belongs to both $F$ and $W$, then $f'$ belongs to both $F$ and $W$. This completes our proof that condition (ii) of Proposition \ref{EquivalentDefinitionsOfModelStructureProposition} is satisfied. 

That condition (iii) of Proposition \ref{EquivalentDefinitionsOfModelStructureProposition} is satisfied given that conditions (iv) and (v) of Definition \ref{ModelStructureDefinition} hold follows formally by duality from the proof we have just given that condition (ii) of Proposition \ref{EquivalentDefinitionsOfModelStructureProposition} holds. 
 
\end{proof} 
\end{section}

model_structure.tex
\begin{section}{Model structure} \label{ModelStructureSection}

Suppose that we have cylinder $\cylinder$ and a co-cylinder $\cocylinder$ in a category $\cl{A}$ such that: 

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] both $\cylinder$ and $\cocylinder$ are equipped with all the structures we have considered in this work and have strictness of identities,

\item[(ii)] $\cylinder$ is left adjoint to $\cocylinder$ and the adjunction between $\cyl$ and $\cocyl$ is compatible with their respective contraction structures.

\end{itemize} %
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We bring all our theory together to prove that we obtain a model structure upon $\cl{A}$ by taking:

\begin{itemize} [itemsep=1em,topsep=1em]

\item[(i)] weak equivalences to be homotopy equivalences with respect to $\cylinder$ or equivalently with respect to $\cocylinder$,

\item[(ii)] fibrations to be fibrations with respect to $\cocylinder$,

\item[(iii)] cofibrations to be normally cloven cofibrations with respect to $\cylinder$.

\end{itemize} %
%
Equally we prove that we obtain a model structure upon $\cl{A}$ by taking:

\begin{itemize} [itemsep=1em,topsep=1em]

\item[(i)] weak equivalences to be homotopy equivalences with respect to $\cylinder$ or equivalently with respect to $\cocylinder$,

\item[(ii)] fibrations to be normally cloven fibrations with respect to $\cocylinder$,

\item[(iii)] cofibrations to be cofibrations with respect to $\cylinder$.

\end{itemize} %
%
An interval $\interval$ with respect to a monoidal structure upon $\cl{A}$ gives rise as in \ref{StructuresUponAnIntervalSection} to a cylinder $\cylinderinterval$ and a co-cylinder $\cocylinderinterval$ in $\cl{A}$ under certain conditions. In this way we also obtain two model structures upon $\cl{A}$ from an interval $\interval$ in a monoidal category equipped with all the structures we have considered in this work and satisfying strictness of identities. 

\begin{assum} Let $\cl{A}$ be a category with finite limits and colimits. \end{assum} 

\begin{rmk} We make this assumption to be consistent with the definition of a model category which was recalled in \ref{ModelCategoryRecollectionsSection}. Our work in fact relies only upon the existence of mapping cylinders and mapping co-cylinders in $\cl{A}$. 

This is a significant difference. Mapping cylinders and mapping co-cylinders exist in the category of chain complexes in an arbitrary additive category for example whereas finite limits and colimits do not. \end{rmk}

\begin{thm} \label{CofibrationsNormallyClovenFibrationsModelStructureTheorem} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{ul}, \Gamma_{lr}, \Gamma_{ur} \big)$ be a cylinder in $\cl{A}$ equipped with: 

\begin{itemize} [itemsep=1em,topsep=1em]

\item[(i)] a contraction structure $p$,

\item[(ii)] an involution structure $v$ compatible with $p$,

\item[(iii)] a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ compatible with $p$,

\item[(iv)] an upper left connection structure $\Gamma_{ul}$, 

\item[(v)] a lower right connection structure $\Gamma_{lr}$ compatible with $p$,

\item[(vi)] an upper right connection structure $\Gamma_{ur}$.

\end{itemize} %
%
Suppose that: 

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] $\Gamma_{lr}$ and $\Gamma_{ur}$ are compatible with $\big( \subdiv, r_{0}, r_{1}, s \big)$, 

\item[(ii)] $\cyl$ preserves mapping cylinders with respect to $\cylinder$,

\item[(iii)] $\cylinder$ has strictness of identities.

\end{itemize} %
%
Let $\cocylinder = \big( \cocyl, e_{0}, e_{1}, c, v', \subdiv',r_{0}', r_{1}', s', \Gamma_{ul}', \Gamma_{lr}' \big)$ be a co-cylinder in $\cl{A}$ equipped with:

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] a contraction structure $c$, 

\item[(ii)] an involution structure $v'$ compatible with $c$, 

\item[(iii)] a subdivision structure $\big(\subdiv', r_{0}', r_{1}', s' \big)$ compatible with $c$, 

\item[(iv)] an upper left connection structure $\Gamma_{ul}'$,

\item[(v)]  a lower right connection structure $\Gamma_{lr}'$ compatible with $c$. 

\end{itemize} %
% 
Suppose that:

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] $\cocyl$ preserves mapping co-cylinders with respect to $\cocylinder$,

\item[(ii)] $\cocylinder$ has strictness of identities.

\end{itemize} %
%
Suppose moreover that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$. 

We obtain a model structure upon $\cl{A}$ by taking: 

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] weak equivalences to be the homotopy equivalences with respect to $\cylinder$ or equivalently by Proposition \ref{HomotopyEquivalenceCylinderIffHomotopyEquivalenceCoCylinderProposition} to be the homotopy equivalences with respect to $\cocylinder$,

\item[(ii)] fibrations to be the normally cloven fibrations with respect to $\cocylinder$,

\item[(iii)] cofibrations to be the cofibrations with respect to $\cylinder$.

\end{itemize} 

\end{thm}

\begin{proof} That the conditions of Proposition \ref{EquivalentDefinitionsOfModelStructureProposition} hold has been established as follows:

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] Proposition \ref{TwoOutOfThreeHomotopyEquivalencesProposition},

\item[(ii)] Proposition \ref{RetractionCofibrationIsCofibrationProposition} and Corollary \ref{RetractionTrivialCofibrationIsTrivialCofibrationCorollary},

\item[(iii)] Corollary \ref{RetractionNormallyClovenFibrationIsNormallyClovenFibrationCorollary} and Corollary \ref{RetractionTrivialNormallyClovenFibrationIsTrivialNormallyClovenFibrationCorollary},

\item[(iv)] Corollary \ref{NormallyClovenFibrationRLPTrivialCofibrationCorollary},

\item[(v)] Corollary \ref{TrivialNormallyClovenFibrationRLPCofibrationCorollary},

\item[(vi)] Corollary \ref{CofibrationFollowedByTrivialNormallyClovenFibrationCorollary},

\item[(vii)] Corollary \ref{TrivialCofibrationFollowedByNormallyClovenFibrationCorollary}. 

\end{itemize}
\end{proof}

\begin{thm} \label{NormallyClovenCofibrationsFibrationsModelStructureTheorem} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p, v, \subdiv, r_{0}, r_{1}, s, \Gamma_{ul}, \Gamma_{lr} \big)$ be a cylinder in $\cl{A}$ equipped with: 

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] a contraction structure $p$,

\item[(ii)] an involution structure $v$ compatible with $p$,

\item[(iii)] a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ compatible with $p$,

\item[(iv)] an upper left connection structure $\Gamma_{ul}$, 

\item[(v)] a lower right connection structure $\Gamma_{lr}$ compatible with $p$.
\end{itemize} %
%
Suppose that: 

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] $\cyl$ preserves mapping cylinders with respect to $\cylinder$,

\item[(ii)] $\cylinder$ has strictness of identities.

\end{itemize} %
%
Let $\cocylinder = \big( \cocyl, e_{0}, e_{1}, c, v', \subdiv',r_{0}', r_{1}', s', \Gamma_{ul}', \Gamma_{lr}', \Gamma_{ur}' \big)$ be a co-cylinder in $\cl{A}$ equipped with:

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] a contraction structure $c$, 

\item[(ii)] an involution structure $v'$ compatible with $c$, 

\item[(iii)] a subdivision structure $\big(\subdiv', r_{0}', r_{1}', s' \big)$ compatible with $c$, 

\item[(iv)] an upper left connection structure $\Gamma_{ul}'$,

\item[(v)] a lower right connection structure $\Gamma_{lr}'$ compatible with $c$,

\item[(vi)] an upper right connection structure $\Gamma_{ur}'$.

\end{itemize} %
% 
Suppose that:

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] $\Gamma_{lr}'$ and $\Gamma_{ur}'$ are compatible with $\big( \subdiv', r_{0}', r_{1}', s' \big)$. 

\item[(ii)] $\cocyl$ preserves mapping co-cylinders with respect to $\cocylinder$,

\item[(iii)] $\cocylinder$ has strictness of identities.

\end{itemize} %
%
Suppose moreover that $\cylinder$ is left adjoint to $\cocylinder$, and that the adjunction between $\cyl$ and $\cocyl$ is compatible with $p$ and $c$.

We obtain a model structure upon $\cl{A}$ by taking:

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] weak equivalences to be the homotopy equivalences with respect to $\cylinder$ or equivalently by Proposition \ref{HomotopyEquivalenceCylinderIffHomotopyEquivalenceCoCylinderProposition} the homotopy equivalences with respect to $\cocylinder$,

\item[(ii)] fibrations to be the fibrations with respect to $\cocylinder$,

\item[(iii)] cofibrations to be the normally cloven cofibrations with respect to $\cylinder$.

\end{itemize} 

\end{thm}

\begin{proof}  That the conditions of Proposition \ref{EquivalentDefinitionsOfModelStructureProposition} hold has been established as follows:

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] Proposition \ref{TwoOutOfThreeHomotopyEquivalencesProposition},

\item[(ii)] Proposition \ref{RetractionNormallyClovenCofibrationIsNormallyClovenCofibrationProposition} and Corollary \ref{RetractionTrivialNormallyClovenCofibrationIsTrivialNormallyClovenCofibrationCorollary},

\item[(iii)] Corollary \ref{RetractionFibrationIsFibrationCorollary} and Corollary \ref{RetractionTrivialFibrationIsTrivialFibrationCorollary},

\item[(iv)] Corollary \ref{FibrationRLPTrivialNormallyClovenCofibrationCorollary},

\item[(v)] Corollary \ref{TrivialFibrationRLPNormallyClovenCofibrationCorollary}, 

\item[(vi)] Corollary \ref{NormallyClovenCofibrationFollowedByTrivialFibrationCorollary}, 

\item[(vii)] Corollary \ref{TrivialNormallyClovenCofibrationFollowedByFibrationCorollary}. 

\end{itemize}
\end{proof}

\begin{assum} Let $\otimes$ be a monoidal structure upon $\cl{A}$. \end{assum}

\begin{cor} \label{CofibrationsNormallyClovenFibrationsIntervalModelStructureCorollary} Let $\interval = \big( I, i_{0}, i_{1}, p, v, S, r_{0}, r_{1}, s, \Gamma_{ul}, \Gamma_{lr}, \Gamma_{ur} \big)$ be an interval in $\cl{A}$ equipped with:

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] a contraction structure $p$,

\item[(ii)] an involution structure $v$ compatible with $p$,

\item[(iii)] a subdivision structure $\big( S, r_{0}, r_{1}, s \big)$ compatible with $p$,

\item[(iv)] an upper left connection structure $\Gamma_{ul}$, 

\item[(v)] a lower right connection structure $\Gamma_{lr}$ compatible with $p$,

\item[(vi)] an upper right connection structure $\Gamma_{ur}$.

