\section{Introduction}


The (seemingly) simple question of how to organize the fibers of a map $f: X \to Y$ has led to significant advancement in mathematics and related disciplines. Recently, this blossoming has been most apparent in topological data analysis (TDA). Indeed, the  case where $Y=\RR$ and~$f$ is simply a function already motivates persistence theory and the use of tools such as Reeb graphs (the higher dimensional analogues of which are called  Reeb spaces). 

In the present, we connect with the work of Edelsbrunner, Harer, and Patel \cite{edelsbrunner2008reeb} on  Reeb spaces for piecewise linear (PL) vector valued functions (we refer to maps with the specific hypotheses imposed as EHP maps).  In particular, we develop simple tools that allow us to refine the stratification of the Reeb space previously given.  In contrast to \emph{ibid}, our stratification depends entirely on the Jacobi set, and we give an explicit description of how this stratification also extends to both the domain and codomain of the piecewise linear function in question.


\begin{thma}
Let $f: M \to \RR^k$ be a EHP map and let  $\WW_f$ denote the corresponding Reeb space.
\begin{enumerate}
\item There is a stratification of $\WW_f$ such that each stratum is connected; and
\item With respect to this filtration the Stein factorization $g: \WW_f \to \RR^k$ is a map of stratified spaces.
\end{enumerate}
\end{thma}

Morally, the stratification of $\WW_f$ is a refinement of that constructed in \cite{edelsbrunner2008reeb}. This is not literally true as we give a slightly different definition of {\em critical simplices}. We make an explicit comparison between resulting Jacobi sets in Appendix \ref{app:critical}. 

A convenient consequence of the generality of our constructions is that they apply in a variety of settings beyond the EHP setup.  In particular, we obtain well-behaved stratifications on the codomains of higher Morse functions. For Morse two-functions, we prove the following theorem.

\begin{thmb}
Let $f: M \to \RR^2$ be a Morse two-function. There is a stratification of $\RR^2$ such that the topology of pre-images is locally constant on top/open strata.
\end{thmb}




This note is but the first of several works on applying stratified and homotopical techniques to the theory and application of persistence modules. Indeed, the perspective of persistence modules as (constructible) cosheaves on stratified spaces, \cite{curry2013sheaves, CP}, drives this work. One motivator for the present work is the observation that sub-posets of the stratifying poset of a space are equivalent to multi-parameter filtrations of the space.  Recent work in multi-parameter persistence often advocates for analyzing such filtrations by focusing on one-dimensional `slices' through the parameter space (e.g., \cite{lesnick2015interactive, kerber2018exact}), and these slices are equivalent to linear sub-posets of the stratifying poset. 

To wit, one consequence of Theorem B is that to any Morse two-function, there is an associated persistence module (over $\RR^2$). This construction explicitly extends the case of an ordinary Morse function $h: M \to \RR$, which is a classical ingress to persistence theory. This example and related higher dimensional analogues will be explained in detail in forthcoming work.

It is our hope that by bringing the tools of higher ($\infty$) categories and homotopical algebra to the setting of persistence theory and  TDA, new discriminating invariants and computational techniques will grow.




\subsection*{Outline}
In \secref{stratspaces}, we present the necessary background on stratified spaces. In particular, we introduce the (connected) ambient stratification and prove several fundamental properties. \secref{reeb} and \secref{higher} contain our main theorems, related to Reeb spaces and higher Morse functions, respectively. Appendix \ref{app:critical} discusses three different notions of criticality for functions on combinatorial manifolds including the one used in the bulk of the paper and that from \cite{edelsbrunner2008reeb}. Finally, Appendix \ref{app:Jsimp} provides omitted details on the stratifications of $\RR^k$ and~$\WW_f$.


\subsection*{Acknowledgements} AS is supported by the National Science Foundation under NIH/NSF DMS 1664858. The authors thank David Ayala for discussion and shared insight.

\section{Stratified Spaces}\label{sec:stratspaces}


Stratified spaces have been used throughout topology for over 60 years with major foundational contributions due to Thom, Mather, Goresky and MacPherson. The paradigm we use is developed extensively in the work of Ayala, Francis, and Tanaka \cite{aft} and first appeared in Appendix A of Lurie \cite{lurie}.


\subsection{Stratified Recollections}

\subsubsection{Poset Topologies and Operations}



\begin{definition}
    A {\em poset} is a pair $(\cP, \le)$ consisting of
    \begin{itemize}
        \item a set $\cP$, and
        \item a relation $\le$ on $\cP$
    \end{itemize}
    such that $\le$ defines a partial order on $\cP$. I.e., for all $p,q,r \in \cP$, the following hold
    \begin{enumerate}
        \item (reflexivity) $p \le p$.
        \item (transitivity) if $p \le q$ and $q \le r$, then $p \le r$.
        \item (anti-symmetry) if $p \le q$ and $q \le p$, then $p=q$.
    \end{enumerate}
\end{definition}

\begin{definition}
Let $(\cP, \le)$ be a poset. We equip $\cP$ with the {\em upward closed topology} as follows: $U \subseteq \cP$ is open if and only if for all $u \in U$, $\cP_{u \le} := \{ p \in \cP ~|~ u \le p\} \subseteq U.$
\end{definition}

\begin{prop}\label{prop:uc}
The upward closed topology defines a functor, $\mathrm{UC} : \Pos \to \Top$.
\end{prop}

\begin{remark}
Note that for $\cP \in \Pos$, the topology $\mathrm{UC} (\cP)$ is rarely Hausdorff.  Two points in $\mathrm{UC} (\cP)$ are only contained in disjoint open sets when they are not comparable in the poset structure. Then if $\mathrm{UC}(\cP)$  is Hausdorff, it is equivalent to the discrete topology, which is certainly not true for general $\cP$.
\end{remark}




\begin{definition}
    Let $(\cP,\le)$ and $(\cQ,\le)$ be posets. We can explicitly describe the \emph{product poset}
    $(\cP \times \cQ,\le)$ by declaring
    $(p,q) \le (p',q')$ if $p\le p'$ and $q \le q'$.
\end{definition}

This construction is categorical.

