Differential Borel equivariant cohomology via connections
Corbett Redden
For a compact Lie group acting on a smooth manifold, we define the differential cohomology of a certain quotient stack involving principal bundles with connection. This produces differential equivariant cohomology groups that map to the Cartan--Weil equivariant forms and to Borel's equivariant integral cohomology. We show the Chern--Weil homomorphism for equivariant vector bundles with connection naturally factors through differential equivariant cohomology.
BorelEquiv.tex
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\title{Differential Borel equivariant cohomology via connections}
\author{Corbett Redden}
\address{Department of Mathematics, LIU Post, Long Island University, 720 Northern Blvd, Brookville, NY 11548, USA}
\email{corbett.redden@liu.edu}
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%\dedicatory{To my wonderful Valentine, Mary}
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\begin{abstract}For a compact Lie group acting on a smooth manifold, we define the differential cohomology of a certain quotient stack involving principal bundles with connection.  This produces differential equivariant cohomology groups that map to the Cartan--Weil equivariant forms and to Borel's equivariant integral cohomology.  We show the Chern--Weil homomorphism for equivariant vector bundles with connection naturally factors through differential equivariant cohomology.
\end{abstract}

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%\begin{center} \fbox{DRAFT!!!!} \end{center}  %Emphasize draft copy
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%Essentially Completed sections: \ref{sec:Background}, \ref{sec:HS}
\section{Introduction}

Differential cohomology, also known as Deligne cohomology \cite{Bry93} or Cheeger--Simons differential characters \cite{CS85}, is a contravariant functor $\HG[]^*$ from the category of manifolds to graded abelian groups.  It sits in the character diagram
\[ \xymatrix@R=3mm@C=6mm{ 0 \ar[dr]&&&&0 \\
&H^{n - 1}(M;\R/\Z) \ar[dr] \ar^{-B}[rr]&& H^{n}(M;\Z) \ar[ur]  \ar[dr]\\
H^{n -1}(M;\R) \ar[ur] \ar[dr]&& \HG[]^{n}(M) \ar[ur] \ar[dr] && H^{n}(M;\R) \\
& {\displaystyle \frac{\Omega^{n-1}(M)}{\Omega^{n-1}(M)_\Z}} \ar[ur] \ar_{d}[rr]&& \Omega^{n}(M)_\Z \ar[ur] \ar[dr]\\
0 \ar[ur]&&&&0
} \]
where the two diagonals are short exact sequences, $B$ is the Bockstein homomorphism, and the subscript $\Z$ denotes closed forms with $\Z$-periods.  Because it captures both the torsion of integral cohomology and the local structure of differential forms, differential cohomology has proven to be useful in a number of contexts.  In particular, the Chern--Weil homomorphism factors through $\HG[]^*$, making it a natural home for secondary invariants of principal bundles with connection.  Differential cohomology has now been generalized in several directions, leading to the notion of a ``differential extension'' of a generalized cohomology theory.

The purpose of this paper is to construct a differential extension of Borel's equivariant integral cohomology.  This was also recently done by K\"ubel \cite{1510.06392v1}, and our functors fit into the same short exact sequences.   Assume that $G$ is a compact Lie group acting smoothly on a manifold $M$.   The equivariant cohomology of $M$, with coefficients in an abelian group $A$, is the cohomology of the homotopy quotient $H_G^*(M;A) := H^*(EG\times_G M;A)$.  The complex of equivariant forms $(\O_G(M), d_G)$ is given by the Weil model $(S\fg^* \otimes \L \fg^* \otimes \O(M))^G_\hor$, whose cohomology is naturally isomorphic to $H^*_G(M;\R)$.  One can equivalently use the Cartan model $\left(S\fg^*\otimes \O(M) \right)^G$ in place of the Weil model, as the two complexes are isomorphic (not merely quasi-isomorphic).  We summarize the paper's main results in two theorems below.   Theorem \ref{thm:Properties} contains the important user-friendly properties of $\HG^*$.  It is simply Propositions \ref{prop:ses}, \ref{prop:GroupChange}, and \ref{prop:Reduction} combined into a single statement.  In fact, we show in Proposition \ref{prop:Uniqueness} that the functor $\HG^*$ is characterized uniquely by properties \ref{thm1diag} and \ref{thm1conn}.  Theorem \ref{thm:equivCCS} states that the equivariant Chern--Weil homomorphism, with equivariant forms defined by the equivariant curvature $\Omega_G$ \cite{MR705039}, factors through $\HG^*$.  For convenience, we restate it here as Theorem \ref{thm:ChernWeil} in the specific case of equivariant complex vector bundles.

 \begin{thm}\label{thm:Properties}For any compact Lie group $G$, there exists a contravariant functor $\HG^*(-)$, from the category of smooth $G$-manifolds to the category of graded abelian groups, satisfying the following.
 \begin{enumerate}[leftmargin=*, label=(\Alph*)]
 \item\label{thm1diag} There exist natural homomorphisms forming the following character diagram.
\[ \xymatrix@R=3mm@C=6mm{ 0 \ar[dr]&&&&0 \\
&H_G^{n - 1}(M;\R/\Z) \ar[dr] \ar^{-B}[rr]&& H_G^{n}(M;\Z) \ar[ur]  \ar[dr]\\
H_G^{n -1}(M;\R) \ar[ur] \ar[dr]&& \HG^{n}(M) \ar_{\cc}[ur] \ar^{\curv}[dr] \ar@{}_>>>>>{\dR}[rr]&& H_G^{n}(M;\R) \\
& {\displaystyle \frac{\Omega^{n-1}_G(M)}{\Omega^{n-1}_G(M)_\Z}} \ar[ur] \ar_{d_G}[rr]&& \Omega^{n}_G(M)_\Z \ar[ur] \ar[dr]\\
0 \ar[ur]&&&&0
} \]
 \item\label{thm1group} For $G_1 \xrightarrow{\phi} G_2$ a Lie group homomorphism, a smooth map $M \xrightarrow{F} N$ that is $\phi$-equivariant induces homomorphisms 
 \[ \HG[G_2]^*(N) \xrightarrow{F^*_\phi}  \HG[G_1]^*(M)\] compatible with the character diagrams.
 \item\label{thm1conn} A principal $G$-bundle with connection $(P,\Theta) \to M$ induces homomorphisms 
 \[ \HG^*(P) \xrightarrow{\Theta^*} \HG[]^*(M)\] 
 compatible with the character diagrams.
 \end{enumerate}
 \end{thm}
 
 
 \begin{thm}\label{thm:ChernWeil}Let $(V,\nabla) \to M$ be a $G$-equivariant Hermitian vector bundle with $G$-invariant connection.  There exist natural classes $\check{c}_k(\Theta_G) \in \HG^{2k}(M)$ satisfying
 \begin{enumerate}
 \item $\cc\left( \check{c}_k(\Theta_G) \right) = c_k \left( EG \times_G V \to EG \times_G M \right) \in H^{2k}_G(M;\Z)$,
 \item $\curv \left( \check{c}_k(\Theta_G) \right) = c_k\left( \Omega_G\right) \in \O^{2k}_G(M)$.
 \end{enumerate}
 \end{thm}

% states that the equivariant Chern--Weil homomorphism 
%We also show that the equivariant Chern--Weil homomorphism factors through $\HG^*$.  If $(Q,\Theta) \to M$ is a $G$-equivariant principal $K$-bundle with $G$-invariant connection, the Chern--Weil homomorphism gives rise to a homomorphism
%\[ H^{2k}(BK;\Z) \longrightarrow \HG^{2k}(M) \]
%refining the usual equivariant characteristic classes and forms.  This is shown in Theorem \ref{thm:equivCCS}, which we state here for the specific case of equivariant complex vector bundles with connection.

The paper is organized as follows.  Appendix \ref{sec:Background}, which appears at the end of the paper, contains a review of principal bundles with connection and the equivariant de Rham complex $\O_G(M)$.  Readers unfamiliar with the Weil model may find it helpful to begin here.  While this appendix is expository and should probably be replaced by a citation to \cite{GuilleminSternberg99}, we use it to introduce notation and to emphasize the relationship between $\W(\fg)$ and differential forms on principal bundles with connection.

In Section \ref{sec:HS} we define $\HG^*(M)$ using the Hopkins--Singer \cite{HS05} cochain model for $\HG[]^*(EG\times_G M)$, but with $\O(EG\times_G M)$ replaced by $\O_G(M)$.  The short exact sequences and character diagram follow immediately.  The advantage of this construction is that one need not leave the world of cochain complexes and topological spaces.  When giving constructions such as the Chern--Weil refinement, however, one is forced to play the tedious game of choosing classifying maps and checking everything is well-defined.

To emphasize the geometric nature of our constructions, we begin working with the differential Borel quotient stack $\En G \times_G M$ in Section \ref{sec:QuotientStack}.  This is defined as a contravariant functor that associates to any test manifold $X$ the groupoid of principal $G$-bundles with connection $(P,\Theta)\to X$, together with an equivariant map $P\to M$.   In \cite{MR3049871}, Freed--Hopkins show that a natural map
\[ \O^n_G(M) \xrightarrow{\iso} \O^n(\En G \times_G M)\] 
is an isomorphism, and we proceed to give the second definition
\[ \HG^n(M) \= \HG[]^n(\En G \times_G M).\]  
The details of this definition involve simplicial sheaves and are contained in Section \ref{Section:DefnProof}.  The virtue is that maps between various stacks induce homomorphisms in differential cohomology, as we demonstrate in Section \ref{sec:Examples}.

For example, the associated bundle construction defines a natural map
\[ \En G_1 \times_{G_1} M \xrightarrow{F_\phi} \En G_2 \times_{G_2} N,\]
which induces the homomorphism \ref{thm1group} from Theorem \ref{thm:Properties}.  Similarly, if $(P,\Theta) \to M$ is a principal $G$-bundle with connection, the pullback gives a natural map 
\[ M \xrightarrow{\Theta} \En G \times_G P,\]
 and this induces the homomorphism in \ref{thm1conn}.  For $(Q,\Theta) \to M$ a $G$-equivariant principal $K$-bundle with $G$-invariant connection, the refined Chern--Weil homomorphism is induced by a natural map 
 \[ \En G \times_G M \longrightarrow \Bn K,\]
 as described in Section \ref{sec:EquivCW}.  Given a $G$-bundle with connection mapping to $M$, we pull back $(Q,\Theta)$ to form a $G\times K$-bundle with connection, and then we quotient by $G$.  The equivariant extensions of the Chern forms appear when forming the connection on the pullback of $(Q,\Theta)$.  We conclude Section \ref{sec:EquivCW} with a discussion of $\HG^n(M)$ as differential characters on $\En G \times_G M$, and we show $\HG^2(M)$ is naturally equivalent to isomorphism classes of $G$-equivariant principal $\R/\Z$-bundles on $M$ with invariant connection.
%  This 
% , the connection includes a term coupling the $G$-connection with the $G$-action on $Q$.  In this way, equivariant Chern forms use the $G$-connection on a test bundle to detect information about the $G$-action on $Q\to M$.   We should note that these constructions are very natural, and they appear implicitly in Bott--Tu's explicit comparison of the two equivariant Chern--Weil homomorphism maps \cite{MR1850463}.  
% 

A  definition of differential equivariant cohomology was previously given by Gomi \cite{MR2147734}, but it did not fully incorporate $\O_G(M)$, leading to groups that are not isomorphic to the ones in this paper.  See \cite{1510.06392v1} for a more detailed discussion.  In fact, many of our results can also be found in K\"ubel's paper, including the short exact sequences and the refined equivariant Chern--Weil homomorphism.  However, the constructions and methods are different.  The two definitions in this paper involve the topological space $EG\times_G M$ and the stack $\En G \times_G M$, and most homomorphisms are induced by bundle constructions.  On the other hand, K\"ubel uses the simplicial manifold $\{ G^\bullet \times M\}$ as the model for the homotopy quotient.  We believe that having multiple viewpoints will prove to be useful, just as it has for ordinary differential cohomology. 
%   , we believe it will be helpful to have these different points of view when analyzing future problems.  
%
%We should make a comment on how this relates to other constructions of differential equivariant cohomology in the literature.  The differential equivariant cohomology functor $\HG^*$ has already appeared independently in recent work by K\"ubel \cite{1510.06392v1}.  Many of the results from our paper, including the short exact sequences and the refined equivariant Chern--Weil homomorphism, are also contained K\"ubel's work.  
%On the other hand, there have also appeared versions of differential equivariant cohomology that do not agree with our construction.  Most notably, Gomi gives a definition in \cite{MR2147734}, but his version does not properly include the equivariant forms, resulting in something that is a quotient of $\HG^*(M)$.  See \cite[Theorem 3.28]{1510.06392v1}, and also the rest of the paper for a nice discussion of the comparisons between different previous versions.