\end{itemize} %
%
Suppose that:

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] $\Gamma_{lr}$ and $\Gamma_{ur}$ are compatible with $\big( S, r_{0}, r_{1}, s \big)$,

\item[(ii)] $I$ and $S$ are exponentiable with respect to $\otimes$,

\item[(iii)] Requirement \ref{SubdivisionRequirement} holds,

\item[(iv)] $\interval$ has strictness of identities.  
 
\end{itemize} %
%
We obtain a model structure upon $\cl{A}$ by taking:

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] weak equivalences to be the homotopy equivalences with respect to $\cylinderinterval$ or equivalently by Proposition \ref{HomotopyEquivalenceCylinderIffHomotopyEquivalenceCoCylinderProposition} the homotopy equivalences with respect to $\cocylinderinterval$,

\item[(ii)] fibrations to be the normally cloven fibrations with respect to $\cocylinderinterval$,

\item[(iii)] cofibrations to be the cofibrations with respect to $\cylinderinterval$.

\end{itemize}

\end{cor}

\begin{proof} Follows immediately from Theorem \ref{CofibrationsNormallyClovenFibrationsModelStructureTheorem} by the observations of \ref{StructuresUponAnIntervalSection}. \end{proof}

\begin{cor} \label{NormallyClovenCofibrationsFibrationsIntervalModelStructureCorollary} Let $\interval = \big( I, i_{0}, i_{1}, p, v, S, r_{0}, r_{1}, s, \Gamma_{ul}, \Gamma_{lr}, \Gamma_{ur} \big)$ be an interval in $\cl{A}$ equipped with:

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] a contraction structure $p$,

\item[(ii)] an involution structure $v$ compatible with $p$,

\item[(iii)] a subdivision structure $\big( S, r_{0}, r_{1}, s \big)$ compatible with $p$,

\item[(iv)] an upper left connection structure $\Gamma_{ul}$, 

\item[(v)] a lower right connection structure $\Gamma_{lr}$ compatible with $p$,

\item[(vi)] an upper right connection structure $\Gamma_{ur}$.

\end{itemize} %
%
Suppose that:

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] $\Gamma_{lr}$ and $\Gamma_{ur}$ are compatible with $\big( S, r_{0}, r_{1}, s \big)$,

\item[(ii)] $I$ and $S$ are exponentiable with respect to $\otimes$,

\item[(iii)] Requirement \ref{SubdivisionRequirement} holds,

\item[(iv)] $\interval$ has strictness of identities.  
\end{itemize} %
%
We obtain a model structure upon $\cl{A}$ by taking:

\begin{itemize}[itemsep=1em,topsep=1em]

\item[(i)] weak equivalences to be the homotopy equivalences with respect to $\cylinderinterval$ or equivalently by Proposition \ref{HomotopyEquivalenceCylinderIffHomotopyEquivalenceCoCylinderProposition} the homotopy equivalences with respect to $\cocylinderinterval$,

\item[(ii)] fibrations to be the fibrations with respect to $\cocylinderinterval$,

\item[(iii)] cofibrations to be the normally cloven cofibrations with respect to $\cylinderinterval$.

\end{itemize}

\end{cor}

\begin{proof} Follows immediately from Theorem \ref{NormallyClovenCofibrationsFibrationsModelStructureTheorem} by the observations of \ref{StructuresUponAnIntervalSection}. \end{proof}

\end{section}

monoidal_category_theory_preliminaries.tex
\begin{section}{Monoidal category theory preliminaries} \label{MonoidalCategoryTheoryPreliminariesSection}

In \ref{StructuresUponAnIntervalSection} we will introduce the notion of an interval in a monoidal category $\cl{A}$. Under natural hypotheses a structured interval in $\cl{A}$ will give rise to a structured cylinder and a structured co-cylinder in $\cl{A}$. 

For an interval in $\cl{A}$ to yield a co-cylinder in $\cl{A}$ we will have need of the notion of an exponentiable object of $\cl{A}$. As we will not assume that our monoidal structures are symmetric, there are two possible definitions. We now explain the terminology that we have chosen to adopt.

\begin{assum} Let $(\cl{A},\otimes)$ be a monoidal category. Let $1$ denote its unit object, and let $\lambda$ denote its natural isomorphism \ar{- \otimes 1, {-.}} \end{assum}

\begin{defn} An object $a$ of $\cl{A}$ is {\em exponentiable} with respect to $\otimes$ if the functor \ar[4]{\cl{A},\cl{A},- \otimes a} admits a right adjoint, which we will denote by \ar{\cl{A},\cl{A}.,{(-)^{a}}} \end{defn}

\begin{rmk} \label{UnitExponentiableRemark} We have that $1$ is exponentiable with respect to $\otimes$, since the natural isomorphism \ar[7]{{\mathsf{Hom}_{\cl{A}}(- \otimes 1,-)},{\mathsf{Hom}_{\cl{A}}(-,-)},{\mathsf{Hom}_{\cl{A}}(\lambda^{-1},-)}} exhibits the identity functor as a right adjoint of $- \otimes 1$. \end{rmk} 

\begin{notn} Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$ such that both $a_{0}$ and $a_{1}$ are exponentiable with respect to $\otimes$. Then for any object $a$ of $\cl{A}$ we have a natural isomorphism \ar[4]{{\mathsf{Hom}_{\cl{A}}(- \times a_{0}, a)},{\mathsf{Hom}_{\cl{A}}(-,a^{a_{0}})},\mathsf{adj}(a_{0})} and a natural isomorphism \ar[4]{{\mathsf{Hom}_{\cl{A}}(- \times a_{1}, a)},{\mathsf{Hom}_{\cl{A}}(-,a^{a_{1}}).},\mathsf{adj}(a_{1})} %
%
Let \ar{{a^{a_{1}}},{a^{a_{0}}},{a^{f}}} denote the arrow of $\cl{A}$ corresponding via the Yoneda lemma to the following natural transformation.  

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=5 em, row sep=4 em, nodes={anchor=center}]
{ 
|(0-0)| \mathsf{Hom}_{\cl{A}}(-,a^{a_{1}}) \& |(1-0)| \mathsf{Hom}_{\cl{A}}(- \otimes a_{1},a) \\ 
|(0-1)| \mathsf{Hom}_{\cl{A}}(-,a^{a_{0}}) \& |(1-1)| \mathsf{Hom}_{\cl{A}}(- \otimes a_{0}, a) \\
};
	
\draw[->] (0-0) to node[auto] {$\mathsf{adj}(a_{1})^{-1}$} (1-0);
\draw[->] (1-0) to node[auto] {$\mathsf{Hom}_{\cl{A}}(- \otimes f,a)$} (1-1);
\draw[->,dashed] (0-0) to (0-1);
\draw[<-] (0-1) to node[auto,swap] {$\mathsf{adj}(a_{0})$} (1-1);

\end{tikzpicture} 

\end{diagram} 

\end{notn} 

\begin{rmk} Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$ such that both $a_{0}$ and $a_{1}$ are exponentiable with respect to $\otimes$. Associating to an object $a$ of $\cl{A}$ the arrow \ar{{a^{a_{1}}},{a^{a_{0}}},{a^{f}}} of $\cl{A}$ defines a natural transformation \ar{{(-)^{a_{1}}},{(-)^{a_{0}}.},{{(-)}^{f}}} \end{rmk}

\begin{defn} The monoidal structure upon $\cl{A}$ defined by $\otimes$ is {\em closed} if for every object $a$ of $\cl{A}$ the functor \ar[4]{\cl{A},\cl{A},a \otimes -} admits a right adjoint. \end{defn}

\begin{rmk} Suppose that the monoidal structure upon $\cl{A}$ defined by $\otimes$ is symmetric. Then this monoidal structure is closed if and only if every object of $\cl{A}$ is exponentiable with respect to $\otimes$. \end{rmk}

\begin{rmk} Suppose that the following diagram in $\cl{A}$ is co-cartesian. \sq{a_{0},a_{1},a_{2},a_{3},f_{0},f_{1},f_{2},f_{3}} Let $a$ be an object of $\cl{A}$ such that the following diagram in $\cl{A}$ is co-cartesian. \sq[{4,3}]{a \otimes a_{0}, a \otimes a_{1}, a \otimes a_{2}, a \otimes a_{3}, a \otimes f_{0}, a \otimes f_{1}, a \otimes f_{2}, a \otimes f_{3}} If $a_{0}$, $a_{1}$, $a_{2}$, and $a_{3}$ are exponentiable with respect to $\otimes$, then the following diagram in $\cl{A}$ is cartesian. \sq{{a^{a_{3}}},{a^{a_{2}}},{a^{a_{1}}},{a^{a_{0}}},{a^{f_{3}}},{a^{f_{2}}},{a^{f_{1}}},{a^{f_{0}}}} \end{rmk} 

\end{section}

structures_upon_a_cylinder_or_a_co-cylinder.tex
\begin{section}{Structures upon a cylinder or co-cylinder} \label{StructuresUponACylinderOrACoCylinderSection}

We introduce the notion of a cylinder or a co-cylinder in a formal category. We define the structures --- contraction, involution, subdivision, and three flavours of connection --- upon a cylinder and a co-cylinder which will play a role in the present work, and introduce axioms expressing their compatibility. We refer the reader to \ref{ExampleSection} for an example of these structures in the category of categories. 

These structures and axioms have previously appeared in the literature. However, the precise definitions and terminology vary from author to author and paper to paper. Thus we collect in one place and from a single point of view all that we will need. Most of our structures and axioms can be found in \S{4} of Chapter I and at the end of \S{3} of Chapter II of the book \cite{KampsPorterAbstractHomotopyAndSimpleHomotopyTheory} of Kamps and Porter, or in \S{2.1} and \S{2.3} of the paper \cite{GrandisCategoricallyAlgebraicFoundationsForHomotopicalAlgebra} of Grandis. Compatibility of right connections with subdivision appears implicitly and in a slightly different context in \S{6.4} of \cite{BrownHigginsSiveraNonabelianAlgebraicTopology}.
 
We also introduce strictness hypotheses which our structures upon a cylinder or a co-cylinder may satisfy. We will come in \ref{HomotopyAndRelativeHomotopySection} to define the notion of a homotopy with respect to a cylinder or a co-cylinder. The first of our strictness hypotheses, which we will refer to as strictness of left or right identities, will ensure that the left or right composition of an identity homotopy with another homotopy $h$ is exactly $h$. 

Strictness of identities will allow us to prove in \ref{FactorisationAxiomsSection} that the mapping cylinder (respectively the mapping co-cylinder) of any arrow $f$ yields a factorisation of $f$ into a normally cloven cofibration followed by a trivial fibration (respectively into a trivial cofibration followed by a normally cloven fibration). It will also be crucial in establishing that the lifting axioms for a model category hold in \ref{LiftingAxiomsSection}.