\begin{prop}
The category $\mathsf{Poset}$ has (finite) products.
\end{prop}

\begin{example}
Let $(\RR, \le)$ be the standard total order on $\RR$. Then $\RR^2 \cong \RR \times \RR$ admits three partial orders
\begin{enumerate}
\item {\em Lexicographic order}: $(a,b) \le (c,d)$ if and only if $a<c$, or $a=c$ and $b \le d$;
\item {\em Product order}: $(a,b) \le (c,d)$ if and only if $a\le c$ and $b\le d$;
\item {\em Closure of direct product}: $(a,b) \le (c,d)$ if and only if $a <c$ and $b<d$, or $a=c$ and $b=d$.
\end{enumerate}
Only the first partial order is a total order. As the name indicates, the product order is the categorical product.
\end{example}


\begin{definition}
Let $(\cP, \le)$ be a poset. The {\em left cone} on $\cP$ is a new poset, $\cP^\triangleleft$, that is defined by adjoining a new minimum element to $\cP$. More rigorously,  
\[
\cP^\triangleleft := \{0\} \coprod\limits_{\{0\} \times \cP} [1] \times \cP.
\]
Similarly, we can define the {\em right cone} on $\cP$,
\[
\cP^\triangleright :=\{1\} \coprod\limits_{\{1\} \times \cP} [1] \times \cP,
\]
which adjoins a new maximum to $\cP$  (where $[n]$ denotes the totally ordered set $\{0 < 1 < \ldots < n\}$).
\end{definition}

\begin{example}
Note that $\emptyset^\triangleleft = \{0\}$, $\{0\}^\triangleleft = [1]$, and more generally, 
$[n]^\triangleleft = [n+1]$.
Let $n \in \NN$, then we have  isomorphisms of posets
\[
[n]^\triangleleft \cong [n+1] \cong [n]^\triangleright .
\] 
\end{example}


\subsubsection{The category of stratified spaces}


\begin{definition}
A {\em stratified topological space} is a triple $(X \xrightarrow{\phi} \cP)$ consisting of
\begin{itemize}
    \item a paracompact, Hausdorff topological space, $X$,
    \item a poset $\cP$, equipped with the upward closed topology, and
    \item a continuous map $X \xrightarrow{\phi} \cP$.
\end{itemize}
\end{definition}

\begin{definition} Given a stratified topological space $\phi : X \to \cP$, and any $p \in \cP$, the {\em $p$-stratum}, 
$X_p$, is defined as
$$X_p := \phi^{-1}(p).$$
\end{definition}

\begin{example}
Any paracompact, Hausdorff space defines a stratified topological space via $\phi : X \to \ast$. We call this the
\emph{trivial stratification}.
\end{example}

\begin{example}
$\RR$ is stratified over $\{- > 0 < +\}$, where $X_- = \{x \in \RR ~|~ x < 0\}$, $X_0 = \{0\}$, and
$X_+ = \{x\in \RR ~|~ x>0\}$ are mapped to $-$, $0$, and $+$, respectively.
We often find it convenient to indicate a stratified space pictorially, by drawing the strata. For example, below
is a depiction of the stratified space $(\RR \to \{- > 0 < +\})$:
    $$\begin{tikzpicture}[auto]
        \node (a) at (0:0) {$\bullet$};
        \node (b) at (0:1.5) {};
        \node (c) at (180:1.5) {};
        \node (d) at (270:.5) {$X_0$};
        \draw (a) to node {$X_+$} (b)
            (a) to  node[swap] {$X_-$} (c);
    \end{tikzpicture}$$
\end{example}



\begin{definition} 
A {\em map of stratified topological spaces} $(\phi : X \to \cP)$ to $(\psi : Y \to \cQ)$ is a pair of 
continuous maps $(f_1, f_2)$ making the following diagram commute.
$$\begin{tikzcd}
    X \arrow[r,"f_1"] \arrow[d,"\phi"] & Y \arrow[d,"\psi"] \\
    \cP \arrow[r,"f_2"] & \cQ.
    \end{tikzcd}$$
A continuous map is an {\em open embedding} if both $f_1$ and $f_2$ are open (topological) embeddings.
\end{definition}


We will let $\mathsf{TStrat}$ denote the category of stratified topological spaces with morphisms given by continuous maps. The category has many nice properties, e.g., it has finite products.


\begin{definition}
A {\em refinement} of a stratified space $(\phi : X \to \cP)$ is a stratified space $(\phi' : X \to \cR)$ and a map of stratified spaces $(f_1, f_2) : (X \to \cR) \to (X \to \cP)$ such that $f_1 : X \to X$ is the identity, and~$f_2 : \cR \to \cP$ is a continuous surjection, diagrammatically we have the following.
$$\begin{tikzcd}
    X \arrow[equal,r] \arrow[d,"\phi'"] & X \arrow[d,"\phi"] \\
    \cR \arrow[->>,r,"f_2"] & \cP.
    \end{tikzcd}$$
\end{definition}


\subsection{Native stratifications}

Certain classes of topological spaces have canonical stratifications: we refer to such stratifications as {\em native}. For concreteness, we focus our attention to simplicial complexes. When referencing open sets, we assume the relevant simplicial complex is geometeric and endowed with a Euclidean topology. We assume the reader has some background with simplicial complexes, and refer them to \cite{edelsbrunner2010computational} for more information. 

\begin{definition}
Let $K$ be a simplicial complex. The {\em native stratification of $K$} is defined by $K \xrightarrow{\varphi_K} \Nat(K)$, where $\Nat(K)$ is the poset such that, for simplices $\tau, \sigma \subset K$, we have the relation $\tau \leq \sigma$ if and only if $\tau$ is a face of $\sigma$. Suppose that $\tau$ is the lowest dimensional simplex containing $x \in K$. Then $\varphi_K(x) = \tau$.
\end{definition}



We note a couple key properties of the native stratification.

\begin{prop}\label{prop:coarsest}
Let $K$ be a simplicial complex. Then $K \xrightarrow{\varphi_K} \Nat(K)$ is a refinement of the coarsest stratification of $K$ such that 
each stratum is a connected manifold.
\end{prop}

\begin{prop}\label{prop:filtration}
Let $K$ be a simplicial complex. There is a map of posets $\dim : \Nat(K) \to \NN_0$ such that the composite $\dim \circ \varphi_K : K \to \NN_0$ is the skeletal filtration.
\end{prop}

The following is immediate from Proposition \ref{prop:uc} and the definition of simplicial map.

\begin{prop}
Let $f :K \to L$ be a simplicial map, then there is a map of posets $\Nat(f) : \Nat(K) \to \Nat (L)$ that extends to a functor $\Nat: \sCpx \to \mathsf{TStrat}$.
\end{prop}



Note that the native stratification makes sense in other categories: $\Delta$-complexes, CW-complexes, simplicial objects, etc. One notable caution is that composition with a ``dimension" map need not result in the corresponding skeletal filtration, but rather only a collapsed filtration e.g., consider the CW structure on $S^2$ consisting of one 0-cell and one 2-cell. 




\subsection{Ambient stratifications} 

We now consider how an embedded stratified space determines a stratification of the ambient space. The example of a simplicial complex---equipped with its native stratification---embedded in some Euclidean space is an illustrative example to track throughout.

Suppose that $(S \xrightarrow{ \varphi } \cQ)$ is a stratified space and $S \hookrightarrow X$ is a topological embedding into the paracompact, Hausdorff space $X$, e.g., $X = \RR^k$. (We will conflate $S$ with its image in $X$.) To extend the  stratification of~$S$ to a stratification of $X$ compatible with the usual topology on $X$, we can append a final element to the poset, $\cQ$, and map the complement,  $X \setminus Y$, to this element. This is exactly the result of taking the right cone on $\cQ$. We will  denote the resulting map $\varphi_S : X \to \cQ^\triangleright$.