%
%While $\HG^*(M)$ can be constructed using only cochain complexes and the topological space $EG\times_G M$ (regarded as a limit of smooth manifolds), the need to regularly choose and check independence of classifying maps is a nuisance that obscures the geometric ideas.  We avoid these complications by using the language of stacks and simplicial presheaves.  The more natural definition is
%\[ \HG^*(M) \= \HG[]^*(\En G \times_G M),\]
%where $\En G \times_G M$ is the quotient stack defined using principal bundles with connection.  This allows us to give constructions in a direct geometric way.  For example, the homomorphism \ref{thm1group} is induced by a natural map 
%\[ \En G_1 \times_{G_1} M \xrightarrow{F_\phi} \En G_2 \times_{G_2} N,\]
%which is given explicitly by the associated bundle construction.  Similarly, if $(P,\Theta) \to M$ is a principal $G$-bundle with connection, the pullback operation for bundles with connection gives a natural map 
%\[ M \xrightarrow{\Theta} \En G \times_G P,\]
% and this induces the homomorphism in \ref{thm1conn}.  For $(Q,\Theta) \to M$ a $G$-equivariant principal $K$-bundle with $G$-invariant connection, the refined Chern--Weil homomorphism is induced by a natural map 
% \[ \En G \times_G M \longrightarrow \Bn K.\]



%The paper is organized as follows.  Appendix \ref{sec:Background} contains a detailed exposition the de Rham theory for equivariant cohomology.  Though there are already several excellent introductions to the subject, we have included this for a few reasons. The first is to make the paper self-contained, for the equivariant forms are the key to getting the differential refinement correct.  Secondly, there are a few equivalent, but different definitions, so writing this all here allows us to specify our conventions without interrupting the main flow for readers already familiar with the Weil algebra.  Finally, we want to emphasize the close relationship between Weil algebra and differential forms on principal bundles with connection, as this is foundational for understanding the isomorphism $\O_G(M) \xrightarrow{\iso} \O(\En G \times_G M)$.

%Section \ref{sec:HS} uses the Hopkins--Singer construction of differential cohomology \cite{HS05} to define $\HG^*(M)$ purely in terms of the usual singular cohomology of the topological space $EG \times_G M$ and $\O_G(M)$.  We use the Hopkins--Singer cochain construction for $EG\times_G M$, but replace $\O(EG\times_G M$ by $\O_G(M)$.  In fact, the construction appears explicitly in \cite{HS05} for the special case where $M=\pt$.  A number of properties follow immediately by the exact same arguments as ordinary differential cohomology.  The advantage of this construction is that one need not leave the relatively comfortable world of nice topological spaces and cochain complexes.  The price one must pay is that $\O_G(M)$ is best understood from the point of view of principal bundles with connection.  While $(EG,\Theta_{EG})\to BG$ is a universal bundle, classifying maps are not unique.  This means that when working with connections, one is forced to play the tedious (though not terribly difficult) game of continually choosing classifying maps and then checking that results do not depend on them.
%
%However, the real main goal of our paper is to show that the equivariant Chern--Weil homomorphism is produced by functorial bundle constructions.  This is in contrast to usual constructions, where you have two different constructions, and then you must put them in the same place and show they agree.  
%
%To accomplish this, we use the language of stacks, which are contravariant functors from the category of manifolds to the category of groupoids.  Basically, stacks are things that pull back, and principal bundles with connection are one of the fundamental examples.  Working in this formalism means that a principal bundle with connection $(P,\Theta)\to X$ is naturally given by a map
%\[ X \xrightarrow{(P,\Theta)} \Bn G,\]
%and the category of bundles with connection is naturally equivalent to the ``mapping space." We fully explain this in Section \ref{sec:QuotientStack}, and we explain the result of Freed--Hopkins.  While $\O_G(M)$ is often thought of as a convenient algebraic replacement for $\O(EG\times_G M)$, there is an actual isomorphism
%\[ \O_G(M) \xrightarrow{\iso} \O(\En G \times_G M),\]
%where the right-hand side is defined appropriately.  We define the quotient stack $\En G \times_G M$, which associates to any manifold $X$ the category of principal $G$-bundles with connection mapping equivariantly to $M$.  
%
%We must embed groupoid-valued functors into simplicial-set valued functors via the nerve construction, and then we define $\HG[]^*(\En G \times_G M)$ appropriately in Section \ref{subsec:sPreDefn} via simplicial presheaves.  The point of this is that our construction is functorial with respect to stack maps, and our main constructions are most easily phrased in terms of stack maps (functorial constructions involving bundles with connection).  The details of this construction are not needed to understand the remainder of the paper (and what we consider the primary points), so we postpone the proofs until Section \ref{Section:DefnProof}.
%
%Section \ref{sec:Examples} gives two main constructions.  They are the associated bundle construction, and pulling back bundles.  The associated bundle construction gives, to $\phi\colon  G_1 \to G_2$ and $F\colon  M\to N$, 
%\[ \En G_1 \times_{G_1} M \xrightarrow{F_\phi} \En G_2 \times{G_2} N.\]
%Similarly for $G$-equivariant $K$-bundles with connection.  
%
%We also show that these structures give a uniqueness theorem for differential equivariant cohomology.  It is not known whether all these conditions are necessary.  
%
%Combining these two results allows us to describe the Chern--Weil homomorphism as being induced by a map between stacks (Section \ref{sec:EquivCW}).  We should note that these constructions are very natural, and they appear implicitly in Bott--Tu's explicit comparison of the two equivariant Chern--Weil homomorphism maps \cite{MR1850463}.  We describe how elements naturally give equivariant differential characters, a sort of differential character on the quotient stack $\En G \times_G M$.  Structures such as these have appeared before as topological terms in gauged sigma models, and we hope that $\HG^*(M)$ may provide a useful way to describing such terms.  Finally, we note familiar interpretations of $\HG^*(M)$ in low degrees.  In particular, $\HG^2(M)$ is naturally equivalent to isomorphism classes of $G$-equivariant principal $U(1)$-bundles with invariant connection on $M$.
%
%Finally, Section \ref{Section:DefnProof} contains the proofs that our definition of $\HG[]^*(\En G \times_G M)$ agrees with our the definition from Section \ref{sec:HS}.  The proof essentially follows by using two existing results.  We consider a simplicial presheaf representing differential cohomology (this has been given in various places in the literature), and it is given as the homotopy pullback between forms (set-valued) and objects representing integral and real cohomology.  We must show that 3 corners are equivalent, and the property of homotopy pullback then gives that the final corner is equivalent.  The forms are taken care of by the Freed--Hopkins result.  The other two corners are computed using the theory of homotopy-invariant sheaves by Bunke--Nikolaus--V\"olkl \cite{1311.3188}.  
%
%


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\section{Cochain model}\label{sec:HS}
Our first definition of $\HG^*(-)$ is given by using the Hopkins--Singer construction for $\HG[]^*(EG\times_G M)$, with $\O(EG\times_G M)$ replaced by the Weil model of equivariant forms $\O_G(M)$.  The definition, short exact sequences, and ring structure all follow immediately from the same arguments given in Sections 2 and 3 of the Hopkins--Singer paper \cite{HS05}, so we will keep the proofs brief.  (In fact, \cite{HS05} also contains our construction for the case where $M=\pt$.)  Readers unfamiliar with the  Weil model of equivariant forms $\O_G(M)$ can find a full exposition in Appendix \ref{sec:Background}.  As noted in Section \ref{subsec:CartanMQ}, the Weil model may be replaced, via the Mathai--Quillen isomorphism, by the isomorphic Cartan model $\left( S\fg^* \otimes \O(M)\right)^G$  if desired.  

%  We give a more geometric definition in Section \ref{sec:characters}.

% In Subsection \ref{subsec:characters}, we give an equivalent but more geometric construction using differential characters on the stack $\EnG \times_G M$.  In Subsection \ref{subsec:Trivializations}, we note that the results from \cite{Redden11a} concerning trivializations apply to the equivariant situation.
%

As discussed in Section \ref{subsec:deRhamModel}, let $(EG, \Theta_{EG})  \to BG$ be a universal principal $G$-bundle with connection, given as a direct limit of smooth finite-dimensional bundles.  If $M$ is a $G$-manifold, the Weil homomorphism induces an inclusion 
\[ \O^*_G(M) \xhookrightarrow{\w(\Theta_{EG})\otimes 1} \O^*(EG\times_G M) \into C^*(EG\times_G M;\R)\]
that is a quasi-isomorphism of cochain complexes.  

For the moment, we will use general coefficients.  Let $\L \subset V$ be a completely disconnected subgroup of a (possibly graded) vector space, and denote $\O^*(-;V) = \O^*(-)\otimes V$.  

\begin{defn}The differential equivariant cochain complex $(\CG(q)^*(M;\L),d)$ is a homotopy pullback in the diagram
\begin{equation}\label{eq:htpysquare} \vcenter{ \xymatrix{\CG(q)^*(M;\L) \ar[d]^{\curv} \ar[r]^-{\cc} &  C^*(EG\times_G M;\L) \ar@{_(->}[d] \\ \O_G^{*\geq q}(M;V) \ar@{^(->}[r] &C^*(EG \times_G M;V). }}\end{equation}
Explicitly, this is defined so that for $k \geq q$,
\begin{gather*}
\CG(q)^k(M;\L) \= C^k(EG \times_G M;\L) \times C^{k-1}(EG\times_G M;V) \times \O_G^k(M;V) \\
d(c,h,\omega) \= (\delta c, \, \omega - c - \delta h, \, \dG \omega),
\end{gather*}
and for $k < q$ we restrict to the subcomplex with $\omega=0$.  

For $\x = (c,h,\omega)$, we call $c$ the \textit{characteristic cocycle} and $\omega$ the \textit{curvature}, and we say $\x$ is \textit{flat} if the curvature is zero.  The degree $n$ differential equivariant cohomology is defined as the abelian group 
\[ \HG^n(M;\L) \=  H^n\left(\CG(n)^*(M;\L) \right) = \frac{\ZG^n(M;\L)}{d\CG^{n-1}(M;\L)_{\flat}} .\]
\end{defn}

%As hinted at in the above diagram, f
%  We denote the subgroup of flat elements 
%\[ \CG^k(M;\L)_{\flat} \= \Ker \curv \subset \CG^k(M;\L),\]
%and  $\CG^*(M;\L)_{\flat < n}$ denotes the sub-complex defined by 
%\[ \CG^k(M;\L)_{\flat < n} \= \begin{cases}  \CG^k(M;\L)_{\flat}  & k < n\\ \CG^k(M;\L) &k\geq n .\end{cases} \]
%
%\begin{defn}The category $\cHG^n(M;\L)$ is the category of $n$-cocycles in the complex $\CG^*(M;\L)_{\flat < n}$.  Explicitly, this means
%\begin{align*} &\Obj \cHG^n(M;\L) = \ZG^n(M;\L), \\
%&\Hom(\x_1, \x_2) = \{ \y \in \CG^{n-1}(M;\L)_{\flat} \> | \> d\y = \x_2 - \x_1 \} \Big/ d\CG^{n-2}(M;\L)_{\flat}.
%\end{align*}
%The set of isomorphism classes $\pi_0\left(\cHG^n(M;\L)\right)$ defines defines the abelian group
%\[ \HG^n(M;\L) \= H^n\left(\CG^*(M;\L)_{\flat < n}\right) = \frac{\ZG^n(M;\L)}{d\CG^{n-1}(M;\L)_{\flat}}. \]
%\end{defn}


As one would expect, the groups $\HG^*(M;\L)$ lie in short exact sequences that are completely analogous to those for ordinary differential cohomology.  We denote the image of $H^n_G(M;\L) \to H^n_G(M;V)$ by $H_G^n(M;V)_\L$, and we denote the subgroup of closed $n$-forms with $\L$-periods by $\O_G^n(M;V)_\L$.  This is defined so that under the de Rham isomorphism, 
\[ \frac{\O^n_G(M;V)_\L}{\dG \O^{n-1}_G(M;V)} \xrightarrow{\iso} H^n_G(M;V)_\L. \]

We will primarily be concerned with $\L=\Z$, $V=\R$, and we use the notations 
\[  \HG^*(M) = \HG^*(M;\Z), \quad  \HG^* = \HG^*(\pt;\Z).\]
In later sections, we will simply use $\Z \subset \R$ to simplify notation, but all results will immediately generalize to general coefficients $\L \subset V$.