The significance of strictness of identities with regard to lifting has to the author's knowledge not previously been observed. Its importance with regard to factorisation was independently explored by van den Berg and Garner in \cite{VanDenBergGarnerTopologicalAndSimplicialModelsOfIdentityTypes}. We particularly draw the reader's attention to Remark 4.3.3 in this paper. Our strictness of left (respectively right) identities condition corresponds to the left (respectively right) unitality condition of van den Berg and Garner. 

The second of our hypotheses, which we will refer to as strictness of left inverses, will ensure that the composition of a homotopy with its inverse is exactly an identity homotopy. It allows us to give a criterion for the compatibility of right connections with subdivision. This property will be vital for us when in \ref{CoveringHomotopyExtensionPropertySection} we investigate the covering homotopy extension property.

\begin{assum} Let $\cl{C}$ be a 2-category, and let $\cl{A}$ be an object of $\cl{C}$. \end{assum}

\begin{defn} A {\em cylinder} in $\cl{A}$ is a 1-arrow \ar{\cl{A},\cl{A},\cyl} of $\cl{C}$ together with a pair of 2-arrows \pair{\id_{\cl{A}},\cyl,i_0,i_1} of $\cl{C}$. \end{defn}

\begin{defn} A {\em co-cylinder} in $\cl{A}$ is a 1-arrow \ar[4]{\cl{A},\cl{A},\cocyl} of $\cl{C}$ together with a pair of 2-arrows \pair{\cocyl,\id_{\cl{A}},e_{0},e_{1}} of $\cl{C}$. \end{defn}

\begin{rmk} Let $\cocylinder = \big(\cocyl,e_{0},e_{1} \big)$ be a co-cylinder in $\cl{A}$. Then $\big( \cocyl, e_{0}^{op}, e_{1}^{op} \big)$ defines a cylinder in $\cl{A}^{op}$, which we denote by $\cocylinder^{op}$. \end{rmk}

\begin{defn} Let $\cylinder = \big( \cyl, i_0, i_1 \big)$ be a cylinder in $\cl{A}$. A {\em contraction structure} with respect to $\cylinder$ is a 2-arrow \ar{\cyl,\id_{\cl{A}},p} of $\cl{C}$ such that the following diagrams in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commute. \twotriangles{\id_{\cl{A}},\cyl,\id_{\cl{A}},i_{0},p,id,\id_{\cl{A}},\cyl,\id_{\cl{A}},i_{1},p,id} \end{defn}

\begin{defn} Let $\cocylinder = \big( \cocyl, e_0, e_1 \big)$ be a co-cylinder in $\cl{A}$. A {\em contraction structure} with respect to $\cocylinder$ is a 2-arrow \ar{\id_{\cl{A}},\cocyl,c} of $\cl{C}$ such that $c^{op}$ equips the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$ with a contraction structure. \end{defn}

\begin{defn} Let $\cylinder = \big( \cyl,i_{0},i_{1} \big)$ be a cylinder in $\cl{A}$. An {\em involution structure} with respect to $\cylinder$ is a 2-arrow \ar{\cyl,\cyl,v} of $\cl{C}$ such that the following diagrams in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commute. \twotriangles{\id_{\cl{A}},\cyl,\cyl,i_{0},v,i_{1},\id_{\cl{A}},\cyl,\cyl,i_{1},v,i_{0}} \end{defn}
 
\begin{defn} Let $\cocylinder = \big( \cocyl,e_{0},e_{1} \big)$ be a co-cylinder in $\cl{A}$. An {\em involution structure} with respect to $\cocylinder$ is a 2-arrow \ar{\cocyl,\cocyl,v} of $\cl{C}$ such that $v^{op}$ defines an involution structure with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$. \end{defn}

\begin{defn} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. An involution structure $v$ with respect to $\cylinder$ is {\em compatible with $p$} if the following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commutes. \tri{\cyl,\cyl,\id_{\cl{A}},v,p,p} \end{defn}

\begin{defn} Let $\cocylinder = \big( \cocyl, e_{0}, e_{1}, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. An involution structure $v$ with respect to $\cocylinder$ is {\em compatible with $c$} if the involution structure $v^{op}$ with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$ is compatible with the contraction structure defined by $c^{op}$. \end{defn}

\begin{defn} Let $\cylinder = \big( \cyl,i_{0},i_{1} \big)$ be a cylinder in $\cl{A}$. A {\em subdivision structure} with respect to $\cylinder$ is a 1-arrow \ar{\cl{A},\cl{A},\subdiv} of $\cl{C}$ together with a pair of 2-arrows \pair{\cyl,\subdiv,r_{0},r_{1}} of $\cl{C}$ such that the diagram \sq{\id_{\cl{A}},\cyl,\cyl,\subdiv,i_{0},r_{0},i_{1},r_{1}}in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ is co-cartesian, and a 2-arrow \ar{\cyl,\subdiv,s} of $\cl{C}$ such that the following diagrams in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commute. \twosq{\id_{\cl{A}},\cyl,\cyl,\subdiv,i_{0},s,i_{0},r_{1},\id_{\cl{A}},\cyl,\cyl,\subdiv,i_{1},s,i_{1},r_{0}} \end{defn}

\begin{defn} Let $\cocylinder = \big( \cocyl,e_{0},e_{1} \big)$ be a co-cylinder in $\cl{A}$. A {\em subdivision structure} with respect to $\cocylinder$ is a 1-arrow \ar{\cl{A},\cl{A},\subdiv} of $\cl{C}$ together with a pair of 2-arrows \pair{\subdiv,\cocyl,r_{0},r_{1},s} of $\cl{C}$ and a 2-arrow \ar{\subdiv,\cocyl,s} of $\cl{C}$, such that $\big( \subdiv,r_{0}^{op},r_{1}^{op},s^{op} \big)$ defines a subdivision structure with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$. \end{defn}

\begin{defn} \label{SubdivisionCompatibleWithContractionDefinition} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p, \subdiv, r_{0}, r_{1}, s \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$ and a subdivision structure $\big(\subdiv,r_{0},r_{1},s \big)$. Let \ar{\subdiv,\id_{\cl{A}},\overline{p}} denote the canonical 2-arrow of $\cl{C}$ such that the diagram \pushout{\id_{\cl{A}},\cyl,\cyl,\subdiv,\id_{\cl{A}},i_{0},r_{0},i_{1},r_{1},p,p,\overline{p}} in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commutes. The subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ is {\em compatible with $p$} if the following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commutes. \tri{\cyl,\subdiv,\id_{\cl{A}},s,\overline{p},p}  \end{defn}

\begin{defn} Let $\cocylinder = \big( \cocyl, e_{0}, e_{1}, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. A subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ with respect to $\cocylinder$ is {\em compatible with $c$} if the subdivision structure $\big( \subdiv, r_{0}^{op}, r_{1}^{op}, s^{op} \big)$ with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$ is compatible with the contraction structure defined by $c^{op}$. \end{defn}

\begin{defn} Let $\cylinder = \big( \cyl, i_0, i_1, \subdiv, r_0, r_1, s \big)$ be a cylinder in $\cl{A}$ equipped with a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$. Then {\em $\cyl$ preserves subdivision} with respect to $\cylinder$ if the following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ is co-cartesian. \sq[{4,3}]{\cyl,\cyl^{2},\cyl^{2},\cyl \circ \subdiv,\cyl \cdot i_0, \cyl \cdot r_0, \cyl \cdot i_1, \cyl \cdot r_1} \end{defn} 

\begin{defn} Let $\cocylinder = \big( \cocyl, e_0, e_1, \subdiv, r_0, r_1, s \big)$ be a co-cylinder in $\cl{A}$ equipped with a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$. Then {\em $\cocyl$ preserves subdivision} with respect to $\cocylinder$ if $\cocyl$ preserves subdivision with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$ equipped with the subdivision structure $\big( \subdiv, r_{0}^{op}, r_{1}^{op}, s^{op} \big)$. \end{defn}

\begin{defn} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. An {\em upper left connection structure} with respect to $\cylinder$ is a 2-arrow \ar{\cyl^{2},\cyl,\Gamma_{ul}} of $\cl{C}$ such that the following diagrams in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commute. 

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=4 em, row sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| \cyl         \& |(1-0)| \cyl^{2} \&[1em] |(2-0)| \cyl         \& |(3-0)| \cyl^{2} \\ 
                     \& |(1-1)| \cyl     \&                           \& |(3-1)| \cyl     \\
|(0-2)| \cyl         \& |(1-2)| \cyl^{2} \&      |(2-2)| \cyl         \& |(3-2)| \cyl^{2} \\
|(0-3)| \id_{\cl{A}} \& |(1-3)| \cyl     \&      |(2-3)| \id_{\cl{A}} \& |(3-3)| \cyl     \\      
};
	
\draw[->] (0-0) to node[auto] {$i_{0} \cdot \cyl$} (1-0);
\draw[->] (1-0) to node[auto] {$\Gamma_{ul}$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$id$} (1-1);
\draw[->] (2-0) to node[auto] {$\cyl \cdot i_{0}$} (3-0);
\draw[->] (3-0) to node[auto] {$\Gamma_{ul}$} (3-1);
\draw[->] (2-0) to node[auto,swap] {$id$} (3-1);
\draw[->] (0-2) to node[auto] {$i_{1} \cdot \cyl$} (1-2);
\draw[->] (1-2) to node[auto] {$\Gamma_{ul}$} (1-3);
\draw[->] (0-2) to node[auto,swap] {$p$} (0-3);
\draw[->] (0-3) to node[auto,swap] {$i_{1}$} (1-3);
\draw[->] (2-2) to node[auto] {$\cyl \cdot i_{1}$} (3-2);
\draw[->] (3-2) to node[auto] {$\Gamma_{ul}$} (3-3);
\draw[->] (2-2) to node[auto,swap] {$p$} (2-3);
\draw[->] (2-3) to node[auto,swap] {$i_{1}$} (3-3);

\end{tikzpicture} 

\end{diagram} \end{defn}

\begin{defn} Let $\cocylinder = \big( \cocyl, e_{0}, e_{1}, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. An {\em upper left connection structure} with respect to $\cocylinder$ is a 2-arrow \ar{\cyl,\cyl^{2},\Gamma_{ul}} of $\cl{C}$ such that $(\Gamma_{ul})^{op}$ defines an upper left connection structure with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$ equipped with the contraction structure $c^{op}$. \end{defn}

\begin{defn} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. A {\em lower right connection structure} with respect to $\cylinder$ is a 2-arrow \ar{\cyl^{2},\cyl,\Gamma_{lr}} of $\cl{C}$ such that the following diagrams in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commute. 