Note that we often want the strata to be connected. In this case we introduce a new stratifying poset $\cQ^{\wwedge}$, which is a refinement of $\cQ^\triangleright$.


\begin{definition}\label{def:connectedstrat}
Let  $(S \xrightarrow{ \varphi } \cQ)$ be a stratified space, $S \hookrightarrow X$ a topological embedding, and $\pi_0 (X \setminus S)= \cA$. Define the poset, $\cQ^{\wwedge}$, as the set $\cQ \amalg \cA$, subject to the following generating relations:
\begin{enumerate}
\item The relations of $\cQ$;
\item For $\ell \in \cQ$ and $\alpha \in \cA$, $\ell \le \alpha$ if and only if $\varphi^{-1}_S (\ell) \subseteq \overline{\alpha}$, i.e., the $\ell$-stratum is in the closure of the connected component indexed by $\alpha$.
\end{enumerate}
There is an obvious extension of the map $\varphi_S$, $\psi_S : X \to \cQ^{\wwedge}$ and we call this stratification the {\em connected ambient stratification}. 
\end{definition}


\begin{example}
Consider the 1-simplex embedded as $K: [0,1] \hookrightarrow \RR$. The poset $\Nat(K)$ consists of three objects: $b_0 \le a \ge b_1$ and $\Nat(K)^{\wwedge}$ is the poset $a_- \ge b_0 \le a \ge b_1 \le a_+$. The stratifying map $\psi_K : \RR \to \Nat(K)^{\wwedge}$ is given by
\[
\psi_K (x) = \begin{cases}a_-  & \text{if } x<0\\ b_0 & \text{if } x=0 \\ a & \text{if } x \in (0,1)\\ b_1& \text{if } x=1\\ a_+ & \text{if } x>1\end{cases} 
\]


\end{example}


\begin{lemma}\label{lem:nativeext}
Let  $(S \xrightarrow{ \varphi } \cQ)$ be a stratified space, $S \hookrightarrow X$ a topological embedding, The maps $X \xrightarrow{\varphi_S} \cQ^\triangleright$ and $X \xrightarrow{\psi_S} \cQ^{\wwedge}$ define stratified spaces that restrict to the $\cQ$ stratification on $S$. Moreover, by collapsing the maximal elements in the poset, the connected ambient stratification refines the conical extension of the $\cQ$ stratification:
\[
\xymatrix{ S\ar[d]^{\varphi} \ar@{^{(}->}[r] & X \ar@{=}[r] \ar[d]^{\psi_S} & X \ar[d]^{\varphi_S} \\ \cQ \ar[r]& \cQ^{\wwedge} \ar@{->>}[r]& \cQ^\triangleright}
\]
\end{lemma}



\begin{definition} Define a category $\mathsf{TStrat}^{\text{embd}}$ as follows:
\begin{itemize}
\item Objects: an object is a triple $(S, X, \iota_S : S \hookrightarrow X)$, where $(S \xrightarrow{\phi} \cP)$ is a stratified space, $X$ is a paracompact Hausdorff space, and~$\iota_S$ is a (topological) embedding. 
\item Morphisms: a morphism $f: (S,X,\iota_S) \to (T,Y, \iota_T)$ is a  map $f: X \to Y$ such that
\begin{enumerate}
\item The image of $S$ under $f$ is contained in $T$; 
\item The restriction $f|_S : (S \xrightarrow{\phi} \cP)  \to (T \xrightarrow{\phi'} \cQ)$ is a stratified map; and
\item The preimage of $T$ is contained in $S$, i.e., $f(X\setminus S) \cap T = \emptyset$.
\end{enumerate}
\end{itemize}
\end{definition}

We will at times condense notation and simply denote an object by $S \subset X$. Note that $\mathsf{TStrat}^{\text{embd}}$ is well defined as the identity is clearly a morphism and composition is inherited from composition in spaces. The conditions on $f: (S \subset X) \to (T \subset Y)$ are equivalent to asking that the following is a pull-back diagram
\[
\xymatrix{ S \ar[d]_{f|_S} \ar@{^{(}->}[r]^{\iota_S} & X \ar[d]^{f} \\ T \ar@{^{(}->}[r]^{\iota_T} &Y}
\]
This rephrasing makes composition apparent as pasting of pullback squares.



\begin{prop}\label{prop:nativemap}
The connected ambient stratification defines a functor $(-)^{\wwedge} : \mathsf{TStrat}^{\text{embd}} \to \mathsf{TStrat}$. In particular, given a morphism $f: (S \subset X) \to (T \subset Y)$, we have a commutative diagram (in spaces) 
\[
\xymatrix{ X \ar[rr]^{f} \ar[d] && Y \ar[d] \\ \cP^{\wwedge} \ar[rr] && \cQ^{\wwedge}}
\]
\end{prop}
\begin{proof}
Lemma \ref{lem:nativeext} shows that we have a well-defined functor at the level of objects.  As the map $f$ is continuous, it takes path components into path components, so we obtain a map at the level of morphisms. Functoriality follows, in a straightforward manner, from composition in $\mathsf{TStrat}$.
\end{proof}



\subsection{Combinatorial Manifolds}

Let us briefly recall some combinatorial and piecewise-linear (PL) topology. A standard reference is \cite{RSPL} or Section 3.9 of \cite{thurston}. (In the latter reference a combinatorial manifold is called a triangulated PL manifold.)


\begin{definition}
A \emph{combinatorial manifold}, $X$, is a PL manifold equipped with a compatible triangulation, i.e., a PL homeomorphism $K  \to X$ for some simplicial complex $K$.
\end{definition}

A PL manifold is a topological manifold equipped with an equivalence class of triangulations, while the combinatorial structure picks out a particular triangulation. It is no surprise that any PL manifold admits a structure of a combinatorial manifold, though the condition is still stronger than being a triangulated topological manifold. A classical theorem of Whitehead is that any smooth manifold admits a unique PL structure, so a triangulated smooth manifold has a unique structure as a combinatorial manifold.


\begin{definition}\label{defn:generic}
Let $X$ be a combinatorial manifold. A PL  (locally affine) function $f: X \to \RR^k$ is \emph{generic} if for each $0<i\le k$ and any $i$-tuple of vertices $\{v_1 , \dotsc , v_i\}$ of $X$ in general position, the images $\{f(v_1) , \dotsc , f(v_i)\}$ are in general position in $\RR^k$.
\end{definition}


The {\em standard} PL disk is given by the cube $I^n$ and its boundary is the {\em standard} PL sphere. In actuality, any convex polyhedron of dimension $n$ presents the standard PL disk as they are all PL homeomorphic. A similar statement can be made for boundaries of convex polyhedrons and PL spheres.

\begin{definition}
Let $X$ and $Y$ be topological spaces. The \emph{join} of $X$ and $Y$, denoted $X \ast Y$, is given by 
$X\times Y \times I/\sim$, where
\[
\forall_{a,a' \in A} \forall_{b,b' \in B} \; (a,b,0) \sim (a, b' , 0) \text{ and } (a,b,1)\sim (a',b,1).
\]
\end{definition}