\begin{prop}\label{prop:ses}The groups $\HG^*(M;\L)$ lie in the short exact sequences
\begin{align}
\label{ses1}\tag{SES 1}  0 \longrightarrow H^{n- 1}_G(M; V/\L) \longrightarrow &\HG^n(M;\L) \overset{\curv}\longrightarrow \O^n_G(M;V)_\L \longrightarrow 0, \\
\label{ses2}\tag{SES 2} 0 \longrightarrow \frac{\O^{n -1}_G(M;V)}{\O^{n -1}_G(M;V)_\L} \longrightarrow &\HG^n(M;\L) \overset{\cc}\longrightarrow H^n_G(M;\L) \longrightarrow 0, \\
\label{ses3}\tag{SES 3} 0 \longrightarrow \frac{H^{n - 1}_G(M;V)}{H^{n - 1}_G(M;V)_\L} \longrightarrow &\HG^n(M;\L) \longrightarrow A^n_G(M;V) \longrightarrow 0,
\end{align}
where $A^n_G(M)$ is the pullback (in sets) of the commutative square
\begin{equation}\label{eq:FundSquare} \vcenter{ \xymatrix{ \HG^n(M;\L) \ar@{->>}[r]^-{\cc} \ar@{->>}[d]^{\curv}& H^n_G(M;\L)  \ar@{->>}[d] \\
\O^n_G(M)_\L \ar@{->>}[r]^-{\dR} & H^n_G(M;V)_\L.
} } \end{equation}
These sequences fit into the character diagram
\[ \xymatrix@R=3mm@C=6mm{ 0 \ar[dr]&&&&0 \\
&H_G^{n - 1}(M;V/\L) \ar[dr] \ar^-{-B}[rr]&& H_G^{n}(M;\L) \ar[ur]  \ar[dr]\\
H_G^{n -1}(M;V) \ar[ur] \ar[dr]&& \HG^{n}(M;\L) \ar_-{\cc}[ur] \ar^-{\curv}[dr] && H_G^{n}(M;V) \\
& {\displaystyle \frac{\Omega^{n-1}_G(M;V)}{\Omega^{n-1}_G(M;V)_\L}} \ar[ur] \ar_-{d_G}[rr]&& \Omega^{n}_G(M;V)_\L \ar[ur] \ar[dr]\\
0 \ar[ur]&&&&0
} \]
where $B$ is the Bockstein homomorphism.
\begin{proof}The proof is straightforward and follows from the exact same arguments as those given in \cite{HS05}.  Alternatively, one can use the homological framework of Harvey--Lawson spark complexes, specifically Proposition 1.3 from \cite{HL06}.  The key point is that $\O_G^*(M) \to C^*(EG\times_G M;\R)$ is an inclusion, as noted in Lemma \ref{lem:WeilEGinj}, and it induces an isomorphism in cohomology.
%The de Rham homomorphism $\O^*_G(M) \to C^*(EG\times_G M;\R)$ is a weak equivalence, and the homotopy fiber of $C^*(EG\times_G M;\Z) \to C^*(EG\times_G M;\R)$ is equivalent to $C^{*-1}(EG\times_G M;\R/\Z)$.
\end{proof}\end{prop}


\begin{prop} \label{prop:HS-Properties} \
\begin{enumerate}[leftmargin=*]
\item\label{item1:HS-Prop} The constructions $\CG^*(M;\L)$ and $\HG^*(M;\L)$ define contravariant functors from $G\GMan$ to cochain complexes and graded abelian groups, respectively.  
\item\label{item2:HS-Prop} An inclusion $\L_1 \into \L_2 \into V$ of totally disconnected subgroups of $V$ induces natural transformations
\[ \HG^*(-;\L_1) \longrightarrow \HG^*(-;\L_2). \]
\item\label{item3:HS-Prop} The cup product and wedge product induce algebra homomorphisms
\[ \HG^{k_1}(M;\L_1) \otimes \HG^{k_2}(M;\L_2) \longrightarrow \HG^{k_1+k_2}(M;\L_1\otimes \L_2),\]
where $\L_1 \otimes \L_2 \subset V_1 \otimes V_2$.
\item \label{item4:HS-Prop} The map $M \overset{\pi}\to \pt$ makes $\HG^*(M;\L)$ into a $\HG^*$-module.
\item\label{item5:HS-Prop} If $V$ is a ring with sub-ring $\L$, then $\HG^*(M;\L)$ is a graded commutative ring, and $\cc$ and $\curv$ are ring homomorphisms.
\end{enumerate}
\end{prop}
\begin{proof}
The constructions $\CG^*(M)$ and $\HG^*(M)$ are clearly functorial with respect equivariant maps $f\colon M\to N$, as demonstrated in Appendix \ref{sec:Background}, thus proving \eqref{item1:HS-Prop}.

An inclusion $\L_1 \into \L_2$ induces natural cochain maps $C^*(-;\L_1) \into C^*(-;\L_2)$, and this gives $\CG^*(-;\L_1) \into \CG^*(-;\L_2)$, thus proving \eqref{item2:HS-Prop}.
%If $M$ and $N$ are $G$-manifolds, an equivariant map $M \overset{f}\to N$ induces an ordinary map $EG\times_G M \to EG \times_G N$, and hence cochain homomorphisms $C^*(EG\times_G N;-) \overset{f^*}\to C^*(EG\times_G M;-)$.  In  and $\O^*_G(N) \overset{f^*}\to \O^*_G(M)$, and hence a homomorphism $\CG^*(N;\L) \overset{f^*}\to \CG^*(M;\L)$.  Therefore, the construction $\HG^*(-;\L)$ is a functor from the category of $G$-manifolds to the category of graded abelian groups.  

%Just as in the non-equivariant case, $\HG^*(M)$ has a natural ring structure induced by the wedge product and cup product.  
For \eqref{item3:HS-Prop}, first note that the homomorphism $\O_G^*(M) \to \O^*(EG\times_G M)$ is a homomorphism of DGAs, as the basic subcomplex of a $G^\star$-algebra is a sub-DGA.  Hence, the product structure from \cite{HS05} immediately defines the product
\begin{align*}
\CG(q_1)^{k_1}(M;\L_1) \otimes \CG(q_2)^{k_2}(M;\L_2) &\overset{\cdot}\longrightarrow \CG(q_1+q_2)^{k_1 + k_2}(M;\L_1\otimes \L_2) \\
\HG^{k_1}(M;\L_1) \otimes \HG^{k_2}(M;\L_2) &\overset{\cdot}\longrightarrow \HG^{k_1 + k_2}(M;\L_1\otimes \L_2)
\end{align*} by 
\[ (c_1, h_1, \omega_1) \cdot (c_2, h_2, \omega_2) \= (c_1 \cup c_2, \> (-1)^{|c_1|} c_1 \cup h_2 + h_1 \cup \omega_2 + B(\omega_1, \omega_2), \> \omega_1\wedge \omega_2). \]
Here $B(\omega_1, \omega_2) \in C^{k_1 + k_2 - 1}(EG\times_G M;\R)$ is any natural cochain homotopy between $\wedge$ and $\cup$, and we denote the product in $\O_G(M)$ by $\wedge$ in a slight abuse of notation.

For \eqref{item4:HS-Prop}, the map $M \overset{\pi}\to \pt$ is $G$-equivariant, so it induces natural ring homomorphisms $\HG^*(\pt;\Z) \overset{\pi^*_G}\to \HG^*(M;\Z)$.  The composition 
\[ \HG^* \otimes \HG^*(M;\L) \overset{\pi^*_G \otimes 1}\longrightarrow \HG^*(M;\Z) \otimes \HG^*(M;\L) \overset{\cdot}\longrightarrow \HG^*(M;\L)\]
 makes $\HG^*(M;\L)$ into an $\HG^*$-module.

Item \eqref{item5:HS-Prop} follows similarly. If $\L$ is a ring, the product in $\HG^*$ can be composed with the product in $\L$, giving 
\[ \HG^*(M;\L) \otimes \HG^*(M;\L) \overset{\cdot}\longrightarrow \HG^*(M; \L\otimes \L) \longrightarrow \HG^*(M;\L).\]
The homomorphisms $\cc$ and $\curv$ preserve products at the cochain level, so the induced maps $\HG^*(M;\L) \overset{\cc}\to H^*_G(M;\L)$ and $\HG^*(M;\L) \overset{\curv}\to \O^*_G(M;V)_\L$ are also ring homomorphisms.
%The geometric interpretation of the $\HG^*$-module structure will be apparent in Section \ref{sec:characters}.  Under the ring homomorphisms ``$\cc$" and ``$\curv$," this recovers the usual $H^*(BG;\Z)$-module and $(S\fg^*)^G$-module structures on $H^*_G(M;\Z)$ and $\O_G^*(M)$, respectively.  
\end{proof}


%Furthermore, a group homomorphism $G_1 \overset{\phi}\to G_2$ induces morphisms
%\[ \vcenter{ \xymatrix{ EG_1 \ar[r]^{E\phi} \ar[d] &EG_2 \ar[d]\\ BG_1 \ar[r]^{B\phi}& BG_2}} \quad \text{ and } \quad \fg_1 \overset{\phi_*}\longrightarrow \fg_2.
%\]  If $E\phi^*(\Theta_{EG_2}) = \Theta_{EG_1}$, these combine to give natural transformations
%\[  \CG[G_2]^*(\bullet) \overset{\phi^*}\longrightarrow \CG[G_1]^*(\bullet) \quad \text{and} \quad
%\HG[G_2]^*(\bullet) \overset{\phi^*}\longrightarrow \HG[G_1]^*(\bullet).\]
%Since, as noted in $blah$, $\HG[G_1](\bullet)$ is independent of the choice of connection on $EG_1$, the homomorphism $\phi^*\colon  \HG[G_1](\bullet) \to \HG[G_2](\bullet)$ depends only on $\phi$.

%In summary, $\HG^*(-)$ is a contravariant functor from $G$-manifolds to $\HG^*$-modules, and it lies in the short exact sequences from Proposition \ref{prop:ses}.  

Let us now use Propositions \ref{prop:ses} and \ref{prop:HS-Properties} to analyze $\HG^*(M)$ in a few simple examples.  

\begin{example}If $G=e$ is the trivial group, then $\O_G(M) \iso \O(M)$, and we can take $EG= \pt$.  This recovers the usual differential cohomology
\[ \HG[e]^*(M;\L) \iso \HG[]^*(M;\L).\]
\end{example}

\begin{example}The point $\pt$ is trivially a $G$-manifold, and the Borel construction is $EG \times_G \pt \iso BG$.  For $G$ compact we have $H^{2k}(BG;\R) \iso \O^{2k}_G(\pt) = S^k(\fg^*)^G$ and $H^{2k+1}(BG;\R) = \O^{2k+1}_G(\pt) = 0$.  The short exact sequences \eqref{ses2} and \eqref{ses1} give isomorphisms
\begin{equation}\label{eq:HGpt} \HG^{2k} \iso H^{2k}(BG;\Z), \quad \quad
\HG^{2k+1} \iso H^{2k}(BG;\R/\Z).\end{equation}
Furthermore, \eqref{ses2} becomes
\begin{align}\label{ses2pt}
0 \longrightarrow \frac{S^k(\fg^*)^G}{S^k(\fg^*)^G_\Z} \longrightarrow &\HG^{2k+1} \longrightarrow H^{2k+1}(BG;\Z) \longrightarrow 0,
\end{align}
where $\frac{S^k(\fg^*)^G}{S^k(\fg^*)^G_\Z}$ can also be written as $H^{2k}(BG;\R)\otimes \R/\Z$.
\end{example}


\begin{example}
If $G=\Gamma$ is a finite group, the Lie algebra is trivial.  Hence, the equivariant forms $\O_\Gamma(M)$ are simply the invariant forms  $\O(M)^\Gamma$.  When $M$ is a point, \eqref{ses2} implies  
\[\HG[\Gamma]^* \iso H^*(B\Gamma;\Z). \]
\end{example}


\begin{example}
For $G = S^1$, we know that $H^*(BS^1;\Z) \iso H^*(BS^1;\R)_{\Z} \iso \Z[t]$, where $|t|=2$.  The above identities \eqref{eq:HGpt} and \eqref{ses2pt} imply
\begin{align*} \HG[S^1]^{2k} &\iso H^{2k}(BS^1;\Z) \iso \Z t^k, \\
\HG[S^1]^{2k+1} &\iso H^{2k}(BS^1;\R)\otimes \R/\Z \iso (\R/\Z) t^k.
\end{align*}
%To denote the shift in degree, we use the a formal variable $\theta$ and the notation $\HG[S^1]^{2k+1} \iso (\R/\Z) t^k  \theta$.
We now show that as a graded commutative ring, this can be written
\[ \HG[S^1]^* \iso \Big( \Z \oplus (\R/\Z)\theta  \Big)[t], \]
where $|\theta|=1$ and $|t|=2$.  The relations $\theta t = t \theta$ and $\theta^2 = 0$ are implied by the adjectives graded commutative.