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=4 em, row sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| \cyl         \& |(1-0)| \cyl^{2} \&[1em] |(2-0)| \cyl         \& |(3-0)| \cyl^{2} \\ 
                     \& |(1-1)| \cyl     \&                           \& |(3-1)| \cyl     \\
|(0-2)| \cyl         \& |(1-2)| \cyl^{2} \&      |(2-2)| \cyl         \& |(3-2)| \cyl^{2} \\
|(0-3)| \id_{\cl{A}} \& |(1-3)| \cyl     \&      |(2-3)| \id_{\cl{A}} \& |(3-3)| \cyl     \\      
};
	
\draw[->] (0-0) to node[auto] {$i_{1} \cdot \cyl$} (1-0);
\draw[->] (1-0) to node[auto] {$\Gamma_{lr}$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$id$} (1-1);
\draw[->] (2-0) to node[auto] {$\cyl \cdot i_{1}$} (3-0);
\draw[->] (3-0) to node[auto] {$\Gamma_{lr}$} (3-1);
\draw[->] (2-0) to node[auto,swap] {$id$} (3-1);
\draw[->] (0-2) to node[auto] {$i_{0} \cdot \cyl$} (1-2);
\draw[->] (1-2) to node[auto] {$\Gamma_{lr}$} (1-3);
\draw[->] (0-2) to node[auto,swap] {$p$} (0-3);
\draw[->] (0-3) to node[auto,swap] {$i_{0}$} (1-3);
\draw[->] (2-2) to node[auto] {$\cyl \cdot i_{0}$} (3-2);
\draw[->] (3-2) to node[auto] {$\Gamma_{lr}$} (3-3);
\draw[->] (2-2) to node[auto,swap] {$p$} (2-3);
\draw[->] (2-3) to node[auto,swap] {$i_{0}$} (3-3);

\end{tikzpicture} 

\end{diagram} \end{defn}

\begin{defn} Let $\cocylinder = \big( \cocyl, e_{0}, e_{1}, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. A {\em lower right connection structure} with respect to $\cocylinder$ is a 2-arrow \ar{\cyl,\cyl^{2},\Gamma_{lr}} of $\cl{C}$ such that $(\Gamma_{lr})^{op}$ defines a lower right connection structure with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$ equipped with the contraction structure $c^{op}$. \end{defn}

\begin{defn} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$. A lower right connection structure $\Gamma_{lr}$ with respect to $\cylinder$ is {\em compatible with $p$} if the following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commutes. \sq{\cyl^{2},\cyl,\cyl,\id_{\cl{A}},\Gamma_{lr},p,p \cdot \cyl,p} \end{defn} 

\begin{defn} Let $\cocylinder = \big( \cocyl, e_{0}, e_{1}, c \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$. A lower right connection structure $\Gamma_{lr}$ with respect to $\cocylinder$ is {\em compatible with $c$} if the lower right connection structure $(\Gamma_{lr})^{op}$ with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$ is compatible with the contraction structure $c^{op}$. \end{defn} 

\begin{rmk} We will not need to consider compatibility of an upper left connection structure with a contraction structure, or compatibility of an upper right connection structure --- to be defined next --- with a contraction structure. \end{rmk}

\begin{defn} Let $\cylinder = \big( \cyl, i_{0}, i_{1}, p, v \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$ and an involution structure $v$. An {\em upper right connection structure} with respect to $\cylinder$ is a 2-arrow \ar{\cyl^{2},\cyl,\Gamma_{ur}} of $\cl{C}$ such that the following diagrams in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commute. 

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=4 em, row sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| \cyl         \& |(1-0)| \cyl^{2} \&[1em] |(2-0)| \cyl         \& |(3-0)| \cyl^{2} \\ 
                     \& |(1-1)| \cyl     \&                           \& |(3-1)| \cyl     \\
|(0-2)| \cyl         \& |(1-2)| \cyl^{2} \&      |(2-2)| \cyl         \& |(3-2)| \cyl^{2} \\
|(0-3)| \id_{\cl{A}} \& |(1-3)| \cyl     \&      |(2-3)| \id_{\cl{A}} \& |(3-3)| \cyl     \\      
};
	
\draw[->] (0-0) to node[auto] {$i_{0} \cdot \cyl$} (1-0);
\draw[->] (1-0) to node[auto] {$\Gamma_{ur}$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$id$} (1-1);
\draw[->] (2-0) to node[auto] {$\cyl \cdot i_{1}$} (3-0);
\draw[->] (3-0) to node[auto] {$\Gamma_{ur}$} (3-1);
\draw[->] (2-0) to node[auto,swap] {$v$} (3-1);
\draw[->] (0-2) to node[auto] {$\cyl \cdot i_{0}$} (1-2);
\draw[->] (1-2) to node[auto] {$\Gamma_{ur}$} (1-3);
\draw[->] (0-2) to node[auto,swap] {$p$} (0-3);
\draw[->] (0-3) to node[auto,swap] {$i_{0}$} (1-3);
\draw[->] (2-2) to node[auto] {$i_{1} \cdot \cyl$} (3-2);
\draw[->] (3-2) to node[auto] {$\Gamma_{ur}$} (3-3);
\draw[->] (2-2) to node[auto,swap] {$p$} (2-3);
\draw[->] (2-3) to node[auto,swap] {$i_{0}$} (3-3);

\end{tikzpicture} 

\end{diagram} \end{defn}

\begin{defn} Let $\cocylinder = \big( \cocyl, e_{0}, e_{1}, c, v \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$ and an involution structure $v$. An {\em upper right connection structure} with respect to $\cocylinder$ is a 2-arrow \ar{\cyl,\cyl^{2},\Gamma_{ur}}of $\cl{C}$ such that $(\Gamma_{ur})^{op}$ defines an upper right connection structure with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$ equipped with the contraction structure $c^{op}$ and the involution structure $v^{op}$. \end{defn}

\begin{rmk} \label{LowerLeftConnectionNotNecessaryRemark} Analogously one can define a {\em lower left connection structure} with respect to a cylinder or a co-cylinder. Everything concerning upper and lower right connections below can equally be carried out for upper and lower left connections. \end{rmk}

\begin{defn} \label{RightConnectionsCylinderCompatibilityDefinition} Let $\cylinder = \big(\cyl,i_0,i_1,p,v,\subdiv,r_0,r_1,s,\Gamma_{lr},\Gamma_{ur} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $v$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s\big)$, an upper right connection structure $\Gamma_{ur}$, and a lower right connection structure $\Gamma_{lr}$. %
%
Let \ar{\subdiv \circ \cyl,\cyl,x} denote the canonical 2-arrow of $\cl{C}$ such that the following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commutes. \pushout[{4,3,-1}]{\cyl,\cyl^{2},\cyl^{2},\subdiv \circ \cyl,\cyl,i_0 \cdot \cyl,r_0 \cdot \cyl,i_1 \cdot \cyl,r_1 \cdot \cyl,\Gamma_{lr},\Gamma_{ur},x} %
%
Then $\Gamma_{lr}$ and $\Gamma_{ur}$ are {\em compatible with $\big( \subdiv, r_{0}, r_{1}, s \big)$} if the following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commutes. \tri[{4,3}]{\cyl^{2},\subdiv \circ \cyl,\cyl,s \cdot \cyl,x,p \cdot \cyl} \end{defn}

\begin{defn} \label{RightConnectionsCoCylinderCompatibilityDefinition} Let $\cocylinder = \big(\cocyl,e_0, e_1, c, v, \subdiv, r_0, r_1, s, \Gamma_{lr}, \Gamma_{ur} \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$, an involution structure $v$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$, an upper right connection structure $\Gamma_{ur}$, and a lower right connection structure $\Gamma_{lr}$. 

Then $\Gamma_{lr}$ and $\Gamma_{ur}$ are {\em compatible with $\big( \subdiv, r_{0}, r_{1}, s \big)$} if the right connections $\Gamma_{lr}^{op}$ and $\Gamma_{ur}^{op}$ with respect to the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$ equipped with the contraction structure $c^{op}$ and the involution structure $v^{op}$ are compatible with the subdivision structure $\big( \subdiv, r_{0}^{op}, r_{1}^{op}, s^{op} \big)$. \end{defn}

\begin{prpn} \label{LowerRightConnectionPlusInvolutionGivesUpperRightConnectionCylinderProposition} Let $\cylinder = \big( \cyl, i_0, i_1, p, v, \Gamma_{lr} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $v$ compatible with $p$, and a lower right connection structure $\Gamma_{lr}$. Then the 2-arrow \ar[7]{\cyl^{2},\cyl,\Gamma_{lr} \circ (v \cdot \cyl)} of $\cl{C}$ defines an upper right connection structure with respect to $\cylinder$. \end{prpn}

\begin{proof} Firstly the following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commutes. 

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=6 em, row sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| \cyl \& |(1-0)| \cyl^{2} \\[2em] 
|(0-1)| \cyl \& |(1-1)| \cyl^{2} \\
};
	
\draw[->] (0-0) to node[auto] {$i_{0} \cdot \cyl$} (1-0);
\draw[->] (1-0) to node[auto] {$v \cdot \cyl$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$i_{1} \cdot \cyl$} (1-1);
\draw[->] (1-1) to node[auto] {$\Gamma_{lr}$} (0-1);
\draw[->] (0-0) to node[auto,swap] {$id$} (0-1);
\end{tikzpicture} 

\end{diagram} Secondly the following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commutes, appealing to the compatibility of $v$ with $p$. \trapeziums[{3,3,1,0}]{\cyl,\cyl^{2},\cyl,\cyl^{2},\id_{\cl{A}},\cyl,\cyl \cdot i_{0}, v \cdot \cyl, \Gamma_{lr}, p, i_{0}, v, \cyl \cdot i_{0}, p}  %
%
Thirdly the following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commutes.  

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=7 em, row sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| \cyl         \& |(1-0)| \cyl^{2} \\
|(0-1)| \id_{\cl{A}} \& |(1-1)| \cyl^{2} \\
                     \& |(1-2)| \cyl     \\
};
	
\draw[->] (0-0) to node[auto] {$i_{1} \cdot \cyl$} (1-0);
\draw[->] (1-0) to node[auto] {$v \cdot \cyl$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$i_{0} \cdot \cyl$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$p$} (0-1);
\draw[->] (0-1) to node[auto,swap] {$i_{0}$} (1-2);
\draw[->] (1-1) to node[auto] {$\Gamma_{lr}$} (1-2);