Morally, the join of spaces is obtained by adjoining all line segments connecting the two spaces (this is literally true for polyhedral spaces embedded in Euclidean space). The join of simplicial complexes is again a simplicial complex, the join of PL manifolds is again PL, and the join of combinatorial manifolds is again a combinatorial manifold.
\begin{definition}
Let $K$ be a simplicial complex.
\begin{itemize}
\item For $\sigma \subseteq K$ a simplex, the \emph{star} of $\sigma$, $\mathop{St} \sigma$ is the set of simplices in $K$ that contain $\sigma$ as a face.
\item For $\sigma \subseteq K$ a simplex, the \emph{link} of $\sigma$, $\Link (\sigma)$, is the closure of the star of $\sigma$ set minus the union of stars of the faces of $\sigma$, i.e., $\Link(\sigma) = \mathop{Cl} \mathop{St} \sigma \setminus \mathop{St} \mathop{Cl} \sigma$.
\item For  $\sigma \subseteq K$ a simplex and $p \in \mathring{\sigma}$, the \emph{link} of $p$, $\Link (p)$, is given by
\[
\Link (p) = \partial \sigma \ast \Link (\sigma).
\]
\end{itemize}
\end{definition}



\begin{lemma}
A simplicial complex is a PL manifold if and only if the link of each simplex is PL homeomorphic to a PL sphere.
\end{lemma}




\begin{lemma}
For $n \in \NN$, let $B^n$ be a PL $n$-disk,  $S^n$ a PL sphere, and $M$ any PL manifold. Assuming the factors are disjoint, we have the following PL homeomorphisms:
\[pt \ast M \cong CM, \quad
B^\ell \ast B^k \cong B^{\ell +k+1}, \; \; \text{ and } \; \; 
S^\ell \ast S^k \cong S^{\ell+k+1}.
\]
\end{lemma}



\section{Reeb Spaces}\label{sec:reeb}
Reeb graphs have long been an object of interest, and, roughly, are graphic summaries of level sets of real valued functions.The Reeb graph is formed by identifying points that lie in the same connected component of a level set.  First introduced in ~\cite{reeb1946points} for use in Morse theory, they have since been widely used in applications (see \cite{biasotti2008reeb} for a survey) and have been studied for their mathematical properties (e.g., \cite{bauer2014measuring, de2016categorified, cole2004loops}) as well as in terms of efficient computation (e.g., \cite{doraiswamy2009efficient, dey2013efficient}) Reeb spaces generalize the notion of Reeb graphs to the case of an arbitrary map between spaces. 

\begin{definition}\label{def:reeb} Let $f: X \to Y$ be a continuous map of spaces.  The \emph{Reeb space} $\WW_f$ is the quotient space defined by the relation: for $a,b \in X$, we have $a \sim b$ whenever $f(a) = f(b)$ and if $a$ and $b$ are in the same connected component of $f^{-1}(f(a))$. 
\end{definition}

Let $q: X \to \WW_f$ be the canonical quotient map. It is often useful to visualize the relationship between these spaces in diagramatic form:
\begin{center}
\begin{tikzcd}[column sep=small]
X \arrow[rr, "f"] \arrow[dr, "q"'] & & Y \\
& \WW_f \arrow[ur, "g"'] & 
\end{tikzcd}
\end{center}

\noindent
where $g$ is the {\em Stein factorization} of $f$, the unique continuous map making the diagram commute. 

\begin{lemma}
Let $\WW_f$ be the Reeb space corresponding to the map $f: X \to Y$. If $X$ is normal, then $\WW_f$ is Hausdorff.
\end{lemma}

\begin{proof}
Since $X$ is normal, it suffices to show that $\sim^C$ is an open subset of $X \times X$. Let $(a,b) \in \sim^C$, meaning $a$ and $b$ do not lie in the same connected component of a level set of $f$. Note that the level set componenets containing $a$ and $b$, which we denote $A$ and $B$, respectively, are disjoint, and as they are connected components, both are closed in $X$. Since $A$ and $B$ are disjoint closed subsets of a normal space, we can find disjoint open sets containing them, which we denote $A'$ and $B'$, respectively. Then $A' \times B' \subseteq X \times X$ is an open set containing $(a,b)$. We claim that $A' \times B' \cap \sim = \emptyset$. Suppose, towards a contradiction, that $(x,y) \in A' \times B' \cap \sim$. Then $x$ and $y$ would be in the same connected component of a level set of $f$ while simultaneously being in $A'$ and $B'$, a contradiction. Since we have found an open neighborhood around an arbitrary point of $\sim^C$, we know $\sim$ is closed in $X \times X$, and thus, $\WW_f= X/\sim$ is Hausdorff.
\end{proof}


As our future applications are based in the realm of computational geometry, we consider the following hypotheses, a subset of those in~\cite{edelsbrunner2008reeb}:



\begin{assumption}[The EHP Hypotheses]
Let $X$ be a combinatorial manifold of dimension $n>k$ and  $f: X \to \RR^k$ be a PL function, so restricted to each simplex, $f$ is affine. Further, assume that $f$ is generic.
\end{assumption}





Note that the condition of normality needed for the above lemma is met by \ehp, so the associated Reeb space $\WW_f$ is Hausdorff. 

Recall the notion of genericity from Definition \ref{defn:generic}; the definition given is a codification of the idea that topological properties should be preserved under small perturbations of the map $f$.
Observe that the condition of genericity of $f$ implies that $f$ restricted to some $i$ simplex for $i \leq k$ is injective, i.e., the image of this simplex is again an $i$-simplex. 

Note that for a point in the interior of a simplex, $p \in \mathring{\sigma}$, the link is naturally presented as a join of spheres $\Link (p) =  S^{n-d-1}\ast S^{d-1} $ where $\dim \sigma = d$ and $\dim K = n$. If $K$ is the triangulation of a smooth manifold, then the second factor in the join can be identified with the unit sphere in tangent bundle to $\sigma$ at $p$, while the first factor is the unit normal sphere at $p$. This observation motivates the following definitions.