To verify this is a ring isomorphism, first note that the characteristic class map gives a ring isomorphism $\HG[S^1]^{2*} \iso \Z[t]$ in even degrees.  We can use the product's definition to determine what happens when multiplying by odd-degree elements.  Choose some cocycle $\alpha \in Z^2(BS^1;\Z)$ representing the generator of $H^2(BS^1;\Z)$.  An even element $n t^k \in \Z t^k \iso \HG[S^1]^{2k}$ can be represented by a cocycle of the form $(\alpha^k, b, t^k)$, and an odd element $[r] \theta t^k \in (\R/\Z) \theta t^k \iso \HG[S^1]^{2k+1}$ can be represented by $(0, r \alpha^k, 0)$.  The cochain-level product yields
\begin{align*}
 (n\alpha^{k_1}, \> b,\> n t^{k_1}) \cdot (0, \> r \alpha^{k_2},\> 0) &= (0, \> nr \alpha^{k_1 + k_2},\> 0), \\
 (0, \> r_1 \alpha^{k_1}, \>0) \cdot (0, \>r_2 \alpha^{k_2}, \>0) &= (0,\>0,\>0).
\end{align*}
Thus, the product of two odd elements is 0, and $(n t^{k_1}) \cdot ([r] \theta t^{k_2}) = [nr] \theta t^{k_1+k_2}$.
\end{example}

\begin{example}
Similarly, the odd cohomology groups of $BU(n)$ vanish, and
\[H^{*}(BU(n);\Z) \iso H^*(BU(n);\R)_\Z \iso \left(S^{*/2} \mathfrak{u} (n)^*\right)^{U(n)}_\Z \iso \Z[c_1, \ldots, c_n]. \]
The same argument given in the previous example shows that 
\begin{align*} \HG[U(n)]^{2k} &\iso H^{2k}(BU(n);\Z), \\
\HG[U(n)]^{2k+1} &\iso H^{2k}(BU(n);\R))\otimes \R/\Z,
\end{align*}
and as graded commutative rings with $|c_i| = 2i$ and $|\theta|=1$, 
\[ \HG[U(n)]^* \iso \Big(\Z\oplus (\R/\Z) \theta \Big) [c_1, \ldots, c_n].\]
\end{example}


%\begin{rem}In the above construction, one can easily replace $\Z$ and $\R$ by any $\L \subset V$, where $\L$ is a completely disconnected subgroup of a vector space $V$.  We denote the resulting cochain complexes and abelian groups by $\CG^*(M;\L)$ and $\HG^*(M;\L)$, respectively.  Proposition \ref{prop:ses} still holds, but with $\O^k_G(M)$ replaced by $\O^k_G(M)\otimes V$, and with $\Z$ and $\R$ replaced uniformly by $\L$ and $V$.

%\[ \xymatrix{ 0 \ar[dr]&&&&0 \\
%&H_G^{\bullet - 1}(M;V/\L) \ar[dr] \ar^{-B}[rr]&& H_G^{\bullet}(M;\L) \ar[ur]  \ar[dr]\\
%H_G^{\bullet -1}(M;V) \ar[ur] \ar[dr]&& \HG^{\bullet}(M;\L) \ar_{\cc}[ur] \ar^{\curv}[dr] \ar@{}^>>>>>{\dR}[rr]&& H_G^{\bullet}(M;V) \\
%& {\displaystyle \frac{\Omega^{\bullet-1}_G(M;V)}{\Omega^{\bullet-1}_G(M;V)_\L}} \ar[ur] \ar_{d_G}[rr]&& \Omega^{\bullet}_G(M;V)_\L \ar[ur] \ar[dr]\\
%0 \ar[ur]&&&&0
%} \]

%The product structure still holds, though it comes from the more general product
%\[ \HG^{k_1}(M;\L_1) \otimes \HG^{k_2}(M;\L_2) \longrightarrow \HG^{k_1+k_2}(M;\L_1\otimes \L_2),\]
%where $\L_1 \otimes \L_2 \subset V_1 \otimes V_2$.
%\end{rem}

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\section{The differential quotient stack}\label{sec:QuotientStack}
From the description of the Weil algebra $\W(\fg)$ in Appendix \ref{sec:Background}, it is clear that equivariant forms $\O_G(M)$ most naturally arise when considering principal bundles {\it with connection}.  While the topological space $EG\to BG$ can be viewed as a universal bundle with connection, it does not naturally represent the {\it category} of principal bundles with connection on general manifolds.  For this reason, it will be more convenient to replace $EG\times_G M$ with the stack $\En G \times_G M$ and study its differential cohomology.  While we are not aware of any work specifically on $\HG[]^*(\En G \times_G M)$,  the general idea of defining differential cohomology via sheaves on manifolds has been widely used. This includes, but is not limited to, \cite{1208.3961, 1311.3188, MR3335251, MR3019405}.

\subsection{Background}\label{subsec:Sheaves}

First, remember that a \textit{groupoid} is a category in which every morphism is invertible.  Any \textit{set} $S$ can be viewed as a category, where the only morphisms are the identity morphisms.  We denote this fully faithful embedding of the category of sets into the category of groupoids by $\Set \subset \Gpd$.

A \textit{stack} $\cM$ (on the site of smooth manifolds) is a contravariant functor from the category of manifolds to the category of groupoids satisfying a descent condition; we denote the category by
\[ \Stack = \Shv_\Gpd \subset \Fun(\Man^\op, \Gpd).\]
This means that associated to every manifold $X$ is a groupoid $\cM(X)$, and a smooth map $X_1 \xrightarrow{f} X_2$ induces a functor 
\[ \cM(X_2) \xrightarrow{f^*} \cM(X_1).\]
There are associative natural transformations $g^* f^* \iso (g\o f)^*$ to deal with composition of functions.  To be a stack, as opposed to a prestack, $\cM$ must also satisfy a sheaf/descent condition that all of our examples will satisfy (see \cite{MR2206877} or Section \ref{Section:DefnProof} for more details).

\begin{example}
Any smooth manifold $N$ defines a stack $\underline{N}$ by associating to $X$ the set 
\[ \underline{N}(X) = C^\infty(X, N) \in \Set,\]
and to a smooth map $X_1 \xrightarrow{F} X_2$ the pullback $C^{\infty}(X_2, N) \xrightarrow{F^*} C^\infty (X_1, N)$.
\end{example}

\begin{example}Principal $G$-bundles with connection form a stack, which we denote $\Bn G$.  To any manifold $X$, let $\Bn G(X)$ be the groupoid whose objects are principal $G$-bundles $P\xrightarrow{\pi}X$ with connection $\Theta \in \O^1(P;\fg)$.  A \textit{morphism} $(P_1,\Theta_1) \overset{\varphi}\to (P_2,\Theta_2)$ is a bundle map preserving the connection; i.e. it is a $G$-equivariant map $\varphi$
\[ \xymatrix{P_1 \ar[rr]^{\varphi} \ar[dr]_{\pi_1}&& P_2 \ar[dl]^{\pi_2} \\ &X}\]
such that $\varphi^*(\Theta_2) = \Theta_1$.  Such a $\varphi$ must be a diffeomorphism, and hence all morphisms in $\Bn G(X)$ are isomorphisms.  Since bundles and connections pull back, a smooth map $f\colon X_1 \to X_2$ induces a functor $f^*\colon  \Bn G(X_2) \to \Bn G(X_1)$.  The groupoid of principal $G$-bundles on $X$, without connection, is defined analogously and denoted $\B G$.
\end{example}

The collection of \textit{morphisms between stacks} $\Shv_{\Gpd}(\cM_1, \cM_2)$ naturally forms a groupoid.  An object $\cM_1 \xrightarrow{\Psi} \cM_2$ is a collection of functors $\cM_1(X) \xrightarrow{\Psi(X)} \cM_2(X)$ for all $X$, together with natural transformations $\Psi(f)\colon  \Psi(X)\o f^* \to f^* \o \Psi(Y)$ for all smooth $f\colon  X\to Y$; morphisms between morphisms are given by natural transformations, which must be invertible since $\cM_2(-)$ is always a groupoid.

%\begin{rem}We can use the nerve construction to include groupoids into simplicial sets.  Therefore, any stack can naturally be viewed as a simplicial sheaf, and we will use this in Section \ref{sec:DefnSPRE}.
%\end{rem}


The Yoneda Lemma states that there is a canonical equivalence of categories
\[ \cM(X) \approx \Shv_{\Gpd}(\underline{X}, \cM),\]
and this defines a faithful embedding of the category of manifolds into the 2-category of stacks.  For this reason, we will usually not distinguish between a manifold $X$ and its associated stack $\underline{X}$.  We can also view an object in the category $\cM(X)$ as a map $X \to \cM$.  Thus, maps between stacks are  a generalization of smooth maps between manifolds; when $M$ and $X$ are both manifolds,
\[ \Shv_{\Gpd}(\underline{X}, \underline{M}) \iso \underline{M}(X) = C^\infty(X, M). \]

Via Yoneda, any bundle $(P,\Theta) \to X$ is naturally equivalent to a map 
\[ X \longrightarrow \Bn G, \]
and the groupoid of bundles with connection is naturally equivalent to the ``category of maps" $X \to \Bn G$.  This makes $\Bn G$ a more convenient classifying object for many purposes than the ordinary topological space $BG$.

\begin{example}Differential forms of degree $k$ define a stack $\O^k$, where 
\[  \O^k(X) \in \Set\]
is a set viewed as a groupoid with only identity morphisms.  A differential form $\omega \in \O^k(X)$ is equivalent to a morphism $X \xrightarrow{\omega} \O^k$.
\end{example}

%\begin{example}The stack $\Bn G$ is defined as the functor that associates to $X$ the groupoid $\Bunc[G](X)$.  
%
%The stack $\En G$ is defined by associating to any manifold the groupoid of $G$-bundles with connection and a global section.  The global section does not need not be flat.   For example, the pullback $\pi^*P \to P$ of any bundle $P \xrightarrow{\pi} X$ has a canonical section, giving an an element in $\En G[P]$.  
%
%
%We analogously define $\B G$ and $\E G$ using the groupoids $\Bun[G](X)$ of principal $G$-bundles without connection.  
%\end{example}


\begin{example}\label{ExDefn:EnGM}If $M$ is a (left) $G$-manifold, the connection quotient stack $\En G \times_G M$ is defined as follows.  An object of $\left( \En G\times_G M \right)(X)$ is a principal $G$-bundle with connection $(P, \Theta) \to X$, together with a $G$-equivariant map $f\colon  P \to M$.  The map $f$ is equivalent to a section $F$ of the associated fiber bundle $P\times_G M$, with the equivalence given by $F(x) = [p, f(p)] \in P\times_G M$.  
\begin{align*}
X \to \En G \times_G M \quad \longleftrightarrow \quad  \vcenter{\xymatrix{ (P,\Theta) \ar[r]^{f} \ar[d] & M \\ X
}} \quad \longleftrightarrow \quad \vcenter{\xymatrix{ (P,\Theta) \times_G M \ar[d] \\ X \ar@/_1pc/[u]_{F} }}
\end{align*}
A morphism 
\[ \left(X \leftarrow (P_1, \Theta_1) \xrightarrow{f_1} M \right) \xrightarrow{\,\varphi\,} \left(X \leftarrow (P_2, \Theta_2) \xrightarrow{f_2} M \right) \] is a connection-preserving bundle isomorphism that covers the maps to $M$; i.e.
\[ \xymatrix{ &(P_1, \Theta_{1}) \ar[dd]_{\varphi}^{\iso} \ar[dl] \ar[dr]^{f_1}\\
X &&M\\
&(P_2, \Theta_{2}) \ar[ul] \ar[ur]_{f_2}
} \]
with $\varphi^*(\Theta_2) = \Theta_1$.

In the case where $M=\pt$, there is only one map $P \xrightarrow{f}\pt$, leading to the natural isomorphism $\En G \times_G \pt \iso \Bn G$.  We also let $\E G \times_G M$ denote the stack defined analogously via bundles without connection.
\end{example}

\begin{rem}Other possible notations for this connection quotient stack could include $M\!  \sslash^\nabla \! G$ and $(M_G )_\nabla$.  We choose to use $\En G\times_G M$ to emphasize the correspondence between maps of stacks and maps of topological spaces.  
%We will later use the notation $M \sslash G$ for the Lie groupoid $[M \leftleftarrows G \times M]$ associated to a group action.
\end{rem}

\subsection{Differential forms on the quotient stack}\label{subsec:FreedHopkins}
We want to study the differential cohomology of $\En G \times_G M$, but it is not immediately clear what that means.  As a first step, let's understand differential forms.  Since $\O^n(X) \iso \Shv_{\Gpd}(X, \O^n)$, it is natural to define 
\begin{equation}\label{defn:dRStack} \O^n(\En G \times_G M) \= \Shv_{\Gpd}(\En G \times_G M, \O^n).\end{equation}
An element of $\O^n(\En G \times_G M)$ is equivalent to functorially assigning, for all manifolds $X$, an element of $\O^n(X)$ to every object in $(\En G \times_G M)(X)$.  Since $\O^n(X)$ has only identity morphisms, isomorphic objects in $(\En G \times_G M)(X)$ must give equal elements in $\O^n(X)$.

Based on the  description of $\O_G(M)$ in Appendix \ref{sec:Background}, there is an obvious homomorphism
\begin{equation}\label{eq:EnGForms} \O_G^n(M) \overset{\iso}\longrightarrow \O^n( \En G \times_G M). \end{equation}
%\begin{thm}[Freed--Hopkins]\label{thm:FreedHopkins} The above map is an isomorphism.
%\end{thm}
While it is not difficult to show this is injective, Freed--Hopkins prove it is in fact an  isomorphism of abelian groups \cite[Theorem 7.28]{MR3049871}.  Though $\O_G(M)$ is usually regarded as an algebraic replacement for $\O(EG\times_G M)$, it is the actual de Rham complex of the stack $\En G \times_G M$.  