\end{tikzpicture} 

\end{diagram} Fourthly the following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commutes.\squareabovetriangle[{4,3,0}]{\cyl,\cyl^{2},\cyl,\cyl^{2},\cyl,\cyl \cdot i_{1},v \cdot \cyl, v, \cyl \cdot i_{1},\Gamma_{lr},id} \end{proof}

\begin{cor} \label{LowerRightConnectionPlusInvolutionGivesUpperRightConnectionCoCylinderCorollary} Let $\cocylinder = \big( \cocyl, e_0, e_1, c, v, \Gamma_{lr} \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$, an involution structure $v$ compatible with $c$, and a lower right connection structure $\Gamma_{lr}$. Then the 2-arrow \ar[8]{\cocyl,\cocyl^{2},{(v \cdot \cocyl) \circ \Gamma_{lr}}} of $\cl{C}$ defines an upper right connection structure with respect to $\cocylinder$. \end{cor}

\begin{proof} Follows immediately from Proposition \ref{LowerRightConnectionPlusInvolutionGivesUpperRightConnectionCylinderProposition} by duality. \end{proof}

\begin{defn} \label{StrictnessLeftIdentitiesCylinderDefinition} Let $\cylinder = \big(\cyl, i_0, i_1, p, \subdiv, r_0, r_1, s \big)$ be a cylinder in $\cl{A}$, equipped with a contraction structure $p$ and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$. Let \ar{\subdiv,\cyl,q_{l}} denote the canonical 2-arrow of $\cl{C}$ such that the following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commutes. \pushout{\id_{\cl{A}},\cyl,\cyl,\subdiv,\cyl,i_{0},r_{0},i_{1},r_{1},id,i_{0} \circ p, q_{l}} Then $\cylinder$ has {\em strictness of left identities} if the following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commutes. \tri{\cyl,\subdiv,\cyl,s,q_{l},id} \end{defn}

\begin{defn} \label{StrictnessLeftIdentitiesCoCylinderDefinition} Let $\cocylinder = \big( \cocyl, e_0, e_1, c, \subdiv, r_0, r_1, s \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$ and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$. Then $\cocylinder$ has {\em strictness of left identities} if the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$ equipped with the contraction structure $c^{op}$ and the subdivision structure $\big( \subdiv^{op}, r_{0}^{op}, r_{1}^{op}, s^{op} \big)$ has strictness of left identities. \end{defn} 

\begin{defn} \label{StrictnessRightIdentitiesCylinderDefinition} Let $\cylinder = \big(\cyl, i_0, i_1, p, \subdiv, r_0, r_1, s \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$ and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$. Let \ar{\subdiv,\cyl,q_{r}} denote the canonical 2-arrow of $\cl{C}$ such that the following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commutes. \pushout{\id_{\cl{A}},\cyl,\cyl,\subdiv,\cyl,i_{0},r_{0},i_{1},r_{1},i_{1} \circ p,id,q_{r}} %
%
Then $\cylinder$ has {\em strictness of right identities} if the following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commutes. \tri{\cyl,\subdiv,\cyl,s,q_{r},id} \end{defn}

\begin{defn} \label{StrictnessRightIdentitiesCoCylinderDefinition} Let $\cocylinder = \big( \cocyl, e_0, e_1, c, \subdiv, r_0, r_1, s \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$ and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$. Then $\cocylinder$ has {\em strictness of right identities} if the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$ equipped with the contraction structure $c^{op}$ and the involution structure $v^{op}$ has strictness of right identities. \end{defn} 

\begin{defn} \label{StrictnessIdentitiesCylinderDefinition} Let $\cylinder = \big(\cyl, i_0, i_1, p, \subdiv, r_0, r_1, s \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$ and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$. Then $\cylinder$ has {\em strictness of identities} if it has both strictness of left identities and strictness of right identities. \end{defn}

\begin{defn} \label{StrictnessIdentitiesCoCylinderDefinition} Let $\cocylinder = \big( \cocyl, e_0, e_1, c, \subdiv, r_0, r_1, s \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$ and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$. Then $\cocylinder$ has {\em strictness of identities} if it has both strictness of left identities and strictness of right identities. \end{defn}

\begin{defn} \label{StrictnessLeftInversesCylinderDefinition} Let $\cylinder = \big(\cyl, i_0, i_1, v, \subdiv, r_0, r_1, s \big)$ be a cylinder in $\cl{A}$ equipped with an involution structure $v$ and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$. Let \ar{\subdiv,\cyl,w} denote the canonical 2-arrow of $\cl{C}$ such that the following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commutes. \pushout{\id_{\cl{A}},\cyl,\cyl,\subdiv,\cyl,i_{0},r_{0},i_{1},r_{1},id,v,w} Then $\cylinder$ has {\em strictness of left inverses} if the following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commutes. \sq{\cyl,\subdiv,\id_{\cl{A}},\cyl,s,w,p,i_1} \end{defn}

\begin{defn} \label{StrictnessLeftInversesCoCylinderDefinition} Let $\cocylinder = \big(\cocyl, e_0, e_1, v, \subdiv, r_0, r_1, s \big)$ be a co-cylinder in $\cl{A}$ equipped with an involution structure $v$ and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$. Then $\cocylinder$ has {\em strictness of left inverses} if the cylinder $\cocylinder^{op}$ in $\cl{A}^{op}$ has strictness of left inverses. \end{defn}

\begin{rmk} We will not need to consider strictness of right inverses, for which there is an analogue of Proposition \ref{CriterionCompatibilityRightConnectionsWithSubdivisionCylinderProposition}. \end{rmk}

\begin{prpn} \label{CriterionCompatibilityRightConnectionsWithSubdivisionCylinderProposition} Let $\cylinder = \big(\cyl, i_0, i_1, p, v, \subdiv, r_0, r_1, s, \Gamma_{lr} \big)$ be a cylinder in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $v$ compatible with $p$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s\big)$, and a lower right connection structure $\Gamma_{lr}$. Suppose that $\cylinder$ has strictness of left inverses. Let $\Gamma_{ur}$ denote the upper right connection structure with respect to $\cylinder$ constructed in Proposition \ref{LowerRightConnectionPlusInvolutionGivesUpperRightConnectionCylinderProposition}. Then $\Gamma_{lr}$ and $\Gamma_{ur}$ are compatible with $\big( \subdiv,r_{0},r_{1},s \big)$. \end{prpn}

\begin{proof} Let \ar{\subdiv,\cyl,w} denote the canonical 2-arrow of $\cl{C}$ of Definition \ref{StrictnessLeftInversesCylinderDefinition}. The following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commutes. \tri[{5,3}]{\cyl^{2},\subdiv \circ \cyl,\cyl^{2},r_{0} \cdot \cyl, w \cdot \cyl, id} Hence the following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commutes. \tri[{5,3}]{\cyl^{2},\subdiv \circ \cyl, \cyl, r_{0} \cdot \cyl, \Gamma_{lr} \circ (w \cdot \cyl), \Gamma_{lr}} The following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ also commutes.  

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=5 em, row sep=4 em, nodes={anchor=center}]
{ 
|(0-0)| \cyl^{2} \& |(1-0)| \subdiv \cyl \\ 
|(0-1)| \cyl \& |(1-1)| \cyl^{2} \\
};
	
\draw[->] (0-0) to node[auto] {$r_{1} \cyl$} (1-0);
\draw[->] (1-0) to node[auto] {$w \cdot \cyl$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$v \cdot \cyl$} (1-1);
\draw[->] (1-1) to node[auto] {$\Gamma_{lr}$} (0-1);
\draw[->] (0-0) to node[auto,swap] {$\Gamma_{ur}$} (0-1);
\end{tikzpicture} 

\end{diagram} Putting the last two observations together we have that the following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commutes. \pushout[{4,3,-1}]{\cyl,\cyl^{2},\cyl^{2},\subdiv \circ \cyl, \cyl, i_{0} \cdot  \cyl, r_{0} \cdot \cyl, i_{1} \cdot \cyl, r_{1} \cdot \cyl, \Gamma_{lr}, \Gamma_{ur}, \Gamma_{lr} \circ (w \cdot \cyl)} %
%
By the universal property of $\subdiv \circ \cyl$ we deduce that $\Gamma_{lr} \circ (w \cdot \cyl=x$, where \ar{S \circ \cyl,\cyl,x} is the canonical 2-arrow of $\cl{C}$ of Definition \ref{RightConnectionsCylinderCompatibilityDefinition}. The following diagram in $\underline{\mathsf{Hom}}_{\cl{C}}(\cl{A},\cl{A})$ commutes. \squareabovetriangle[{5,3,0}]{\cyl^{2},\subdiv \circ \cyl,\cyl,\cyl^{2},\cyl,s \cdot \cyl,w \cdot \cyl,p \cdot \cyl, i_{1} \cdot \cyl, \Gamma_{lr},id} %
%
We conclude that the following diagram commutes, as required. \tri[{4,3}]{\cyl^{2},\subdiv \circ \cyl,\cyl,s \cdot \cyl,x,p \cdot \cyl}  
\end{proof}

\begin{cor} \label{CriterionCompatibilityRightConnectionsWithSubdivisionCoCylinderCorollary} Let $\cocylinder = \big(\cocyl, e_0, e_1, c, v, \subdiv, r_0, r_1, s, \Gamma_{lr} \big)$ be a co-cylinder in $\cl{A}$ equipped with a contraction structure $c$, an involution structure $v$ compatible with $c$, a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s\big)$, and a lower right connection structure $\Gamma_{lr}$. Suppose that $\cocylinder$ has strictness of left inverses. Let $\Gamma_{ur}$ denote the upper right connection with respect to $\cocylinder$ of Corollary \ref{LowerRightConnectionPlusInvolutionGivesUpperRightConnectionCoCylinderCorollary}. Then $\Gamma_{lr}$ and $\Gamma_{ur}$ are compatible with $\big( \subdiv, r_{0}, r_{1}, s \big)$. \end{cor}

\begin{proof} Follows immediately from Proposition \ref{CriterionCompatibilityRightConnectionsWithSubdivisionCylinderProposition} by duality. \end{proof}

\end{section}

structures_upon_an_interval.tex
\begin{section}{Structures upon an interval} \label{StructuresUponAnIntervalSection} 

We define the notion of an interval in a monoidal category $\cl{A}$ and introduce --- exactly in parallel with \ref{StructuresUponACylinderOrACoCylinderSection} --- structures with which this interval may be able to be equipped. An interval $\interval$ in $\cl{A}$ gives rise under natural hypotheses to a cylinder $\cylinderinterval$ and a co-cylinder $\cocylinderinterval$ in $\cl{A}$. Structures upon $\interval$ pass to structures upon $\cylinderinterval$ and $\cocylinderinterval$. 

Moreover $\cylinderinterval$ is left adjoint to $\cocylinderinterval$. Whereas in an abstract setting we work with cylinders and co-cylinders, we often in practise construct a cylinder and co-cylinder via an interval in a monoidal category. 

Also in parallel with \ref{StructuresUponACylinderOrACoCylinderSection} we introduce a strictness of left identities, a strictness of right identities hypothesis, and a strictness of left inverses hypothesis.If these hypotheses hold for $\interval$ they hold for $\cylinderinterval$ and $\cocylinderinterval$. 

We refer the reader to \ref{MonoidalCategoryTheoryPreliminariesSection} for our conventions regarding exponential objects in a monoidal category, and for other preliminary observations to which we shall appeal.

\begin{assum} Let $(\cl{A},\otimes)$ be a monoidal category. Let $1$ denote its unit object, and let $\lambda$ denote its natural isomorphism \ar{- \otimes 1, {-.}} %
%
Let $\alpha$ denote its natural isomorphism \ar{{(- \otimes -) \otimes -}, {- \otimes (- \otimes -).}} \end{assum}

\begin{defn} An {\em interval} in $\cl{A}$ is an object $I$ of $\cl{A}$ together with a pair of arrows \pair{1,I,i_0,i_1} of $\cl{A}$. \end{defn} 

\begin{rmk} We let \ar{\cl{A},\cl{A},{(-)^{1}}} denote the identity functor --- see Remark \ref{UnitExponentiableRemark}. 