\begin{definition}\label{def:jacobi} Let $f: X \to \RR^k$ satisfy \ehp, and suppose that $X$ is $n$-dimensional.
\begin{itemize}
\item[{\rm (a)}] A point $p \in X$ is \emph{regular} if there exists a PL homeomorphism $\Link (p) \cong S^{n-k-1} \ast S^{k-1}$ such that
\[
f |_{\Link (p)} : S^{n-k-1} \ast S^{k-1} \to f(p) \ast S^{k-1}\]
respects the join decomposition and is a PL homeomorphism of spheres on the second factor.
\item[{\rm (b)}] The \emph{Jacobi set} of $X$ is the collection of critical points, i.e., those points that are not regular. The Jacobi set of the map $f$ is denoted $J_f$.
\end{itemize}
\end{definition}


\begin{remark} 
In the case that $f$ is a Morse one-function on a smooth manifold,  $J_f$ is exactly the critical points of $f$. 
\end{remark}

\begin{example}\label{ex:icos}
Let $X$ be an octohedron in $\RR^3$, and let $f: X \to \RR$ be the standard height function for a direction ${u} \in S^{2}$ that satisfies the EHP hypothesis (namely, this means all vertices of $X$ have a unique height with respect to $u$). Consider a point $p$, as in \figref{icos} We see that 
\[
\Link(p) = \Link ([am]) \ast \partial [am] = \{b, d\} \ast \{a,m\}
\] is the join of two copies of $S^0$, and the image $f|_{\Link(p)} = f(p) \ast \{f(a), f(m)\}$ is indeed a PL homeomorphism onto the second $S^0$ factor. Thus, we see that $p$ is regular.  Instead, consider $\Link(m) = \{a,c\} \ast \{d,b\}$. Although this link is also the join of two copies of $S^0$, since $f(m)$ is maximal, the image of $f|_{\Link(m)} \neq f(m) \ast S^0$. Thus, we see that $m$ is critical.
\end{example}


\begin{figure}[h!]
\centering
\includegraphics[scale=.35]{icosohedron}
\begin{caption}{The leftmost images illustrates why $p \in X$, an interior point, is regular and the rightmost image illustrates why $m \in X$ is a critical simplex, as described in \exref{icos}\label{fig:icos}}. 
\end{caption}
\end{figure}




\begin{prop}
Let $f: X \to \RR^k$ satisfy \ehp. The Jacobi set $J_f$ is an embedded simplicial complex in $X$ of dimension at most $k-1$.
\end{prop}

\begin{proof}
As the function $f$ is affine restricted to any given simplex, $\sigma$, if $p \in \mathring{\sigma}$ is critical than for any $q \in \mathring{\sigma}$, $q$ is also critical, i.e., criticality is constant on simplices. Also, by genericity of $f$, if $\sigma$ is a critical simplex, then $\dim \sigma \le k-1$.

To complete the proof, it is sufficient (and necessary) to show that criticality is a closed condition or equivalently that regularity is an open condition. Let $\tau \subset \sigma$ be a codimension one face, $x \in \mathring{\tau}$, and $p \in \mathring{\sigma}$. We claim that if $x$ is regular, then $p$ is regular. Indeed, we have
\[
\Link (x) \cong \partial \tau \ast (x \amalg p) \ast S^{n-k} \quad \text{ and } \quad \Link (p) \cong \partial \sigma \ast S^{n-k}.
\]
There is a PL homeomorphism $h: \Link(x) \cong \Link(p)$ that is the identity on the last factor of the join and such that $h(x) = p$.
Now if $x$ is regular via the decomposition $\Link(x) \cong S^{n-k-1} \ast S^{k-1}$, then the resulting decomposition of $\Link (p)$ proves that $p$ is regular. By induction on codimension, if the simplex $\sigma$ is critical, then any point in its closure is critical as well.
\end{proof}

 Lemma \ref{lem:nativeext} then determines two stratifications of $X$: one determined by $\Nat(J_f)^{\triangleright}$ and a refinement thereof given by $\Nat (J_f)^{\wwedge}$.


\begin{remark}
In \cite{edelsbrunner2008reeb} the authors define critical simplices using reduced homology. We discuss this definition (and another) of the Jacobi set in Appendix \ref{app:critical}.
\end{remark}



\subsection{The stratification of $\RR^k$}  


Next, we will show that $f(J_f)$ can determine similar stratifications of $\RR^k$.

Since $J_f$ contains only $(k-1)$ and lower dimensional simplices and since $f$ is generic, we see that $f$ takes simplices to simplices, hence $f(J_f)$ is a collection of $(k-1)$-simplices in $\RR^k$. Even so, there is no guarantee that these $(k-1)$-simplices will not intersect in $\RR^k$, which precludes us from immediately endowing $f(J_f)$ with a simplicial structure. Nonetheless, we have the following.

\begin{prop}\label{prop:Jsimp}
If $f:X \to \RR^k$ is an EHP map, then $f(J_f)$ admits a refinement into simplices so that, for $\tau, \sigma \in f(J_f)$ where $\dim(\tau) \leq \dim(\sigma)$, whenever a point $x \in \tau$ and $x \in \sigma$, we have $\tau \subseteq \sigma$ as subsets of $\RR^k$.
\end{prop}

The proof of this proposition is not exceedingly difficult, but is a bit tedious so we have relegated it to Appendix \ref{app:Jsimp}. In what follows, we may abuse notation and use $f(J_f)$ to mean $f(J_f)$ taken with this refinement.
Then, we can define the following.

\begin{definition} \label{def:cP}
Let $f:X \to \RR^k$ be an EHP map, and suppose that~$f(J_f)$ has been refined into simplices as described in \propref{Jsimp}. We define the poset $\cP_f$, where, for $\tau, \sigma \in f(J_f)$, we have $\tau \leq \sigma$ if and only if $\tau \subseteq \sigma$ as subsets of $\RR^k$. We stratify $\RR^k$ via the map $\varphi_{f(J_f)}$, where, if $\tau$ is the lowest dimensional simplex containing $x \in f(J_f)$, then $\varphi_{f(J_f)} (x) = \tau$. 
\end{definition}

Given the stratification $\RR^k \xrightarrow{\varphi_{f(J_f)}} \cP_f$ as above, we can stratify $\RR^k$ via
$\RR^k \xrightarrow{\psi_{f(J_F)}} \cP_f^{\wwedge}$ (see \defref{connectedstrat}) so that each stratumn is a connected manifold.

\subsection{The stratification of $\WW_f$}\label{ssec:reebstrat}
Since $\WW_f$ is a quotient of $X$ we would like to use the universal property to stratify $\WW_f$. However, the stratifying map $\psi_{J_f} :X \to \Nat (J_f)^{\wwedge}$ is not constant on the fibers of the quotient map $q$. (Nor does the map $f$ ---in general---satisfy the conditions under which $\Nat^{\wwedge}$ is functorial.) 