We now explain the homomorphism \eqref{eq:EnGForms}.  To a map $X \to \En G \times_G M$, which is an object of $(\En G \times_G M)(X)$, the composition $\Theta^* \o f^*_G = \Theta^* \otimes f^*$ defines a homomorphism $\O^n_G(M) \to \O^n(X)$, as indicated in the following diagram.  
\begin{align*}
\vcenter{ \xymatrix{(P,\Theta) \ar[r]^f \ar[d] & M \\ X} }  \quad &\mapsto \quad \vcenter{\xymatrix{ \O^n_G(P) \ar[d]^{\Theta^*} & \O^n_G(M) \ar[l]_{f_G^*} \\ \O^n(P)_{\basic} \iso \O^n(X)}  }\end{align*}
More explicitly, for $\omega \otimes \eta \in \W^{2i,j}(\fg)$ and $\psi \in \O^k(M)$, then 
\[ \omega \otimes \eta \otimes \psi \mapsto \omega(\Omega^{\wedge i}) \wedge \eta( \Theta^{\wedge j}) \wedge f^*\psi \in \pi^* \O^{2i+j+k}(X).\]
This construction is invariant under morphisms in $(\En G \times_G M)(X)$.  If 
\[ \xymatrix{ &(P_1, \Theta_{P_1}) \ar[dd]_{\phi}^{\iso} \ar[dl]_{\pi_1} \ar[dr]^{f_1}\\
X &&M\\
&(P_2, \Theta_{P_2}) \ar[ul]^{\pi_2} \ar[ur]_{f_2}
} \]
with $\phi^*(\Theta_2) = \Theta_1$, it immediately follows that 
\[ \phi^*\big( \Theta_2^* \otimes f_2^* \big) = \phi^*\Theta_2^* \otimes (f_2 \o \phi)^* = \Theta_1^* \otimes f_1^*.\]
Since $\phi$ covers the identity on $X$, the two homomorphisms $\O^n_G(M) \to \O^n(X)$ are equal.

The de Rham differential induces a universal map of stacks $d\colon  \O^n \to \O^{n+1}$.  This gives a differential $\O^n(\En G \times_G M) \xrightarrow{d} \O^{n+1}(\En G \times_G M)$, and $(\O^*(\En G \times_G M), d)$ is naturally isomorphic to $(\O^*_G(M), d_G)$ as a cochain complex.

\begin{rem}It is important to note that the cochain complex $\O(\En G \times_G M)$ is given by 
\[ \cdots \xrightarrow{d} \Shv_\Gpd (\En G \times_G M, \O^{n}) \xrightarrow{d}  \Shv_\Gpd (\En G \times_G M, \O^{n+1}) \xrightarrow{d} \cdots.\]
Each degree is computed individually, with $\O^n$ being set valued, as opposed to some version of maps maps to the complex $\O^*$, where there are non-trivial morphisms.
%as opposed to some version of $\Shv_{\Gpdinfty}\left( \N(\En G \times_G M), \DK( \O^*) \right)$.
\end{rem}

\begin{example}In the case where $M=\pt$, we have that $(S \fg^*)^G \iso \O(\Bn G)$, which is concentrated in even degrees.  A bundle with connection $(P,\Theta) \to X$ is naturally viewed as a map $X \to \Bn G$;  the induced map on differential forms is the usual map from Chern--Weil theory
\[ \xymatrix@R=0pt{ (S^n \fg^*)^G \ar[r]^{\iso} & \O^{2n}(\Bn G) \ar[r]^-{(P,\Theta)^*} &\O^{2n}(X) \\
\omega \ar@{|->}[rr]&& \omega(\Omega^{\wedge n}). } \]
\end{example}


\subsection{Differential cohomology via sheaves}\label{subsec:sPreDefn}
We now briefly outline the construction of $\HG[]^n(\En G \times_G M)$ and state the main properties.  Because the details are not important to understanding the constructions in Sections \ref{sec:Examples} and \ref{sec:EquivCW}, we postpone them until Section \ref{Section:DefnProof}.

Amongst its many constructions, ordinary cohomology can be represented by homotopy classes of maps to a  space.  For $A$ an abelian group, there exists an Eilenberg--MacLane space $K(A,n) \in \Top$ with 
\[ H^n(X;A) \iso [X, K(A,n)] = \ho \Top(X, K(A,n)).\]
While differential cohomology cannot be represented by a fixed topological space, it can be represented by a sheaf of ``spaces'' on the site of manifolds \cite{MR3019405,1208.3961, MR3335251, 1311.3188}.  We work with simplicial sets for convenience, as they form a natural home for groupoids, topological spaces, and chain complexes.  Letting $\Gpdinfty$ denote the $(\infty, 1)$-category of simplicial sets spanned by Kan complexes,  define $\K(\Z,n) \in \Shv_{\Gpdinfty}$ to be a sheaf fitting into a homotopy-commutative diagram of the form 
\begin{equation}\label{eq:SheafSquare} \vcenter{\xymatrix{ \K(\Z, n) \ar[d] \ar[r] & \cK(\Z,n) \ar[d] \\
\N(\O^n_{\cl})   \ar[r] & \cK(\R,n). }}\end{equation}
Here, $\cK(A,n)$ is an object in  $\Shv_{\Gpdinfty}$ representing ordinary cohomology.  

We can consider the sheaf of groupoids $\En G \times_G M$ as a sheaf of $\infty$-groupoids via the nerve construction $\N$.  The cohomology groups and differential cohomology groups are defined by considering maps between sheaves in the homotopy category:
\begin{align*} H^n(\En G \times_G M;A) &\= \ho \Shv_{\Gpdinfty}(\N(\En G \times_G M), \cK(A,n)),\\
\HG[]^n(\En G \times_G M) &\= \ho \Shv_{\Gpdinfty}(\N(\En G \times_G M), \K(\Z,n)) .\end{align*}



%
%   by  we use Ordinary cohomology can be represented by homotopy classes of maps to a space, and differential forms can be represented by sheaves on manifolds.  One can combine these to represent differential cohomology by a sheaf with values in $\Gpdinfty$. 
%Here we consider $\Top(X,Y)$ as the {\it space} of continuous maps $X\to Y$. 
%
%In fact, we are actually considering $\Top$ as the $(\infty, 1)$-category of topological spaces homotopy equivalent to CW complexes, which is equivalent to $\Gpdinfty$, the $(\infty, 1)$-category subcategory of simplicial sets spanned by Kan complexes.   
%
% This construction has appeared elsewhere under various names  In our notation, $\K(\Z,n) \in \Shv_{\Gpdinfty}$ is formed as the homotopy pullback in a diagram of the form

%Note that $\N(\O^n_{\cl})$ is given by promoting the set-valued sheaf $\O^n_{\cl}$ of closed differential forms to a simplicial sheaf via $\Set \into \Gpd \into \sSet$.  


The following theorem is based on \eqref{eq:EnGForms} and the work of Bunke--Nikolaus--V\"olkl on homotopy-invariant sheaves \cite{1311.3188}.  The proof is given at the end of Section \ref{Section:DefnProof}.

\begin{thm}\label{thm:TwoDefnsAgree}
The two definitions of $\HG^*(M)$ are naturally equivalent,
\[ \HG[]^n(\En G \times_G M) \iso H^n\left( \CG^*(n)(M;\Z) \right).\]
\end{thm}

This leads to the following commutative square.
\begin{equation}\label{eq:StackSquare} \vcenter{\xymatrix{ \HG[]^n(\En G \times_G M) \ar[d] \ar[r] & H^n(\En G \times_G M; \Z) \iso H^n_G(M;\Z) \ar[d] \\ 
\O^n_G(M)_\cl \iso \O^n_{\cl}(\En G \times_G M) \ar[r] & H^n(\En G \times_G M; \R) \iso H^n_G(M;\R) } }\end{equation}
The isomorphism in the bottom-left corner is given by the Freed--Hopkins isomorphism \eqref{eq:EnGForms}.  The two isomorphisms on the right are induced by the map $EG\times_G M \to \En G\times_G M$, which induces an isomorphism in cohomology (Propositon \ref{prop:StackCohIso}).  In the following sections, we will use these two properties to check that maps of stacks induce the desired maps at the level of differential forms and cohomology.


\begin{example}Since $\O^{2k-1}(\Bn G) = 0$, the short exact sequence \eqref{ses2} gives an isomorphism
\[ \HG[]^{2k} (\Bn G) \xrightarrow{\iso} H^{2k}(BG;\Z).\]
The Cheeger--Chern--Simons refinement of Chern--Weil theory \cite{CS85} can then be repackaged in the following way.  A principal $G$-bundle with connection $(P,\Theta) \to X$ is equivalent to a map 
\[ X \xrightarrow{(P,\Theta)} \Bn G,\]
and this induces a homomorphism
\[ \HG[]^{2k}(X) \xleftarrow{(P,\Theta)^*} \HG[]^{2k}(\Bn G) \iso H^{2k}(BG;\Z).\]
Hence, any universal characteristic class in $H^{2k}(BG;\Z)$ has a canonical differential refinement.
\end{example}



%\[ \Gpd \xhookrightarrow{\N} \sSet \xhookleftarrow{\DK} \operatorname{Ch}_+ .\]



%We will also begin to work freely with the language of simplicial sets, simplicial presheaves, and model categories.  The paper \cite{MR3049871} contains an excellent introduction to these concepts, along with further references.  


%Via the nerve construction $\N\colon  \Gpd \into  \sSet$, we have the simplicial presheaf $\N(\En G \times_G M) \in \sPre$.  Just as $\O^n(\En G \times_G M)$ was defined in the previous Section \ref{subsec:FreedHopkins}, we make the following definitions, which may be better thought of as abbreviations.
%
%The full definitions appear as Definitions \ref{} and \ref{}.
%
%Homotopy-commutative diagram
%\[ \xymatrix{ \K(\Z, n) \ar[d] \ar[r] & \cK(\Z,n) \ar[d] \\
%\O^n   \ar[r] & \cK(\R,n) }\]
%
%\begin{defn}\label{defn:StackCoh}Let $\cM$ be a stack, and let $A$ be a fixed abelian group.  We define the ordinary cohomology groups (with coefficients in $A$) and the differential cohomology groups, respectively, of $\cM$ as 
%\begin{align*} 
%H^n(\cM;A) &\= \ho \sPre \left( \N(\cM), \cK(A,n) \right), \\
%\HG[]^n(\cM) &\= \ho \sPre \left( \N(\cM), \K(\Z,n) \right). \end{align*}
%\end{defn}
%
%\begin{prop}\label{prop:StackSquare}
%\[ \xymatrix{ \HG[]^n(\En G \times_G M) \ar[d] \ar[r] & H^n(\En G \times_G M; \Z) \ar[d] \\ 
%\O^n(\En G \times_G M) \ar[r] & H^n(\En G \times_G M; \R) } \]
%\end{prop}

%
%As traditional constructions in equivariant cohomology take place in the complex of equivariant forms, or they are induced at the level of cohomology by maps between the homotopy quotient spaces, Propositions \ref{prop:FreedHopkins} and \ref{prop:StackCohIso} are sufficient to check that maps between stacks recover these classical constructions.
%
%While proofs of these statements will be given in Section \ref{Section:DefnProof}, we proceed to give the main idea of the proof.  The main idea of the proof is that $\K(\Z,n)$ is a homotopy pullback, and so is $\CG^*(n)(M;\Z)$.  By the result of Freed--Hopkins, we know the bottom left corner.  We can also use results from Bunke--Nikolaus--Volkl \cite{1311.3188}, which show that since $\cK(A,n)$ is a homotopy-invariant sheaf, we have can replace the sheaf $\En G \times_G M$ by its homotopification.  This functor gives 
%\[ \ho \sPre\left(\N( \En G\times_G M), \cK(A,n) \right) \iso \ho \Top \left( EG \times_G M, K(A,n) \right).\]
%We then compare our alternate version with the cochain complex definition from Section \ref{sec:HS}, and we see that we have a weak equivalence on the three corners.  Therefore it gives a weak equivalence on the homotopy pullback.  