We will frequently implicitly identify the functor \ar[4]{\cl{A},\cl{A},- \otimes 1} with the identity functor via the natural isomorphism $\lambda$. \end{rmk}

\begin{defn} \label{CylinderAndCoCylinderFromInterval} Let $\interval = \big( I, i_0, i_1 \big)$ be an interval in $\cl{A}$. Regarding $\cl{A}$ as an object of the 2-category of categories, we denote by $\cylinderinterval$ the cylinder in $\cl{A}$ defined by the functor \ar[4]{\cl{A},\cl{A},- \otimes I} and the natural transformations \pair[{5,0.75}]{\id_{\cl{A}},{- \otimes I.},- \otimes i_{0}, - \otimes i_{1}} 

If $I$ is exponentiable with respect to $\otimes$ we denote by $\cocylinderinterval$ the co-cylinder in $\cl{A}$ defined by the functor \ar{\cl{A},\cl{A},{(-)^{I}}} and the natural transformations \pair[{4,0.75}]{{(-)^{I}},{\id_{\cl{A}}.},{(-)^{i_{0}}},{(-)^{i_{1}}}} \end{defn}

\begin{prpn} \label{CylinderAndCoCylinderFromIntervalAreAdjointProposition} Let $\interval = \big( I, i_0, i_1 \big)$ be an interval in $\cl{A}$. Suppose that $I$ is exponentiable with respect to $\otimes$. Then the cylinder $\cylinderinterval$ in $\cl{A}$ is left adjoint to the co-cylinder $\cocylinderinterval$ in $\cl{A}$. \end{prpn} 

\begin{proof} Let \ar{a_{0} \otimes I,a_{1},h} be an arrow of $\cl{A}$. Since $I$ is exponentiable with respect to $\otimes$, we have a natural isomorphism \ar{{\mathsf{Hom}_{\cl{A}}(- \otimes I, a_{1})},{\mathsf{Hom}_{\cl{A}}(-,(a_{1})^{I}).},\sim} In particular, we have an isomorphism \ar{{\mathsf{Hom}_{\cl{A}}(a_{0} \otimes I,a_{1})},{\mathsf{Hom}_{\cl{A}}(a_{0}, (a_{1})^{I}).},\sim} %
%
Let us denote it by $\mathsf{adj}$. 

By definition of \ar[4]{{(a_{1})^{I}},{a_{1},},{(a_{1})^{i_{0}}}} the following diagram in the category of sets commutes. 

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=10 em, row sep=4 em, nodes={anchor=center}]
{ 
|(0-0)| \mathsf{Hom}_{\cl{A}}(a_{0},(a_{1})^{I}) \& |(1-0)| \mathsf{Hom}_{\cl{A}}(a_{0} \otimes I,a_{1}) \\ 
|(0-1)| \mathsf{Hom}_{\cl{A}}(a_{0},a_{1}) \& |(1-1)| \mathsf{Hom}_{\cl{A}}(a_{0} \otimes 1, a_{1}) \\
};
	
\draw[->] (0-0) to node[auto] {$\mathsf{adj}^{-1}$} (1-0);
\draw[->] (1-0) to node[auto] {$\mathsf{Hom}_{\cl{A}}(a_{0} \otimes i_{0},a_{1})$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$\mathsf{Hom}_{\cl{A}}(a_{0},(a_{1})^{i_{0}})$} (0-1);
\draw[<-] (0-1) to node[auto,swap] {$\mathsf{Hom}_{\cl{A}}(\lambda^{-1}(a_{0}),a_{1})$} (1-1);

\end{tikzpicture} 

\end{diagram} %
%
Thus the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{a_{0},a_{0} \otimes I,{(a_{1})^{I}},a_{1},a_{0} \otimes i_{0},h,\mathsf{adj}(h),{(a_{1})^{i_{0}}}} 

Similarly by definition of \ar[4]{{(a_{1})^{I}},{a_{1},},{(a_{1})^{i_{1}}}} the following diagram in the category of sets commutes. 

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=10 em, row sep=4 em, nodes={anchor=center}]
{ 
|(0-0)| \mathsf{Hom}_{\cl{A}}(a_{0},(a_{1})^{I}) \& |(1-0)| \mathsf{Hom}_{\cl{A}}(a_{0} \otimes I,a_{1}) \\ 
|(0-1)| \mathsf{Hom}_{\cl{A}}(a_{0},a_{1}) \& |(1-1)| \mathsf{Hom}_{\cl{A}}(a_{0} \otimes 1, a_{1}) \\
};
	
\draw[->] (0-0) to node[auto] {$\mathsf{adj}^{-1}$} (1-0);
\draw[->] (1-0) to node[auto] {$\mathsf{Hom}_{\cl{A}}(a_{0} \otimes i_{1},a_{1})$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$\mathsf{Hom}_{\cl{A}}(a_{0},(a_{1})^{i_{1}})$} (0-1);
\draw[<-] (0-1) to node[auto,swap] {$\mathsf{Hom}_{\cl{A}}(\lambda^{-1}(a_{0}),a_{1})$} (1-1);

\end{tikzpicture} 

\end{diagram} Hence the following diagram in $\cl{A}$ commutes. \sq[{4,3}]{a_{0},a_{0} \otimes I,{a_{1}^{I}},a_{1},a_{0} \otimes i_{1},h,\mathsf{adj}(h),{a_{1}^{i_{1}}}} 

\end{proof}

\begin{defn} Let $\interval =\big( I, i_0, i_1 \big)$ be an interval in $\cl{A}$. A {\em contraction structure} with respect to $\interval$ is an arrow \ar{I,1,p} of $\cl{A}$ such that the following diagrams in $\cl{A}$ commute. \twotriangles{1,I,1,i_{0},p,id,1,I,1,i_{1},p,id} \end{defn}

\begin{rmk} Let $\interval = \big( I, i_0, i_1 \big)$ be an interval in $\cl{A}$. Suppose that $1$ is a final object of $\cl{A}$, for example if the monoidal structure upon $\cl{A}$ is cartesian. Then the canonical arrow \ar{I,1,} of $\cl{A}$ defines a contraction structure with respect to $\interval$. \end{rmk}

\begin{rmk} \label{IntervalWithContractionGivesCylinderWithContractionAndCocylinderWithExpansionRemark} Let $\interval = \big( I, i_0, i_1, p \big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$. Then the natural transformation \ar[4]{- \otimes I, \id_{\cl{A}},- \otimes p} equips the cylinder $\cylinderinterval$ in $\cl{A}$ with a contraction structure. 

If $I$ is exponentiable with respect to $\otimes$, the natural transformation \ar{\id_{\cl{A}},{(-)^{I}},{(-)^{p}}} equips the co-cylinder $\cocylinderinterval$ in $\cl{A}$ with a contraction structure. \end{rmk}

\begin{prpn} \label{AdjunctionBetweenCylinderAndCoCylinderFromIntervalCompatibleWithContractionAndExpansionProposition} Let $\interval = \big( I, i_0, i_1,p \big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$. Suppose that $I$ is exponentiable with respect to $\otimes$. We regard the cylinder $\cylinderinterval$ in $\cl{A}$ as equipped with the contraction structure defined by $- \otimes p$, and regard the co-cylinder $\cocylinderinterval$ in $\cl{A}$ as equipped with the contraction structure defined by $(-)^{p}$. 

Recall by Proposition \ref{CylinderAndCoCylinderFromIntervalAreAdjointProposition} that $\cylinderinterval$ is left adjoint to $\cocylinderinterval$. We moreover have that the adjunction between $- \otimes I$ and $(-)^{I}$ is compatible with $- \otimes p$ and $(-)^{p}$. \end{prpn}

\begin{proof} Let \ar{a_{0},a_{1},f} be an arrow of $\cl{A}$. Since $I$ is exponentiable with respect to $\otimes$ we have a natural isomorphism \ar{{\mathsf{Hom}_{\cl{A}}(- \otimes I, a_{1})},{\mathsf{Hom}_{\cl{A}}(-,(a_{1})^{I}).},\sim} %
%
In particular we have an isomorphism \ar{{\mathsf{Hom}_{\cl{A}}(a_{0} \otimes I,a_{1})},{\mathsf{Hom}_{\cl{A}}(a_{0}, (a_{1})^{I}).},\sim} 

Let us denote this isomorphism by $\mathsf{adj}$. By definition of \ar[4]{a_{1},{(a_{1})^{I},},{(a_{1})^{p}}} the following diagram in the category of sets commutes. 

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=10 em, row sep=4 em, nodes={anchor=center}]
{ 
|(0-0)| \mathsf{Hom}_{\cl{A}}(a_{0},a_{1}) \& |(1-0)| \mathsf{Hom}_{\cl{A}}(a_{0} \otimes 1,a_{1}) \\ 
|(0-1)| \mathsf{Hom}_{\cl{A}}(a_{0},(a_{1})^{I}) \& |(1-1)| \mathsf{Hom}_{\cl{A}}(a_{0} \otimes I, a_{1}) \\
};

\draw[->] (0-0) to node[auto] {$\mathsf{Hom}_{\cl{A}}(\lambda(a_{0}),a_{1})$} (1-0);
\draw[->] (1-0) to node[auto] {$\mathsf{Hom}_{\cl{A}}(a_{0} \otimes p,a_{1})$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$\mathsf{Hom}_{\cl{A}}(a_{0},(a_{1})^{p})$} (0-1);

\draw[<-] (0-1) to node[auto,swap] {$\mathsf{adj}$} (1-1);

\end{tikzpicture} 

\end{diagram} Thus the following diagram in $\cl{A}$ commutes. \tri{a_{0},a_{1},{(a_{1})^{I}},f,{(a_{1})^{p}},\mathsf{adj}\big( f \circ ( a_{0} \otimes p) \big)} \end{proof}  

\begin{defn}  Let $\interval =\big( I, i_0, i_1 \big)$ be an interval in $\cl{A}$. An {\em involution structure} with respect to $\interval$ is an arrow \ar{I,I,v}of $\cl{A}$ such that the following diagrams in $\cl{A}$ commute. \twotriangles{1,I,I,i_{0},v,i_{1},1,I,I,i_{1},v,i_{0}} \end{defn}

\begin{rmk} Let $\interval = \big( I, i_0, i_1, v \big)$ be an interval in $\cl{A}$ equipped with an involution structure $v$. Then the natural transformation \ar[5]{- \otimes I, - \otimes I, - \otimes v} equips the cylinder $\cylinderinterval$ in $\cl{A}$ with an involution structure. 

If $I$ is exponentiable with respect to $\otimes$ then the natural transformation \ar{{(-)^{I}},{(-)^{I}},{(-)^{v}}} equips the co-cylinder $\cocylinderinterval$ in $\cl{A}$ with an involution structure. \end{rmk}

\begin{defn} Let $\interval = \big( I, i_{0}, i_{1}, p \big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$. An involution structure $v$ with respect to $\interval$ is {\em compatible with $p$} if the following diagram in $\cl{A}$ commutes. \tri{I,I,1,v,p,p} \end{defn}

\begin{rmk} Let $\interval = \big( I, i_{0}, i_{1}, p, v \big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$ and an involution structure $v$ compatible with $p$. Then the involution structure $- \otimes v$ with respect to the cylinder $\cylinderinterval$ in $\cl{A}$ is compatible with the contraction structure $- \otimes p$.