Consider the composition 
\[
 \varphi_{f(J_f)} \circ f  |_{J_f} : J_f \to \cP_f .
\]
This stratification is a refinement of $\Nat(J_f)$. For clarity, let us denote a copy of $\cP_f$ by $\widetilde{\cP}_f$ and notate the preceding map by $\Psi_{J_f}$, so we have $\Psi_{J_f} : J_f \to \widetilde{\cP}_f$. We can now apply the connected ambient construction to stratify $\WW_f$:
\[
\Psi_{\WW_f} : \WW_f \to \widetilde{\cP}_f^{\wwedge}.
\]
As the Stein factorization $g: \WW_f \to \RR^k$ is continuous---hence preserving connected components---it induces a natural map of posets $\overline{g} : \widetilde{\cP}_f^{\wwedge} \to \cP_f^{\wwedge}$. Moreover, the following is an easy tour through definitions.


\begin{prop}\label{prop:steinstrat}
The Stein factorization $g: \WW_f \to \RR^k$ is a map of stratified spaces, i.e., the following commutes in the category of topological spaces
\[
\xymatrix{ \WW_f \ar[r]^g \ar[d]_{\Psi_{\WW_f}} & \RR^k \ar[d]^{\psi_{\RR^k}} \\ \widetilde{\cP}_f^{\wwedge} \ar[r]^{\overline{g}} & \cP_f^{\wwedge}}
\]
\end{prop}

\begin{example}
Consider the height function on the upright torus (equipped with a triangulation), $h: T^2 \to \RR$ with  Jacobi set, $J_h$, the four Morse critical points.  \figref{torusheight} depicts the associated Stein factorization and stratifying posets. In this case, one could also conclude that the Stein factorization is a stratified map by applying Proposition \ref{prop:nativemap}. 
\end{example}

\begin{figure}[h!] 
\centering
\includegraphics[scale=.40]{torusheight}
\begin{caption}{The Stein factorization as a stratified map for the height function on the upright torus. This is an example of the diagram in \propref{steinstrat}.}
\end{caption}
\label{fig:torusheight}
\end{figure}


 
\begin{prop}
If If the EHP map $f:X \to \RR^k$ is injective on the $(k-1)$-skeleton of $X$, then $f:X \to \RR^k$ and $q: X \to \WW_f$ are stratified maps.
\end{prop}

\begin{proof}
If $f$ is injective on the $(k-1)$-skeleton of $X$, then none of the images of $(k-1)$-simplices of $J_f$ will intersect transversely (i.e., $f(J_f)$ has a simplicial complex structure), so $\cP_f^{\wwedge} = \Nat(f(J_f))^{\wwedge}$. Furthermore, we have $\Nat(f(J_f)) = \Nat(J_f)$ meaning that restricting $f$ and $q$ to elements of $\Nat(J_f)^{\wwedge}$ defines surjections onto $\Nat(f(J_f))^{\wwedge}$. 
\end{proof}

\begin{example}
Consider a generic projection $\pi: \Delta^3 \to \RR^2$ as in the leftmost image of \figref{delta3}.  Let $p$ be an interior point of a one-simplex as indicated. The link of $p$ is the two-sphere defined by the boundary of $\Delta^3$, and no join decomposition of it into $S^0 \ast S^1$ will satisfy the regularity condition (this can be verified by barycentrically subdividing $\Delta^3$ several times and recalling that subdivision induces a PL homeomorphism of links). Therefore, $p$---and hence the edge containing $p$---is in the Jacobi set of $\pi$. The link of $r$ is the same two-sphere, but a join decomposition of it does sataisfy the regularity condition, meaning that $r$---and hence the edge containing $r$---is regular. The Jacobi set is the simplicial circle formed by the four edges and four vertices that lie on the boundary of $\pi(\Delta^3)$ and is indicated by pink bolded lines. The Reeb space is shown on the right, and is homeomorphic to the 2-disc, $\WW_f \cong D^2$: it is stratified with four zero-simplices, four one-simplices (the images of those in the Jacobi set), and the open disc as its top stratum.
\end{example}



\begin{figure}[h!] 
\centering
\includegraphics[scale=.5]{delta3}
\begin{caption}{The quotient map and stratification of the Reeb space corresponding to a generic projection $\pi : \Delta^3 \to \RR^2$. The Jacobi set, $J_\pi$, is highlighted as a subcomplex of the domain.\label{fig:delta3}}
\end{caption}
\end{figure}






\subsection{Comparing to the EHP stratification}

A main result of \cite{edelsbrunner2008reeb} is defining the so called ``canonical stratification'' of Reeb spaces, which we will refer hereafter as the \emph{EHP stratification}. The EHP stratification is described using an algorithmic construction. First, $X$ is decomposed by coarsening a decomposition of $X$ into prisms created by preimages of the affine hulls of images of $(k-1)$-simplices. This decomposition is inherited by $\WW_f$, and the EHP stratification of $\WW_f$ is the skeletal stratification of a triangulation of the decomposition. As noted in \cite{edelsbrunner2008reeb} a precise version of this algorithm is limited to $k < 5$, since a Boolean subroutine used to determine when two triangulated spaces of dimension $k-1$ are homeomorphic is undecidable for $k \geq 5$. 

Although not explicitly considered in \cite{edelsbrunner2008reeb}, the process by which $\WW_f$ is triangulated could be further used to triangulate both $X$ and $\RR^k$ by star triangulating various convex subspaces. One of the more obvious differences between the EHP stratification of $\WW_f$ and our stratification in \ssref{reebstrat} is that, by construction, each stratum of $\widetilde{\cP}_f^{\wwedge}$ is connected, whereas this is not true in general for a stratum of the EHP stratification. This difference is relatively minor, and we expect under further hypotheses (e.g., $J_f$ is defined equivalently when using homological-criticality vs. criticality and $f(J_f)$ is already a simplicial complex without further refinement) that the stratification of \cite{edelsbrunner2008reeb} and the corresponding skeletal filtration obtained via Proposition \ref{prop:filtration} agree.


 
\section{Stratifying Targets of (Higher) Morse Functions}\label{sec:higher}

In this section, we extend the methods of \secref{reeb} to stratify the target of more general Morse functions. In the present section we do not assume \ehp. Our main results are in \ssref{targetstrat} and focus specifically Morse two-functions, but we additionally discuss how Morse n-functions could be treated similarly in \ssref{generalizations}. To build up to higher Morse functions, we start in the warmup setting of Morse one-functions.

\subsection{Morse One-Functions}

Consider the {\em upright} height function on the torus $h: T^2 \to \RR$ (\figref{upright}).

\begin{figure}[h!] 
\centering
\includegraphics[scale=.5]{upright}
\begin{caption}{The four critical values of $h$ are shown on the upright $T^2$ along with their corresponding images in $\RR$. The direction of $h$ is indicated by the leftmost arrow.}
\end{caption}
\label{fig:upright}
\end{figure}


\noindent
There are four critical points and we will stratify $T^2$ via the stable manifolds with respect to gradient flow of the function $-h$ (so we flow down), i.e., for a critical point $c \in T^2$, the corresponding stratum, $X_c$, is given by
\[
X_c = \left \{ p \in T^2 \mid \lim_{t \to \infty} \Phi (t,p) = c \right \},
\]
where $\Phi$ is the gradient flow of $-h$.



\begin{definition}
Let $f: M \to \RR$ be a Morse function on a compact smooth manifold. The {\em stable$^{-}$ stratification} of $M$ is determined by the stratifying map $s^{-} : M \to \cC^{-}$, where $\cC^{-}$ is the poset of critical values $\{c_\alpha \}$ with $c_\alpha$ and $c_\beta$ are related if there is a flow line from $c_\alpha$ to $c_\beta$.
\end{definition}



The following is just a rephrasing of standard results in Morse theory, see \cite{liviu}.