%\begin{rem}While we work with the Weil model of equivariant forms, the Cartan model $\left(S \fg^* \otimes \O(M) \right)^G$ can be used as well.
%\end{rem}

%
%Using the bi-grading,
%\[ \W^{i,j}(\fg) \otimes \O^k(M) \longrightarrow \O^{i,j}(P) \otimes \O^k(P),\]
%the restriction of the image to horizontal forms $\O^{*,0}(P)$ is spanned by elements from $\W^{i,0} \otimes \O(M)$.  This suggests the Cartan model and Mathai--Quillen isomorphism, which we briefly explain.
%
%Following \cite{MR1218928}, there is a $G$-equivariant algebra automorphism on $\W(\fg) \otimes \O(M)$ induced by the derivation $\iota_{\theta_\fg}$.  It maps between the contraction $\iota$ acting on both factors and just the $\W$ factor.  Therefore, conjugating by the automorphism gives a new differential and hence an isomorphism between two $G^\star$-algebra structures on $\W(\fg)\otimes \O(M)$.
%\[ \xymatrix{
%\big( \W(\fg) \otimes \O(M) \big)_{\basic} \ar[dr] \ar@<1ex>[dd]^{e^\gamma}_{\iso} \\
%& \O(\En G \times_G M) \\
%\big( S \fg^* \otimes \O(M) \big)^G \ar@<2ex>[uu]^{e^{-\gamma}} \ar[ur] } 
%\quad 
%\xymatrix{
%\big( \W(\fg) \otimes \O(M) \big)_{\basic} \ar[dr]^{\Theta^* \otimes f^*} \ar@<1ex>[dd]^{e^\gamma}_{\iso} \\
%& \O^{*,0}(P)^G \iso \O(\S) \\
%\big( S \fg^* \otimes \O(M) \big)^G \ar@<2ex>[uu]^{e^{-\gamma}} \ar[ur]^{\Theta^* \otimes f^*}
%}\]
%



%
%\newpage
%\section{The Borel construction}\label{}
%
%In this section, we explain the geometry behind several important constructions by defining maps at the level of stacks for the Borel quotient.
%
%We begin by fixing some conventions.  Throughout, $G$ will be assumed to be a Lie group, though the cases of interest will be when $G$ is compact.  Let $G\GMan$ denote the category of $G$-manifolds.  An object $M \in G\GMan$ is a smooth manifold $M$ equipped with a left action of $G$.  A morphism $f:M_1 \to M_2$ in $G\Man$ is a $G$-equivariant map between manifolds.  We also consider principal $G$-bundles $P \to X$, where $G$ acts on $P$ by a right action.  Of course, one can switch between left and right actions via $g \cdot x = x \cdot g^{-1}$, so we can also consider $P \in G\GMan$.
%
%\subsection{Stacks}\label{subsec:Stacks}
%A \textit{stack} $\cM$ (in smooth manifolds) is a contravariant functor from the category of manifolds to the category of groupoids, and it must satisfy certain gluing and descent conditions \cite{MR2206877}.  We do not spell out these conditions since we will not work with stacks in general.  Using the inclusion of groupoids into simplicial sets, we can consider any stack as a simplicial presheaf on manifolds.
%
%If $X$ is a manifold, then $\cM(X)$ is the associated a category where all morphisms are isomorphisms.  A map $X_1 \overset{f}\to X_2$ induces a functor
%\[ \cM(X_2) \overset{f^*}\longrightarrow \cM(X_1),\]
%and there are associative natural transformations $g^* f^* \iso (g\o f)^*$ to deal with composition of functions.
%
%Any manifold $N$ defines a stack $\underline{N} = \Map(\bullet, N)$, where the set $\Map(X, N)$ is considered a category with no non-trivial morphisms.  The Yoneda Lemma states that there is a canonical equivalence of categories
%\[ \cM(N) \approx \Hom_{Stacks}(\underline{N}, \cM),\]
%and this defines a faithful embedding of the category of manifolds into the 2-category of stacks.  For this reason, we will not distinguish between a manifold $N$ and its associated stack $\underline{N}$.  We also use the suggestive notation $\Map(X, \cM)$ to denote the category $\cM(X) \approx \Hom_{Stacks}(\underline{X}, \cM)$, with a ``map" $X \to \cM$ denoting an object in the category $\cM(X)$.
%
%The stack $\Bn G$ is defined as the functor that associates to $M$ the groupoid $\Bunc[G](M)$.  By definition, this means
%\[ \Map(M, \Bn G) = \Bunc[G](M).\]
%The stack $\En G$ is defined by associating to any manifold the groupoid of $G$-bundles with connection and trivialization.  The natural projection $\En G \to \Bn G$ is the ``universal bundle with connection;" any $(P,\Theta) \to M$ is by definition maps in the following diagram.
%\[ \xymatrix{(P,\Theta) \ar[r]\ar[d] &\En G \ar[d] \\
%M \ar[r] & \Bn G} \]
%We analogously define $\BG$ and $\EG$ using the groupoids $\Bun[G](\bullet)$ of principal $G$-bundles without connection.
%
%
%The language of stacks gives us a ``classifying space" $\Bn G$ that is more functorial than $BG$, but at the expense of leaving the category of topological spaces.  Luckily, this will not pose a serious problem for us.  Since $(EG, \Theta_{EG}) \to BG$ is a principal $G$-bundle with connection, there are natural maps
%\[ \xymatrix{ EG \ar[r]\ar[d] &\En G \ar[d]\\
%BG \ar[r] &\Bn G . } \]
%The fact that $EG\to BG$ is a universal bundle implies that any map $M \to \BnG$ factors through $BG$.  Several constructions we use are more naturally described using stacks, but practically speaking there will be no real difference between $\En G \to \Bn G$ and $EG \to BG$.
%
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\begin{comment}

\section{Equivariant Differential Characters}\label{sec:characters}

We now rewrite the definition of $\HG^*(M;\L)$ using the language of differential characters.  This accomplishes two things.  First, it gives an appealing geometric description.  Second, by defining characters using the stack $\En G \times_G M$, we can perform later constructions at the level of stacks and avoid continually choosing classifying maps.

Note: (possibly move this to previous section), there is a natural injection $\HG^*(M) \into \HG[]^*(EG\times_G M)$ given by the Weil homomorphism,
\[ \xymatrix{ 0 \ar[r] & H_G^{k-1}(M;\R/\Z) \ar[d]^{=} \ar[r] &\HG^k(M) \ar[d]^{\Theta_{EG}^*} \ar[r] &\O^k_G(M)_\Z \ar@{^(->}[d]^{\Theta_{EG}^*}\ar[r]&0 \\
0 \ar[r] & H^{k-1}(EG \times_G M; \R/\Z) \ar[r]  &\HG[]^k(EG\times_G M) \ar[r] & \O^k (EG\times_G M)_\Z \ar[r] & 0
} \]
First note that one could define the differential characters, of degree $k$, as
\[ \{ \rchi \in \Hom(Z_{k-1}(EG \times_G M), V/\L) \> |\> \rchi \o \partial \in \O^k_G(M;V) \}. \]
There is a natural isomorphism from $\HG^k(M;\L)$ to these degree $k$ characters, where $[(c,h,\omega)]$ determines a character by evaluating $h$ on $(k-1)$-cycles and quotienting by $\L$.

We now proceed to give an equivalent description of $\HG^*(M;\L)$ as differential characters on the stack $\EnG \times_G M$.  This will allow us to avoid choosing classifying maps when constructing natural operations in $\HG^*$.

  The following definition is described in greater detail in Appendix \ref{app:StackChains}.

\subsection{Singular chains}

\begin{defn}An equivariant singular $k$-chain, denoted $\sigma \in C_k(\EnG \times_G M)$, is a formal finite sum of elements in $\Map(\Delta_k, \EnG \times_G M)$, with $\Z$ coefficients.  The collection of singular $k$-chains $C_k(\EnG \times_G M)$ is naturally a category, where a morphism $\sigma_1 \overset{\phi}\to \sigma_2$ is a formal finite sum of morphisms in $\Map(\Delta_k, \EnG \times_G M)$.
\end{defn}

More concretely, a $k$-chain $\sigma$ is a formal sum of bundles with connection over the $k$-simplex $\Delta_k$, equipped with equivariant maps to $M$; we write this as
\[ \sigma = \sum_i n_i \sigma_i = \sum_i n_i \left( \vcenter{ \xymatrix{ (P_i, \Theta_{P_i}) \ar[d] \ar[r]& M \\ \Delta_k  }}\right)_. \]
The morphisms are similarly generated by the morphisms in $\Map(\Delta_k, \EnG\times_G M)$.

The functor $\partial\colon  C_k(\EnG \times_G) \to C_{k-1}(\EnG \times_G M)$ is given by restricting a chain to the various faces of the $k$-simplex.  The inclusion in the $i$-th face $d^i\colon  \Delta_{k-1} \to \Delta_k$ induces 
\[ \Map(\Delta_k, \EnG \times_G M) \overset{d^{i*}}\longrightarrow \Map(\Delta_{k-1}, \EnG \times_G M),\]
and we define $\partial \= \sum_i (-1)^i d^{i*}$.  Equivariant cycles on $M$ are equivariant chains $\sigma$ satisfying $\partial \sigma = 0$.  They are denoted $Z_k(\EnG \times_G M)$, and they form a subcategory of $Z_k(\EnG \times_G M)$.

One usually thinks of ordinary cycles $Z_k(M)$ as maps of closed manifolds, possibly with singularities, to $M$.  This is the same idea we have for equivariant cycles, as explained in this fundamental example.

\begin{example}Let $\Sigma_k$ be a smooth manifold of dimension $k$.  Choose a triangulation of $\Sigma_k$. 
%\reminder{That behaves nicely with respect to the boundary.}  
Such a triangulation naturally induces the functors given by the horizontal arrows in the following diagram.
\[ \xymatrix{\Map(\Sigma_k, \EnG\times_G M) \ar[r] \ar[d]^{\partial}& C_k(\EnG \times_G M) \ar[d]^{\partial}\\
\Map(\partial \Sigma_k, \EnG \times_G M) \ar[r]& C_{k-1}(\EnG \times_G M)}\]
When $\partial \Sigma_k = 0$, this induces a natural functor
\[ \Map(\Sigma_k, \EnG \times_G M) \longrightarrow Z_k(\EnG\times_G M).\]
To see this, consider $\Sigma_k \overset{f}\to \EnG\times_G M$.  The equivariant chain is obtained by  restricting $f$ to the simplices of $\Sigma_k$ and summing over the simplices.  If $f_1 \overset{\phi}\to  f_2$ is a morphism in $\Map(\Sigma, \EnG \times_G M)$, the sum of $\phi$ restricted to each simplex is naturally a morphism in $C_k(\EnG \times_G M)$.  It is clear that this is compatible with the boundary operators.  
\end{example}

\begin{lemma}The homomorphism $\O^k_G(M;V) \to C^k(\EnG \times_G M;V/\L)$ is an injection.
\begin{proof}
\[\xymatrix{ &C^*(EG\times_G M;V/\L)  \\
\O^*_G(M) \ar[rr] \ar@{^(->}[ur]&&  C^*(\EnG\times_G M;V/\L) \ar[ul]} \]
The homomorphism $\O^k_G(M;V) \to  C^k(EG \times_G M;V/\L)$ is an injection, and it factors through $C^k(\EnG \times_G M;V/\L)$.
\end{proof}\end{lemma}

\begin{defn}If $M$ is a $G$-manifold, the equivariant differential characters of degree $k$ are defined
\[ \HG[]^k( \EnG \times_G M;\L) \= \left\{ \rchi \in \Hom\left(Z_{k-1}(\EnG \times_G M), V/\L\right) \> | \> \rchi \o \partial \in \O_G^k(M;V) \right\}. \]
\end{defn}

\begin{example}Fix $\rchi \in \HG[]^k(\EnG \times_G M;\L)$.  The character $\rchi$ then assigns
\[ \rchi \left( \vcenter{ \xymatrix{ (P,\Theta_P) \ar[r] \ar[d] &M\\ \Sigma}} \right) \in V/\L
 \]
Additive, and $\rchi \left( \partial X\leftarrow \partial(Q,\Theta_Q) \to M\right)$ given by integrating the image of $\omega$ in the following sequence
\[ \O_G^k(M;V) \overset{f^*}\to \O_G^k(Q;V) \to \O^k(X;V)\]
over $X$.
\end{example}

\begin{prop}The kernel of the curvature map 
\[ \Ker\left( \HG[]^k(\EnG \times_G M;\L) \to \O^k_G(M;V)  \right) \iso H^{k-1}_G(M;V/\L).\]
\begin{proof}This essentially follows from the unsurprising fact $H_{k-1}(\EnG \times_G M) \iso H_{k-1}(EG \times_G M)$, where the homology of $\EnG \times_G M$ is suitably interpreted.
%We show that $\Ker \curv \iso \Hom H_{k-1}(\EnG \times_G M), V/\L)$, and $H_{k-1}(\EnG \times_G M) \iso H_{k-1}(EG \times_G M)$.  
The flat characters are precisely those that do not depend on the homology class.  Hence, we have that $\Ker \curv$ is isomorphic to $\Hom( H_{k-1}(\EnG \times_G M), V/\L)$.  Since the space of connections is contractible (affine), we see that $H_{k-1}(\EnG \times_G M) \iso H_{k-1}(\EG \times_G M)$ (without connections).  Furthermore, any equivariant chain is isomorphic to the pullback of $\Sigma \to EG\times_G M$, and it is boundable if and only if the chain $\Sigma \to EG\times_G M$ is the boundary of a $k$-chain.  Hence, this is isomorphic to
\[ \Hom(H_{k-1}(EG\times_G M), V/\L) \iso H^{k-1}(EG\times_G M;V/\L) = H^{k-1}_G(M;V/\L).\]
The middle isomorphism comes from the universal coefficient theorem and the fact that $V/\L$ is a divisible group.
\end{proof}
\end{prop}

There map $EG \times_G M \to \EnG \times_G M$ induces a natural homomorphism
\[ \HG[]^*(\EnG\times_G M;\L) \longrightarrow \HG^*(M;\L),\]
and we will show this is an isomorphism.  To do so, we will construct a homomorphism in the opposite direction by choosing classifying maps for principal bundles.