If $I$ is exponentiable with respect to $\otimes$ then the involution structure $(-)^{v}$ with respect to the co-cylinder $\cocylinderinterval$ in $\cl{A}$ is compatible with the contraction structure $(-)^{p}$. \end{rmk}

\begin{defn} Let $\interval = \big( I, i_0, i_1 \big)$ be an interval in $\cl{A}$. A {\em subdivision structure} with respect to $\interval$ is an object $S$ of $\cl{A}$ together with a pair of arrows \pair{I,S,r_{0},r_{1}} of $\cl{A}$ such that the square \sq{1,I,I,S,i_{0},r_{0},i_{1},r_{1}} in $\cl{A}$ is co-cartesian, and an arrow \ar{I,S,s} of $\cl{A}$ such that the following diagrams in $\cl{A}$ commute. \twosq{1,I,I,S,i_{0},s,i_{0},r_{1},1,I,I,S,i_{1},s,i_{1},r_{0}} \end{defn}

\begin{requirement} \label{SubdivisionRequirement} Let $\interval = \big( I, i_{0}, i_{1} \big)$ be an interval in $\cl{A}$, and let $\big( S, r_{0}, r_{1}, s \big)$ be a subdivision structure with respect to $\interval$. Then for any object $a$ of $\cl{A}$ the square \sq{a,a \otimes I, a \otimes I, a \otimes S, a \otimes i_{0}, a \otimes r_{0}, a \otimes i_{1}, a \otimes r_{1}} in $\cl{A}$ is co-cartesian. \end{requirement}

\begin{rmk} \label{SubdivisionRequirementRemark} Requirement \ref{SubdivisionRequirement} is satisfied whenever the monoidal structure on $\cl{A}$ is closed. It also holds for example in the category of topological spaces equipped with its cartesian monoidal structure --- this monoidal structure is not closed. 

Since by Proposition \ref{CylinderAndCoCylinderFromIntervalAreAdjointProposition} we have that $\cylinderinterval$ is left adjoint to $\cocylinderinterval$, Requirement \ref{SubdivisionRequirement} is equivalent to the dual requirement that for any object $a$ of $\cl{A}$ the square \sq{a^{S},a^{I},a^{I},a,a^{r_{0}},a^{i_{0}},a^{r_{1}},a^{i_{1}}} in $\cl{A}$ is cartesian. \end{rmk}

\begin{rmk} Let $\interval = \big( I, i_{0}, i_{1} \big)$ be an interval in $\cl{A}$, and let $\big( S, r_{0}, r_{1}, s \big)$ be a subdivision structure with respect to $\interval$ such that Requirement \ref{SubdivisionRequirement} holds. Then the functor \ar[4]{\cl{A},\cl{A},- \otimes S} and the natural transformations \pair[{5,0.75}]{- \otimes I, - \otimes S, - \otimes r_{0}, - \otimes r_{1}} and \ar[5]{- \otimes I, - \otimes S, - \otimes s} define a subdivision structure with respect to the cylinder $\cylinderinterval$ in $\cl{A}$. Moreover $- \otimes I$ preserves subdivision with respect to $\cylinderinterval$ and $\big(- \otimes S, - \otimes r_{0}, - \otimes r_{1}, - \otimes s \big)$. 

If both $I$ and $S$ are exponentiable with respect to $\otimes$ then the functor \ar[4]{\cl{A},\cl{A},{(-)^{S}}} and the natural transformations \pair[{5,0.75}]{{(-)^{S}}, {(-)^{I}}, {(-)^{r_{0}}}, {(-)^{r_{1}}}} and \ar[5]{{(-)^{S}}, {(-)^{I}}, {(-)^{s}}} defines a subdivision structure with respect to the co-cylinder $\cocylinderinterval$ in $\cl{A}$. Moreover $(-)^{I}$ preserves subdivision with respect to $\cocylinderinterval$ and $\big( {(-)^{S}}, {(-)^{r_{0}}}, {(-)^{r_{1}}}, {(-)^{s}} \big)$.  \end{rmk} 

\begin{defn} Let $\interval = \big( I, i_{0}, i_{1}, p, S, r_{0}, r_{1}, s \big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$ and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$. Let \ar{S,1,\overline{p}} denote the canonical arrow of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout{1,I,I,S,1,i_0,r_0,i_1,r_1,p,p,\overline{p}} %
%
The subdivision structure $\big( S, r_{0}, r_{1}, s \big)$ is {\em compatible with $p$} if the following diagram in $\cl{A}$ commutes. \tri{I,S,1,s,\overline{p},p} \end{defn}

\begin{rmk} Let $\interval = \big( I, i_{0}, i_{1}, p, S, r_{0}, r_{1}, s\big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$ and a subdivision structure $\big( \subdiv, r_{0}, r_{1}, s \big)$ compatible with $p$. Suppose that Requirement \ref{SubdivisionRequirement} holds. Then the subdivision structure $\big( - \otimes S, - \otimes r_{0}, i- \otimes r_{1}, - \otimes s \big)$ with respect to the cylinder $\cylinderinterval$ in $\cl{A}$ is compatible with the contraction structure $- \otimes p$. 

If both $I$ and $S$ are exponentiable with respect to $\otimes$ then the subdivision structure $\big( {(-)^{S}}, {(-)^{r_{0}}}, {(-)^{r_{1}}}, {(-)^{s}} \big)$ with respect to the co-cylinder $\cocylinderinterval$ in $\cl{A}$ is compatible with the contraction structure $(-)^{p}$. \end{rmk}

\begin{notn} Let $\big( I, i_0, i_1 \big)$ be an interval in $\cl{A}$. We denote by $I^{2}$ the object $I \otimes I$ of $\cl{A}$. \end{notn}

\begin{rmk} Let $\big( I, i_0, i_1 \big)$ be an interval in $\cl{A}$. We will frequently implicitly identify the functor $(- \otimes I) \otimes I$ with the functor $- \otimes I^{2}$ via the natural isomorphism $\alpha$. Similarly we will frequently implicitly identify the functor  $(- \otimes I) \otimes 1$ with the functor $- \otimes (I \otimes 1)$ via $\alpha$. \end{rmk} 

\begin{defn} Let $\interval = \big( I, i_0, i_1, p \big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$. An {\em upper left connection structure} with respect to $\interval$ is an arrow \ar{I^{2},I,\Gamma_{ul}}of $\cl{A}$ such that the following diagrams in $\cl{A}$ commute.  \twotriangles[{4,3,0}]{I,I^{2},I,i_{0} \otimes I, \Gamma_{ul},id,I,I^{2},I,I \otimes i_{0},\Gamma_{ul},id} %
%
\twosq[{4,3,0}]{I,I^{2},1,I,i_{1} \otimes I,\Gamma_{ul},p,i_{1},I,I^{2},1,I,I \otimes i_{1},\Gamma_{ul},p,i_{1}} \end{defn}

\begin{rmk} Let $\interval = \big( I, i_0, i_1, p, \Gamma_{ul} \big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$ and an upper left connection structure $\Gamma_{ul}$. Let us regard the cylinder $\cylinderinterval$ in $\cl{A}$ as equipped with the contraction structure $- \otimes p$, Then the natural transformation \ar[5]{- \otimes I^{2}, - \otimes I, - \otimes \Gamma_{ul}} equips $\cylinderinterval$ with an upper left connection structure. 

Suppose that $I$ is exponentiable with respect to $\otimes$. Let us regard the co-cylinder $\cocylinderinterval$ in $\cl{A}$ as equipped with the contraction structure $(-)^{p}$. Then the natural transformation \ar[4]{{(-)^{I}},{(-)^{I^{2}}},{(-)^{\Gamma_{ul}}}} equips $\cocylinderinterval$ with an upper left connection structure. \end{rmk}

\begin{defn} Let $\interval = \big( I, i_0, i_1, p \big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$. A {\em lower right connection structure} with respect to $\interval$ is an arrow \ar{I^{2},I,\Gamma_{lr}} of $\cl{A}$ such that the following diagrams in $\cl{A}$ commute.  \twotriangles[{4,3,0}]{I,I^{2},I,i_{1} \otimes I, \Gamma_{lr},id,I,I^{2},I,I \otimes i_{1},\Gamma_{lr},id} %
%
\twosq[{4,3,0}]{I,I^{2},1,I,i_{0} \otimes I,\Gamma_{lr},p,i_{0},I,I^{2},1,I,I \otimes i_{0},\Gamma_{lr},p,i_{0}} \end{defn}

\begin{rmk} Let $\interval = \big( I, i_0, i_1, p, \Gamma_{lr} \big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$ and a lower right connection structure $\Gamma_{lr}$. Let us regard the cylinder $\cylinderinterval$ in $\cl{A}$ as equipped with the contraction structure $- \otimes p$. Then the natural transformation \ar[5]{- \otimes I^{2}, - \otimes I, - \otimes \Gamma_{lr}} equips $\cylinderinterval$ with a lower right connection structure. 

Suppose that $I$ is exponentiable with respect to $\otimes$. Let us regard the co-cylinder $\cocylinderinterval$ in $\cl{A}$ as equipped with the contraction structure $(-)^{p}$. Then the natural transformation \ar[4]{{(-)^{I}},{(-)^{I^{2}}},{(-)^{\Gamma_{lr}}}} equips $\cocylinderinterval$ with a lower right connection structure. \end{rmk}

\begin{defn} Let $\interval = \big( I, i_{0}, i_{1}, p \big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$. A lower right connection structure $\Gamma_{lr}$ with respect to $\interval$ is {\em compatible with $p$} if the following diagram in $\cl{A}$ commutes. \sq{I^{2},I,I,1,\Gamma_{lr},p,I \otimes p,p} \end{defn} 

\begin{rmk} Let $\interval = \big( I, i_{0}, i_{1}, p, \Gamma_{lr} \big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$ and a lower right connection structure $\Gamma_{lr}$. If $\Gamma_{lr}$ is compatible with $p$, then the lower right connection structure $- \otimes \Gamma_{lr}$ with respect to the cylinder $\cylinderinterval$ in $\cl{A}$ is compatible with the contraction structure $- \otimes p$. 