\begin{prop}
The stable$^-$ stratification is well defined, i.e., the triple $(M \xrightarrow{s^-} \cC^-)$ defines a stratified space. Moreover, this stable$^-$ stratification is Euclidean, i.e., each stratum is diffeomorphic to $\RR^n$ for some $n$.
\end{prop}

One should note that this stratification is rather wild. In particular, it is often desirable that an intersection of the closure of strata is contained in a stratum of lower dimension; this is violated in the upright torus example as the intersection of the closures of the strata corresponding to the critical points of index one is the just the upper index one critical point. The situation can be remedied by restricting to the case that the map $f: M \to \RR$ is {\em Morse--Smale}.

\begin{definition}
A Morse function $f: M \to \RR$ is {\em Morse--Smale} if for each pair of critical points $p,q \in M$ such that $f(p)>f(q)$, the the unstable manifold of $p$ intersects the stable manifold of $q$ transversely.
\end{definition}


An illustrative example of a Morse-Smale function is the height function on the {\em tilted torus} (\figref{tilted}). 

\begin{figure}[h!]
\centering
\includegraphics[scale=.5]{tilted}
\begin{caption}{The four critical values of $h$ are shown on the tilted $T^2$ along with their corresponding images in $\RR$. The direction of $h$ is indicated by the leftmost arrow.}\label{fig:tilted}
\end{caption}
\end{figure}


\begin{definition}[Lurie A.5.5]
Let $X \to \cP$ be a stratified space and $x \in X_p \subseteq X$ a point. The space $X$ is \emph{conically stratified at} $x$ if there exists an open neighborhood, $U_x$, of $x$ and a stratified homeomorphism $U_x \cong Z \times CY$ where $Z$ is a topological space and $CY$ is the cone on a space $Y$ stratified by $\cP_{>p}$.
\end{definition}

\begin{remark}
Note that any $C^0$ stratified space of \cite{aft} is conical.
\end{remark}

\begin{prop}
Let $M$ be a closed smooth manifold and $f: M \to \RR$  a Morse--Smale function. The stable$^-$ stratification of $M$ is conical.
\end{prop}

\begin{proof}
Indeed this statement is Proposition 2 of Laudenbach \cite{laud}.  The key idea is that when $f$ is Morse--Smale and the gradient vector field admits a normal form, then the closure of $X_c$ is a submanifold with at worst conical singularities.  The ``normal form" (or ``Special Morse" in {\it ibid}.) means  the vector field is the gradient of $-f$ with respect to local coordinates as in the Morse Lemma. This condition is satisfied by construction in the current setting.
\end{proof}

\begin{remark}
In the case that $f: M \to \RR$ is Morse-Smale, one can prove moreover that the stable$^-$ stratification is {\em conically smooth}. Conical smoothness is a desirable condition that has significant implications for the behavior of links and the exit/entrance path category \cite{aft, lurie, CP}. In the case $n=1$, there is a canonical identification of a link with the corresponding moduli space of gradient flows, see \cite{ralph, AyalaGrady}.
\end{remark}

\subsection{Morse two-functions}\label{ssec:morse2}
Recently Morse two-functions have been a critical tool in low dimensional topology, especially in dimension four. The work of Gay and Kirby is of particular note and \cite{GayKirby} is a good introduction to the relevant ideas.

\begin{definition}
A smooth proper function $f: M \to \RR^2$ is a \emph{Morse two-function} if it is locally a generic Morse homotopy. Equivalently, $f$ is smooth whose only singularities are folds and cusps.
\end{definition}

That $f: M \to \RR^2$ is locally a generic Morse homotopy means that each point $p \in M$ admits a neighborhood, $U_p$, such that $U_p \cong I \times X$ for $X$ a smooth manifold of dimension $\dim (M)-1$. Further, restricted to $U_p$, $f(t,p) = (t, g_t (p))$ where $g_t$ is a 1-parameter family of Morse functions. Finally, the family $g_t$ is \emph{generic}, so the only permitted singularities occur at the transverse crossing of critical values or at birth/death singularities. If $M$ is compact, then there are only finitely many crossings and birth/death singularities in the singular locus $\cL_f \subset \RR^2$. We illustrate these phenomena in the example below, see also Section 2 of~\cite{GayKirby}.

\begin{example}\label{ex:s2}
Suppose that we have a family of diffeomorphisms $g_t: S^2 \to S^2$ such as the one indicated at the bottom of \figref{morse2}. Then, for every $t \in I$, let $f_t: g_t(S^2) \to \RR$ be the standard height function. Since $g$ is smooth, we can define a Morse-2 function $f: I \times S^2 \to \RR^2$ where $f(t, s) = (t, f_t(g_t(s))$. See \figref{morse2} for an illustration of this example.
\end{example}

\begin{figure}[h!] 
\centering
\includegraphics[scale=.5]{morse2}
\begin{caption}{Above is the image of the image of the critical values for the Morse-2 function from \exref{s2} with four specific images of this diffeomorphism are shown at the bottom. Recall that $f(t,s)=(t, f_t(g_t(s))$ (where $f_t$ is the height function on $g_t(S^2)$). The four critical values for $t=1/4$ are indicated by yellow circles, and the heights at which they occur is recorded by heights of points in the singular locus at $t=1/4$. Note that there only values of $t$ for which $g_t(S^2)$ does not have four critical values is $t=0,1$ (where there are only a minimum and maximum) and $t = 1/2$ (where there is a minimum, saddle point, and two maximums).}\label{fig:morse2}
\end{caption}
\end{figure}

It follows from general \emph{Cerf/Stability Theory} that Morse two-functions are generic in the space of smooth maps \cite{GG}. Moreover, following Section 2 of \cite{BH}, we can assume our Morse two-functions are compatible with coordinate projections of $\RR^2$.

\begin{lemma}\label{lem:cerf}
In the space of Morse two-functions, $\mathrm{Morse}(M, \RR^2)$, those functions, $f$,  that satisfy
\begin{enumerate}[label=(\roman*)]
\item The composite  $\mathrm{proj}_x \circ f : M \to \RR$, is Morse; and
\item The composite $\mathrm{proj}_y \circ f$, restricted to the fibers of $\mathrm{proj}_x \circ f$ is Morse;
\end{enumerate}
constitue an open, dense subspace.
\end{lemma}

\begin{example}[\exref{s2} continued]
Consider the map $f: I \times S^2 \to \RR^2$ where $f(t,s) = (t, f_t(g_t(s)))$ records the height of point $s$ under $g_t$, a diffeomorphism of $S^2$ as in \exref{s2}.  First, we observe that that the composite $\mathrm{proj}_x \circ f$ is simply the identity function on $I$, and is therefore Morse.
Next, we observe that the composite $\mathrm{proj}_y \circ f$ restricted to $t_*$ (some fiber of $\mathrm{proj}_x \circ f$) is equivalent to the height function on $g_{t_*}(S^2)$, and is therefore also Morse.
Thus, $f$ satisfies both conditions of \lemref{cerf}.
\end{example}