%To see this, use the description of $\HG^k(M;\L)$ as elements in $\Hom(Z_{k-1}(EG\times_G M), V/\L)$.  The map $EG\times_G M \to \EnG \times_G M$ induces 
%\[ \Hom( Z_{k-1}(\EnG\times_G M), V/\L) \to \Hom(Z_{k-1}(\EG\times_G M),V/\L)\]
%compatible with 
%$EG \times_G M \to \EnG \times_G M$ induces natural homomorphism $\HG[]^k(\EnG \times_G M;\L) \to \HG^k(M;\L)$.  We wish to show this is an isomorphism.  To do so, we will use choices (a la Hopkins--Singer) to create map going other direction, and this will be compatible with curvature map.  Then, we use the Five Lemma.
%
%\reminder{This is not correct, for I did not define the reduction map at the cochain level in $\CG^*$.  Construct the map the other way, and use the homological fact that $\HG^k(M) \iso $ characters on $EG\times_G M$ varying by element in $\O_G(M)$.}

%There is a natural homomorphism
%\[ \HG^*(M;\L) \to \HG[]^*(\EnG \times_G M;\L)\]
%defined via the following process.  Suppose we are given an element $\x \in \HG^*(M;\L)$.  We define an equivariant character via the following process.  Let
%\[ (P_\Sigma, \Theta) \overset{f} \to M \]
%be an equivariant $(k-1)$-cycle.  Then,
%\[ \HG^k(M) \overset{f^*}\to \HG^k(P) \overset{/G}\to \HG[]^k(\Sigma^{k-1}) \to V/\L.\]
%To any equivariant $(k-1)$-cycle, the element $\x$ associates an element in $V/\L$.  In fact, one could construct this map by way $EG\times_G M \to \EnG \times_G M$, but our proof yields a more general result.

\begin{prop}The map $\HG[]^*(\EnG \times_G M;\L) \to \HG^*(M;\L)$ is an isomorphism.
\begin{proof}
%
%
%\reminder{This next part is not correct.}
%We check that this map is compatible with the curvature map.  Hence, the curvature $\HG[]^k(\EnG\times_G M;\L) \to \O_G^k(M;V)_\L$ is surjective.  We also check that the homomorphism is compatible with the flat equivariant characters.  Therefore, by the five-lemma, the middle map is an isomorphism
%\[ \xymatrix{ 0 \ar[r] & \ar[r] H_G^{k-1}(M;V/\L) \ar[r] \ar[d]^{=}& \HG[]^k(\EnG\times_G M;\L) \ar[r]\ar[d]^{\iso}& \O^k_G(M;V)_\L\ar[r]\ar[d]^{=}& 0 \\
%0 \ar[r]& \ar[r]  H_G^{k-1}(M;V/\L) \ar[r]& \HG^k(M;\L) \ar[r] & \O^k_G(M;V)_\L \ar[r] & 0} \]
\end{proof}\end{prop}

In other words, an equivariant character $\rchi \in \HG[]^k( \EnG \times_G M;\L)$ does the following: for a principal $G$-bundle with connection $(P_{\Sigma}, \Theta) \to \Sigma$ equipped with equivariant map $f$
\[ \xymatrix{(P_{\Sigma}, \Theta) \ar[r]^{f} \ar[d] &M \\ \Sigma_{k-1}}\]
such that $(P,f)$ is a cycle, you get an element of $V/\L$.  Secondly, this quantity is invariant under gauge transformations.  Furthermore, if $(P_{\Sigma},\Theta_P) = \partial(\wt{P_{\Sigma}}, \Theta_{\wt{P}})$, then
\[ \rchi((P_\Sigma, \Theta)\overset{f}\to M) = \int_{\wt{\Sigma}} \iota_{(\Theta,\Omega)} f^*(\omega) \]
is given by pulling back the equivariant form, using the Weil homomorphism, and integrating the resulting differential form.


The $\HG^*$-module structure on $\HG^*(M)$ can be seen as follows.  The projection map $M \to \pt$ induces $\EnG \times_G M \overset{\pi}\to \BnG$.  Geometrically, this map is simply the following.  A map $\Sigma \to \EnG \times_G M$ is a bundle $(P,\Theta) \to \Sigma$ with equivariant map $P \to M$.  Forgetting the map to $M$, we have a bundle $(P,\Theta) \to \Sigma$, which is $\Sigma \to \BnG$.


\end{comment}

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


\section{Constructions}\label{sec:Examples}
We now explain how some important constructions, frequently described using classifying spaces or equivariant forms,are naturally given by explicit geometric constructions involving bundles with connection.


\subsection{Associated bundles}\label{subsec:Associated}
Let $\phi\colon  G_1 \to G_2$ be any Lie group homomorphism, with $\phi_*\colon  \fg_1\to \fg_2$ the associated Lie algebra homorphism.  The associated bundle construction, which functorially makes any $G_1$-bundle with connection $(P,\Theta)$ into a $G_2$-bundle with connection, induces morphisms
\[ \xymatrix{ \En G_1 \ar[d] \ar[r] &\En G_2 \ar[d]\\
\Bn G_1 \ar[r] & \Bn G_2.}\]
The most common example is when $\phi$ is an inclusion, which is usually referred to as extending the structure group.  For more details on the associated bundle construction, see Chapter II.6 of \cite{KN63}, or Section 1 of \cite{Fre95}.

If $(P,\Theta) \in \Bn G_1 (X)$, the associated $G_2$-bundle is defined 
\begin{equation}\label{eq:AssocBundle} P_\phi \= P \times_{\phi} G_2 = (P\times G_2)/G_1 = \left( P \times G_2 \right)\big/ \big((pg,f) \sim (p,\phi(g)f)\big).\end{equation}
To define the induced connection $\Theta_\phi$ on $P_\phi$, first note that the natural map $P \xrightarrow{\varphi} P_\phi$, given by $p \mapsto [(p,1)]$, is $\phi$-equivariant.  Hence, the horizontal subspaces of $TP$ are mapped equivariantly into $TP_\phi$, and the image extends uniquely to an equivariant horizontal distribution in $TP_\phi$.  The induced connection $\Theta_\phi$ can also be described as the unique connection on $P_\phi$  that is compatible with $\Theta$ in the sense that 
\begin{equation}\label{eq:AssocConn} \varphi^*(\Theta_\phi) = \phi_* (\Theta) \in \O^1(P;\fg_2). \end{equation}
Therefore, the associated bundle construction is compatible with the Weil homomorphism, giving the commutative diagram
\begin{equation}\label{eq:WeilGroupChange} \vcenter{ \xymatrix{ \W(\fg_1) \ar[r]^{\Theta^*} &\O(P) \\
\W(\fg_2) \ar[u]_{\phi^*} \ar[r]^{\Theta_\phi^*} & \O(P_\phi). \ar[u]_{\varphi^*}
} }\end{equation}

%Or, as noted in equation (1.17) from \cite{Fre95}, one can explicitly define the induced connection 1-form by
%\[ (\Theta_\phi)_{(p,g_2)} \= \Ad_{g_2^{-1}}  \phi_* (\Theta) + \theta_{\fg_2} \in \O^1(P \times G_2; \fg_2 )_{G_1\text{-}\basic}. \]

\begin{prop}\label{prop:GroupChange}Let $F\colon  M \to N$ be a $\phi$-equivariant map, where $\phi\colon  G_1 \to G_2$ is a Lie group homomorphism.
\begin{enumerate}
\item \label{GroupChange1} The associated bundle construction gives a natural morphism
\[ \En G_1 \times_{G_1} M \xrightarrow{F_\phi} \En G_2 \times_{G_2} N.\]

\item \label{GroupChange2}The homomorphism induced by $F_\phi$ on differential forms is naturally isomorphic to $\phi^* \otimes F^*$ in the Weil model.
\[ \xymatrix@R=0cm{ \O_{G_2}(M) \ar[r]^{\iso} & \O(\En G_2 \times_{G_2} N) \ar[r]^{F_\phi^*} & \O(\En G_1 \times_{G_1} M)  & \O_{G_1}(M) \ar[l]_{\iso}\\
\cap &&& \cap \\
\W(\fg_2) \otimes \O(N) \ar[rrr]^{\phi^* \otimes F^*} &&& \W(\fg_1) \otimes \O(M) } \]

\item \label{GroupChange3}The homomorphism induced by $F_\phi$ on ordinary cohomology is naturally isomorphic to the homomorphism induced by the maps of classifying spaces
\[ EG_1 \times_{G_1} M \xrightarrow{} (EG_1 \times_\phi G_2) \times_{G_2} N \xrightarrow{} EG_2 \times_{G_2} N.\]

\item \label{GroupChange4} The induced homomorphism 
\[ \HG[G_2]^*(N) \xrightarrow{F_\phi^*} \HG[G_1]^*(M) \]
recovers the expected homomorphisms in equivariant cohomology and equivariant forms.
\end{enumerate}\end{prop}

Before giving the proof of Proposition \ref{prop:GroupChange}, we wish to first introduce notation for three special cases.  In two of the cases, the general construction simplifies, so we also show what happens when the stack is evaluated on a general manifold $X$. 
\begin{example} Suppose that $M$ and $N$ are $G$-manifolds, and $M \xrightarrow{F} N$ is $G$-equivariant.  The morphism of stacks
\begin{align*}&\En G \times_G M &&\xrightarrow{F_G} &&\En G \times_G N \\ 
&\vcenter{\xymatrix@R=.6cm{ (P,\Theta) \ar[r]^{f} \ar[d] & M \\ X
}}  &&\longmapsto  &&\vcenter{\xymatrix@R=.6cm{ (P,\Theta) \ar[d] \ar[r]^{f} & M \ar[r]^{F} &N \\ X  }}
\end{align*}
induces the homomorphism of abelian groups
\[ \HG^*(N) \xrightarrow{F_G^*} \HG^*(M).\]

\end{example}

\begin{example} Let $M$ be a $G_2$-manifold.  A homomorphism $G_1 \xrightarrow{\phi} G_2$ naturally defines a $G_1$-action on $M$ via $g_1 \cdot M \= \phi(g_1)\cdot M$, and the identity map $M \xrightarrow{\1}M$ is $\phi$-equivariant.  The associated bundle construction then induces
\begin{align*} \En G_1 \times_{G_1} M &\xrightarrow{\1_\phi} \En G_2 \times_{G_2} M, \\
\HG[G_1]^*(M) &\xleftarrow{\1_\phi^*} \HG[G_2]^*(M).
\end{align*}
\end{example}

\begin{example} Suppose that $K \vartriangleleft G$ is a normal subgroup, and suppose the quotient map $M \xrightarrow{q} M/K$ is a smooth map between manifolds.  Then, $q$ is equivariant with respect to the quotient group homomorphism $G \xrightarrow{/K} G/K$.  Quotienting $M$ and $G$ by $K$ induces the stack morphism 
\begin{align*}
&\En G \times_G M &&\xrightarrow{q_{/K}} &&\En (G/K) \times_{G/K} M/K \\ 
&\vcenter{ \xymatrix@R=.7cm{ (P,\Theta) \ar[d] \ar[r]^{f} &M \\ X }}  &&\xmapsto{\quad\,\,} &&\vcenter{\xymatrix@R=.7cm{(P,\Theta) \ar@{..>}[d] \ar@{..>}[r]^{f} &M \ar@{..>}[d]^{q} \\ (P_{/K}, \Theta_{/K}) \ar[r]^{f_{/K}} \ar[d] &M/K \\ X }}
\end{align*}
(with dotted arrows indicating maps no longer being used) and a homomorphism 
\[ \HG[G/K]^*(M/K) \xrightarrow{q_{/K}^*} \HG^*(M). \]  
If $K$ acts freely, this induces the usual isomorphism
\begin{align}\label{eq:QuotientFree}H^*_{G/K}(M/K;-) \xrightarrow[q_{/K}^*]{\iso} H^*_G(M;-) \end{align}
in equivariant cohomology.
\end{example}