Suppose that $I$ is exponentiable with respect to $\otimes$. Then the lower right connection structure $(-)^{\Gamma_{lr}}$ with respect to the co-cylinder $\cocylinderinterval$ in $\cl{A}$ is compatible with the contraction structure $(-)^{p}$. \end{rmk} 

\begin{rmk} We will not need to consider compatibility of an upper left connection structure upon an interval with contraction, or compatibility of an upper right connection structure --- to be defined next --- upon an interval with contraction. \end{rmk}

\begin{defn} Let $\interval = \big( I, i_0, i_1, p, v \big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$ and an involution structure $v$. An {\em upper right connection structure} with respect to $\interval$ is an arrow \ar{I^{2},I,\Gamma_{ur}} of $\cl{A}$ such that the following diagrams in $\cl{A}$ commute. 

\begin{diagram}

\begin{tikzpicture} [>=stealth]

\matrix [ampersand replacement=\&, matrix of math nodes, column sep=5 em, row sep=3 em, nodes={anchor=center}]
{ 
|(0-0)| I \& |(1-0)| I^{2} \&[2em] |(2-0)| I \& |(3-0)| I^{2} \\ 
          \& |(1-1)| I     \&                \& |(3-1)| I     \\
|(0-2)| I \& |(1-2)| I^{2} \&      |(2-2)| I \& |(3-2)| I^{2} \\
|(0-3)| 1 \& |(1-3)| I     \&      |(2-3)| 1 \& |(3-3)| I     \\      
};
	
\draw[->] (0-0) to node[auto] {$I \otimes i_{0}$} (1-0);
\draw[->] (1-0) to node[auto] {$\Gamma_{ur}$} (1-1);
\draw[->] (0-0) to node[auto,swap] {$id$} (1-1);
\draw[->] (2-0) to node[auto] {$i_{1} \otimes I$} (3-0);
\draw[->] (3-0) to node[auto] {$\Gamma_{ur}$} (3-1);
\draw[->] (2-0) to node[auto,swap] {$v$} (3-1);
\draw[->] (0-2) to node[auto] {$i_{0} \otimes I$} (1-2);
\draw[->] (1-2) to node[auto] {$\Gamma_{ur}$} (1-3);
\draw[->] (0-2) to node[auto,swap] {$p$} (0-3);
\draw[->] (0-3) to node[auto,swap] {$i_{0}$} (1-3);
\draw[->] (2-2) to node[auto] {$I \otimes i_{1}$} (3-2);
\draw[->] (3-2) to node[auto] {$\Gamma_{ur}$} (3-3);
\draw[->] (2-2) to node[auto,swap] {$p$} (2-3);
\draw[->] (2-3) to node[auto,swap] {$i_{0}$} (3-3);

\end{tikzpicture} 

\end{diagram} \end{defn}

\begin{rmk} Let $\interval = \big( I, i_0, i_1, p, v, \Gamma_{ur} \big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $v$, and an upper right connection structure $\Gamma_{ur}$. Let us regard the cylinder $\cylinderinterval$ in $\cl{A}$ as equipped with the contraction structure $- \otimes p$ and the involution structure $- \otimes v$. Then the natural transformation \ar[5]{- \otimes I^{2}, - \otimes I, - \otimes \Gamma_{ur}} equips $\cylinderinterval$ in $\cl{A}$ with an upper right connection structure. 

Suppose that $I$ is exponentiable with respect to $\otimes$. Let us regard the co-cylinder $\cocylinderinterval$ in $\cl{A}$ as equipped with the contraction structure $(-)^{p}$ and the involution structure $(-)^{v}$. Then the natural transformation \ar[4]{{(-)^{I}},{(-)^{I^{2}}},{(-)^{\Gamma_{ur}}}} equips $\cocylinderinterval$ in $\cl{A}$ with an upper right connection structure. \end{rmk}

\begin{rmk} Analogously one can define a {\em lower left connection structure} with respect to an interval. Everything concerning upper and lower right connections below can equally be carried out for upper and lower left connections. We will not need this. \end{rmk}

\begin{defn} \label{RightConnectionsIntervalCompatibilityDefinition} Let $\interval = \big( I, i_0, i_1, p, v, S, r_0, r_1, s, \Gamma_{lr}, \Gamma_{ur} \big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $v$, a subdivision structure $\big( S, r_{0}, r_{1}, s \big)$, a lower right connection structure $\Gamma_{lr}$, and an upper right connection structure $\Gamma_{ur}$. Let \ar{I \otimes S,I,x} denote the canonical arrow of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout[{4,3,-1}]{I,I^{2},I^{2},I \otimes S,I,I \otimes i_{0},I \otimes r_0,I \otimes i_1,I \otimes r_1, \Gamma_{lr},\Gamma_{ur},x} Then $\Gamma_{lr}$ and $\Gamma_{ur}$ are {\em compatible with $\big( S, r_{0}, r_{1}, s \big)$} if the following diagram in $\cl{A}$ commutes. \tri[{4,3}]{I^{2},I \otimes S,I,I \otimes s,x,I \otimes p} \end{defn}

\begin{rmk} Let $\interval = \big( I, i_0, i_1, p, v, S, r_0, r_1, s, \Gamma_{lr}, \Gamma_{ur} \big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$, an involution structure $v$, a subdivision structure $\big( S, r_{0}, r_{1}, s \big)$, a lower right connection structure $\Gamma_{lr}$, and an upper right connection structure $\Gamma_{ur}$. Suppose that $\Gamma_{lr}$ and $\Gamma_{ur}$ are compatible with $\big( S, r_{0}, r_{1}, s \big)$ and that Requirement \ref{SubdivisionRequirement} holds. 

Then the right connections $- \otimes \Gamma_{lr}$ and $- \otimes \Gamma_{ur}$ are compatible with the subdivision structure $\big( - \otimes S, - \otimes r_{0}, - \otimes r_{1}, - \otimes s \big)$ upon the cylinder $\cylinderinterval$ in $\cl{A}$ equipped with the contraction structure $- \otimes p$. 

Suppose that both $I$ and $S$ are exponentiable with respect to $\otimes$. Then the right connections $(-)^{\Gamma_{lr}}$ and $(-)^{\Gamma_{ur}}$ are compatible with the subdivision structure $\big( (-)^{S}, (-)^{r_{0}}, (-)^{r_{1}}, (-)^{s} \big)$ upon the co-cylinder $\cocylinderinterval$ in $\cl{A}$ equipped with the contraction structure $(-)^{p}$. \end{rmk}

\begin{defn} \label{StrictnessLeftIdentitiesIntervalDefinition} Let $\interval = \big( I, i_0, i_1, p, S, r_0, r_1, s \big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$ and a subdivision structure $\big( S, r_{0}, r_{1}, s \big)$. Let \ar{S,I,q_{l}} denote the canonical arrow of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout{1,I,I,S,I,i_{0},r_{0},i_{1},r_{1},id,i_{0} \circ p,q_{l}} %
%
Then $\interval$ has {\em strictness of left identities} if the following diagram in $\cl{A}$ commutes. \tri{I,S,I,s,q_{l},id} \end{defn}

\begin{rmk} Let $\interval = \big( I, i_0, i_1, p, S, r_0, r_1, s \big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$ and a subdivision structure $\big( S, r_{0}, r_{1}, s \big)$. Suppose that Requirement \ref{SubdivisionRequirement} holds and that $\interval$ has strictness of left identities. 

Then the cylinder $\cylinderinterval$ in $\cl{A}$ equipped with the contraction structure $- \otimes p$ and the subdivision structure $\big( - \otimes S, - \otimes r_{0}, - \otimes r_{1}, - \otimes s \big)$ has strictness of left identities. 

If both $I$ and $S$ are exponentiable with respect to $\otimes$ then the co-cylinder $\cocylinderinterval$ in $\cl{A}$ equipped with the contraction structure $(-)^{p}$ and the subdivision structure $\big( (-)^{S}, (-)^{r_{0}}, (-)^{r_{1}}, (-)^{s} \big)$ has strictness of left identities. \end{rmk} 

\begin{defn} \label{StrictnessRightIdentitiesIntervalDefinition} Let $\interval = \big( I, i_0, i_1, p, S, r_0, r_1, s \big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$ and a subdivision structure $\big( S, r_{0}, r_{1}, s \big)$. Let \ar{S,I,q_{r}} denote the canonical arrow of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout{1,I,I,S,I,i_{0},r_{0},i_{1},r_{1},i_{1} \circ p,id,q_{r}} % 
%
Then $\interval$ has {\em strictness of right identities} if the following diagram in $\cl{A}$ commutes. \tri{I,S,I,s,q_{r},id} \end{defn}

\begin{rmk} Let $\interval = \big( I, i_0, i_1, p, S, r_0, r_1, s \big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$ and a subdivision structure $\big( S, r_{0}, r_{1}, s \big)$. Suppose that Requirement \ref{SubdivisionRequirement} holds and that $\interval$ has strictness of right identities. 

Then the cylinder $\cylinderinterval$ in $\cl{A}$ equipped with the contraction structure $- \otimes p$ and the subdivision structure $\big( - \otimes S, - \otimes r_{0}, - \otimes r_{1}, - \otimes s \big)$ has strictness of right identities. 

If both $I$ and $S$ are exponentiable with respect to $\otimes$ then the co-cylinder $\cocylinderinterval$ in $\cl{A}$ equipped with the contraction structure $(-)^{p}$ and the subdivision structure $\big( (-)^{S}, (-)^{r_{0}}, (-)^{r_{1}}, (-)^{s} \big)$ has strictness of right identities. \end{rmk} 

\begin{defn}  Let $\interval = \big( I, i_0, i_1, p, S, r_0, r_1, s \big)$ be an interval in $\cl{A}$ equipped with a contraction structure $p$ and a subdivision structure $\big( S, r_{0}, r_{1}, S \big)$. Then $\interval$ has {\em strictness of identities} if it has both strictness of left identities and strictness of right identities. \end{defn}

\begin{defn} \label{StrictnessLeftInversesDefinition} Let $\interval = \big( I, i_0, i_1, v, S, r_0, r_1, s \big)$ be an interval in $\cl{A}$ equipped with an involution structure $v$ and a subdivision structure $\big( S, r_{0}, r_{1}, s \big)$. Let \ar{S,I,w} denote the canonical arrow of $\cl{A}$ such that the following diagram in $\cl{A}$ commutes. \pushout{1,I,I,S,I,i_{0},r_{0},i_{1},r_{1},id,v,w} Then $\interval$ has {\em strictness of left inverses} if the following diagram in $\cl{A}$ commutes. \sq{I,S,1,I,s,w,p,i_1} \end{defn}

\begin{rmk} Let $\interval = \big( I, i_0, i_1, v, S, r_0, r_1, s \big)$ be an interval in $\cl{A}$ equipped with an involution structure $v$ and a subdivision structure $\big( S, r_{0}, r_{1}, s \big)$.Suppose that Requirement \ref{SubdivisionRequirement} holds. Then the cylinder $\cylinderinterval$ in $\cl{A}$ equipped with the involution structure $- \otimes v$ and the subdivision structure $\big( - \otimes S, - \otimes r_{0}, - \otimes r_{1}, - \otimes s \big)$ has strictness of left inverses. 

If both $I$ and $S$ are exponentiable with respect to $\otimes$ then the co-cylinder $\cocylinderinterval$ in $\cl{A}$ equipped with the involution structure $(-)^{v}$ and the subdivision structure  $\big( (-)^{S}, (-)^{r_{0}}, (-)^{r_{1}}, (-)^{s} \big)$ has strictness of left inverses. \end{rmk}

\begin{rmk} We will not need to consider strictness of right inverses. \end{rmk}

\end{section}

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  category is a homotopy category}, Arch. {M}ath. ({B}asel), 23 (1972),
  pp.~435--441.
\newblock Review:
  \url{http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0261.18015}.

\bibitem{ThomasonCatAsAClosedModelCategory}
{\sc R.~W. Thomason}, {\em Cat as a closed model category}, Cahiers {T}opologie
  {G}{\'e}om. {D}iff{\'e}rentielle, 21 (1980), pp.~305--324.
\newblock
  \url{http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1980__21_3_305_0}.
  Review:
  \url{http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0473.18012}.

\bibitem{VanDenBergGarnerTopologicalAndSimplicialModelsOfIdentityTypes}
{\sc B.~van~den Berg and R.~Garner}, {\em Topological and simplicial models of
  identity types}, A{C}{M} {T}rans. {C}omput. {L}og., 13 (2012), pp.~3:1 --
  3:44.
\newblock Preprint: \url{http://arxiv.org/abs/1007.4638}.

\end{thebibliography}