\begin{nonexample}[A Morse Homotopy Non-Example]
Consider the map $f: I \times T^2 \to \RR^2$ where $f(t,s) = (t, f_t(g_t(s)))$ records the height of $s \in T^2$ under a diffeomorphism $g_t$ that rotates $T^2$ in such a way that $g_{t_1}(T^2)$ is a `horizontal torus' (like pushing the upright torus in \figref{upright} back to lay flat) for some $t_1 \in I$. Then $f$ does not satisfy the conditions of \lemref{cerf},  since $\mathrm{proj}_y \circ f$ restricted to the fiber $t_1$ has a degenerate critical point (both the maximum and minimum of $g_{t_1}(T^2)$ are entire circles rather than single points), and is not Morse.
\end{nonexample}
As our primary interest moving forward is the stratification of the target of a Morse two-function, we defer stratifying the domain to the following two (open) tasks. Let $f: M \to \RR^2$ be a Morse two-function.
\begin{enumerate}
\item Describe $M$ as a stratified manifold such that stratification is obtained as a refinement of that determined by the horizontal Morse function $\mathrm{proj}_x : \RR^2 \to \RR$, e.g., by using $\mathrm{proj}_y \circ f$ restricted to the fibers of the horizontal function;
\item Define an analog of the Morse-Smale condition for Morse two-functions such that the stratification of $M$ is conical/conically smooth.
\end{enumerate}

\subsubsection{Remark on gradient flow}\label{ssec:flow}

A key tool in Morse theory is that of gradient like vector fields. Of course for a vector valued function with target at least two dimensional,  $f: M \to \RR^n$, the analogue of the gradient, the Jacobian, does not define a vector field. However, by choosing a \emph{horizontal distribution} we can still define parallel transport along paths. Our presentation is a brief summary of Sections 4.1 and 4.2 of \cite{BH}.


Let $f: M \to B$ be smooth. A \emph{connection} for $f$ is a subset $\cH \subset TM$ that is a point-wise orthogonal complement to $\ker (df)$ with respect to some metric on $M$. (This is a generalization of an Ehresmann connection in the case that $M \to B$ is a fiber bundle.) Given a path $\gamma : [0,1] \to B$, we would like obtain a lift $\widetilde{\gamma} : [0,1] \to M$ such that $\gamma'(t) \in \cH$ for all $t \in [0,1]$. Moreover, we want the lift $\widetilde{\gamma}$ to be unique in order to define parallel transport, monodromy, etc.  There are some technical difficulties that arise in this most general setting (significantly parallel transport may only be partially defined), but for paths of regular values we obtain familiar results.

\begin{prop}[\cite{BH}]\label{prop:transport}
Let $f: M \to B$ be smooth, $\cH \subset TM$ a connection for $f$ and $\gamma: [0,1] \to B$ a path such that for each $t \in [0,1]$ $\gamma(t) \in B$ is a regular value of $f$. Then,
\begin{itemize}
\item[(a)] There is a well defined parallel transport map along $\gamma$, $\Pi_\gamma^{\cH} : f^{-1} (\gamma(0)) \to f^{-1} (\gamma(1))$;
\item[(b)] The map $\Pi_\gamma^{\cH}$ is a diffeomorphism;
\item[(c)] The map $\Pi_\gamma^{\cH}$ is independent, up to isotopy, of the choice of $\cH$ and depends only on the homotopy class of $\gamma$ relative to its endpoints within the connected component of $B \setminus \cL_f$, where $\cL_f$ is the singular locus of $f$.
\end{itemize}

\end{prop}








\subsection{The stratification of the target}\label{ssec:targetstrat}

Given a Morse function $f: M \to \RR$, there is an associated persistence module/cosheaf over $\RR$ stratified by critical values, see \cite{CP}. We now describe the target stratification determined by a Morse two-function $f: M \to \RR^2$. (As noted above, we defer the full discussion of the associated persistence module to a follow up paper.) The idea is quite simple: if $\cL_f \subset \RR^2$ is the singular locus of $f$,  we equip it with a simplicial structure $\fs \cL_f$ and we consider the stratification determined by $\Nat (\fs \cL_f)^{\wwedge}$. 

\subsubsection{Simplicial Structure}

Let $\cL_f \subset \RR^2$ be the singular locus of a Morse two-function $f: M \to \RR^2$.  We describe a recipe for replacing $\cL_f$ by an embedded simplicial complex, $\fs \cL_f$, of dimension 1. The output is such that there is a well defined native stratification $\Nat (\fs \cL_f)$.
We build up $\fs \cL_f$ in the following way; 
\begin{enumerate}
\item zero-simplex at each crossing;
\item zero-simplex at each birth-death;
\item zero-simplex at each vertical tangency;
\item appropriate one-simplices using segments of $\cL_f$ (parametrized by arc length).
\end{enumerate}


\begin{lemma}[Uniqueness]
The stratification $\Nat (\fs \cL_f)$ is the coarsest stratification of $\cL_f$ such that points of vertical tangency are strata and such that each stratum is a connected manifold.
\end{lemma}

\subsubsection{The stratification}

For $f: M \to \RR^2$ a Morse two-function, equip $\RR^2$ with the $\Nat (\fs \cL_f)^{\wwedge}$ stratification. This stratification is quite well-behaved as the following proposition indicates.



\begin{prop}
Let $f: M \to \RR^2$ be a Morse two-function. The topology of preimages only changes when crossing the singular locus $\cL_f$.
\end{prop}

\begin{proof}
Let $p,q \in \RR^2 \setminus \cL_f$ be in the same connected component. Since each stratum is a connected manifold, there is a (smooth) path $\gamma: [0,1] \to \RR^2 \setminus \cL_f$ with $\gamma(0) = p$ and $\gamma (1) = q$. By construction, $\gamma (t)$ is a regular value for all $t \in [0,1]$. Let $\cH \subset TM$ be a connection for $f$ (the space of such is nonempty), and $\Pi_\gamma^\cH : f^{-1} (p) \to f^{-1} (q)$ the associated parallel transport operator. Proposition \ref{prop:transport} (b) implies that $\Pi_\gamma^\cH$ is a diffeomorphism.
\end{proof}


\subsection{Generalizations}\label{ssec:generalizations}

In \cite{GayKirby}, Morse two-functions can have as codomain any (oriented) surface $\Sigma$. The key idea is that any point $q \in \Sigma$ admits a neighborhood diffeomorphic to $I \times I$ such that restricted to the preimage of this neighborhood the function is a generic Morse homotopy. Note that at this generality there is no global notion of ``index," but rather only locally defined indices for the Morse homotopy.



One could give a definition of Morse $n$-function for higher $n$ by iterating the definitions of Gay and Kirby \cite{GayKirby} or by simply fixing singularity types as in \cite{GG}. Again stratifying the domain of such a function is subtle, but the technique described above readily extends to stratify the target of any (reasonable) Morse $n$-function.