\begin{proof}[Proof of Proposition \ref{prop:GroupChange}]
Part \eqref{GroupChange1} is given by constructing, for every $X$ and natural with respect to maps $X\to Y$, the following functor.
\begin{align*}  &\left(\En G_1 \times_{G_1} M\right) (X) &&\xrightarrow{F_\phi(X)} &&\left(\En G_2 \times_{G_2} N\right) (X) \\
&\vcenter{\xymatrix{ (P,\Theta) \times_{G_1} M \ar[d] \\ X \ar@/_1pc/[u]_{s} }}  &&\xmapsto{\quad \quad}
&&\vcenter{\xymatrix{ (P_\phi, \Theta_\phi) \times_{G_2} N \ar[d] \\ X \ar@/_1pc/[u]_{\left(\overline{\varphi \times F}\right) \o s}  }}
 \end{align*}
To explain in a bit more detail, the associated bundle construction associates to $(P,\Theta)$ the bundle $(P_\phi, \Theta_\phi)$, along with a $\phi$-equivariant map $P \xrightarrow{\varphi} P_\phi$.  Since $M \xrightarrow{F} N$ is also $\phi$-equivariant, the map
\[ P \times M \xrightarrow{ \varphi \times F} P_\phi \times N   \]
descends to the quotient
\[ P \times_{G_1} M \xrightarrow{ \overline{\varphi \times F}} P_\phi \times_{G_2} N,  \]
and we precompose with original section $s$.  The associated bundle construction is functorial, so any morphism $\Phi$ in $\left( \En G_1 \times_{G_1} M\right)(X)$ will induce a morphism $F_\phi(\Phi)$ in $\left( \En G_1 \times_{G_1} M\right)(X)$, as evidenced by the following commutative diagram.
\[  \xymatrix@R=.5cm{ &(P,\Theta) \times_{G_1} M \ar[dl] \ar[dd]_{\iso}^{\Phi} \ar[r]_{\overline{\varphi \times F}}& (P_\phi, \Theta_\phi) \times_{G_2} N \ar[dd]_{\iso}^{\Phi_\phi} \\
X \ar@/^1pc/[ur]^{s} \ar@/_1pc/[dr]_{s'} \\
&(P',\Theta') \times_{G_1} M  \ar[ul] \ar[r]^{\overline{\varphi' \times F}} & (P'_\phi, \Theta'_\phi) \times_{G_2} N }\]
Hence, the associated bundle construction induces a morphism of stacks.

For part \eqref{GroupChange2}, we simply trace through the construction of $F_\phi$ and the Weil homomorphism.  Let $\omega \in \O (\En G_2 \times_{G_2} N)$, which may be written as a sum of homogeneous elements $\alpha \otimes \psi \in \left(\W(\fg_2) \otimes \O(N)\right)_{\basic}$.  Then, calculate $F_\phi^*\omega \in \O(\En G_1 \times_{G_1} M)$ by evaluating on a test manifold $X$.  By definition, 
\begin{align*} &\left( \En G_1 \times_{G_1} M\right)(X) & &\xrightarrow{F_\phi} &&\left(\En G_2 \times_{G_2} N \right)(X) &&\xrightarrow{\omega} &&\O(X) \\
&\vcenter{ \xymatrix{ (P,\Theta) \times_{G_1} M \ar[d] \\ X \ar@/_1pc/[u]_{s} } }  &&\longmapsto &&
 \vcenter{ \xymatrix{ (P_\phi, \Theta_\phi) \times_{G_2} N \ar[d] \\ X \ar@/_1pc/[u]_{(\overline{\varphi \times F}) \o s}  }} &&\longmapsto & s^*  \Big( (\overline{\varphi \times F})^*& \left( \alpha(\Theta_{\phi}) \otimes \psi \right) \Big)
\end{align*}
As noted in \eqref{eq:WeilGroupChange}, the associated bundle construction is compatible with the Weil homomorphism, and therefore
\[
(\varphi \times F)^* \big( \alpha(\Theta_\phi) \otimes \psi \big) = \alpha( \varphi^*\Theta_\phi) \otimes F^*\psi 
= \alpha( \phi_*\Theta) \otimes F^*\psi = (\phi^*\alpha) (\Theta) \otimes F^*\psi.
\]
Consequently, $F_\phi^* \omega$ gets mapped to $s^*\big( (\phi^*\alpha) (\Theta) \otimes F^*\psi\big) \in \O(X)$, and this is the same element that $\phi^*\alpha \otimes F^*\psi \in \O_{G_1}(M)$ maps to.  

To see part \eqref{GroupChange3}, choose a classifying map for the $G_2$-bundle $EG_1 \times_\phi G_2$, which gives the following diagram.  We do not need to assume the classifying map is connection preserving.
\[ \xymatrix{ EG_1 \times_{G_1} M \ar[r] \ar[d] \ar@/^1pc/[rr]^{\Psi} & EG_1 \times_\phi G_2 \ar[r] \ar[d]& EG_2 \ar[d] \\ BG_1 \ar[r]^{=} & BG_1 \ar[r] & BG_2 
}\]
For any $X\xrightarrow{f}BG_1$, there is an isomorphism of associated bundles
\[ f^* (\Psi^* EG_2) \iso (f^*EG_1)_\phi.\]
This implies the diagram
\[  \xymatrix{ EG_1 \times_{G_1} M \ar[r] \ar[d] & EG_2 \times_{G_2} M \ar[d] \\ \E G_1 \times_{G_1} M \ar[r] & \E G_2 \times_{G_2} N } \]
commutative up to isomorphism, or homotopy-commutative.  As shown in Proposition \ref{prop:StackCohIso}, the vertical maps induce isomorphisms in ordinary cohomology.  Therefore, the induced map on the cohomology of the stacky homotopy quotients
\[ H^*(\En G_2 \times_{G_2} N;A) \xrightarrow{F^*_\phi} H^*(\En G_1 \times_{G_1} M;A) \] 
is given by the usual map between the homotopy quotient spaces.

Finally, part \eqref{GroupChange4} follows immediately from parts \eqref{GroupChange2} and \eqref{GroupChange3}, and \eqref{eq:StackSquare}.
\end{proof}

\subsection{Pulling back equivariant bundles}\label{subsec:Reduction} As discussed in the previous section, if $G$ acts freely on a manifold $P$, there is a natural homomorphism $\HG[]^*(P/G) \xrightarrow{q_{/G}^*} \HG^*(P)$ inducing the standard isomorphism in cohomology.  We now describe the one-sided inverse to this map.  First we outline the  result, then we give the construction, and then we check the details.

To obtain a map in the opposite direction, one must choose a connection $\Theta$ on $P$.  Using this and denoting $M=P/G$, one naturally obtains a morphism of stacks
\[ M \overset{\Theta}\longrightarrow \En G\times_G P \]
by pulling back the bundle $(P,\Theta)$, as indicated below when the stack is evaluated on a manifold $X$.
\begin{align*}
\vcenter{ \xymatrix@C=.5cm{ X \ar[r]^{f} &M }} &\longmapsto \vcenter{ \xymatrix{ f^*(P,\Theta) \ar[d] \ar[r]^{\wt{f}} &(P,\Theta) \ar@{..>}[d] \\ X \ar@{..>}[r]^{f} & M } }
\end{align*}
Proposition \ref{prop:Reduction} will imply that the induced map 
\[ \HG^*(P) = \HG[]^*(\En G \times_G P) \xrightarrow{\Theta^*} \HG[]^*(M) \]
recovers the isomorphism inverse to $q_{/G}^*$ at the level of cohomology, and it recovers the Weil homomorphism 
\begin{align*}
\O_G(P) = \big( \W(\fg)\otimes \O(P) \big)_{\basic} & \xrightarrow{\Theta^*} \O(P)_{\basic} \iso \O(M) \\
\alpha \otimes \psi \quad \quad &\longmapsto \quad \alpha(\Theta) \otimes \psi
\end{align*} 
at the level of differential forms.

The above is actually a special case of what happens when we have an equivariant principal bundle with connection.  Suppose that $(Q,\Theta) \to M$ is a $G$-equivariant principal $K$-bundle with $G$-invariant connection.  This implies that $Q$ is a $(G\times K)$-manifold, $M$ is a $G$-manifold, and the map $Q\to M$ is equivariant.  (Note that in the following discussion, $G$ will always act on the base manifold $M$, and $K$ will be the structure group for a principal bundle.)  The above construction generalizes to give a stack morphism
\[ \En G \times_G M \xrightarrow{\Theta_G} \En (G\times K) \times_{G\times K} Q, \]
and this gives the expected classical maps in cohomology and equivariant forms.  In particular, we get the following maps between short exact sequences.
\[ \xymatrix{ 0 \ar[r]& \frac{\O^{*-1}_{G\times K}(Q)}{\O^{*-1}_{G\times K}(Q)_\Z} \ar[r] \ar@/_1pc/[d]_{\Theta_G^*}&\HG[G\times K]^*(Q) \ar@/_1pc/[d]_{\Theta_G^*}\ar[r] & H^*_{G\times K}(Q;\Z) \ar[r] \ar@{<->}[d]^{\iso} &0 \\
0 \ar[r]& \frac{\O^{*-1}_G(M)}{\O_G^{*-1}(M)_\Z} \ar[r] \ar@/_1pc/[u]_{q_{/K}^*}&\HG[G]^*(M) \ar[r] \ar@/_1pc/[u]_{q_{/K}^*} & H_G^*(M;\Z) \ar[r] &0 
}\]
\[ \xymatrix{ 0 \ar[r] & H_{G\times K}^{*-1}(Q;\R/\Z) \ar@{<->}[d]^{\iso} \ar[r] & \HG[G\times K]^*(Q) \ar@/_1pc/[d]_{\Theta_G^*} \ar[r] & \O^*_{G\times K}(Q)_\Z \ar@/_1pc/[d]_{\Theta_G^*} \ar[r] &0 \\
0 \ar[r] & H_G^{*-1}(M;\R/\Z) \ar[r] & \HG[G]^*(M) \ar@/_1pc/[u]_{q_{/K}^*} \ar[r] & \O_G^*(M)_\Z \ar@/_1pc/[u]_{q_{/K}^*} \ar[r] &0 
} \]

Let us now give the precise construction in slightly greater generality.  Assume that $K$ is a normal subgroup of $\wt{G}$ with quotient $\wt{G}/K \iso G$, and furthermore assume that we have fixed a splitting of the Lie algebra, i.e.
\begin{align}\label{eq:KGassumption} \begin{split} 1 \to K \into &\wt{G} \to G \to 1 \\ \wt{\fg} \iso &\fg \oplus \fk. \end{split} \end{align}
Such a decomposition of Lie algebras must exist since our groups are compact.  Of primary interest is when $\wt{G} = G\times K$, but we wish to also allow examples such as $SU(n) \into U(n) \to U(1)$ or $U(1) \into {\rm Spin}^c(n) \to SO(n)$.  


% Assume that $K$ acts freely on $Q$, so that $Q \xrightarrow{q} M$ is a principal $K$-bundle.  A  connection $\Theta$ naturally defines a section of $q_{/K}$ by pulling back $(Q,\Theta)$.
%%\[ \xymatrix{ \En K \times_K P  \ar[d]_{q_{/K}} \\ M. \ar@/_1pc/[u]_{\Theta} } \]
%%The section $\Theta$ is given by pulling back $(P,\Theta)$.
%\begin{align*}
%M &\overset{\Theta}\longrightarrow \En K\times_K Q \\
% \vcenter{ \xymatrix@C=.5cm{ X \ar[r]^{f} &M }} &\longmapsto \vcenter{ \xymatrix{ f^*(Q,\Theta) \ar[d] \ar[r]^{\wt{f}} &(Q,\Theta) \ar@{..>}[d] \\ X \ar@{..>}[r]^{f} & M } }
%\end{align*}
%We proceed to generalize this for $G$-equivariant principal $K$-bundles with connection.  In the following, the base manifolds $M$ will always be $G$-manifolds, and $K$ will be the structure group of an equivariant bundle over $M$.
%
%Assume that $K$ is a normal subgroup of $\wt{G}$ with quotient $\wt{G}/K \iso G$, and fix a Lie algebra splitting
%\[ \wt{\fg} \iso \fg \oplus \fk.\]
%(Such a splitting must exist since our groups are compact.)  Of primary interest is when $\wt{G} = G\times K$, but it is no more difficult to also allow examples such as $SU(n) \into U(n) \to U(1)$.


\begin{defn}For $M\in G\GMan$, let $\wt{G}\GBunc[K](M)$ be the groupoid of $\wt{G}$-equivariant principal $K$-bundles on $M$ with invariant connection.  An object $(Q,\Theta) \in \wt{G}\GBunc[K](M)$ is a principal $K$-bundle with connection on $M$ such that:
\begin{itemize}
%\item $(P,\Theta) \in \Bunc[K](M)$,
\item $Q \in \wt{G}\GMan$ and $Q \overset{\pi}\to M$  is equivariant,
\item $\Theta \in \big( \O^1(Q) \otimes \fk\big)^{\wt{G}}$.
\end{itemize}
For $(Q_i, \Theta_i) \in \wt{G}\GBunc[K](M)$, a morphism $(Q_1,\Theta_1) \overset{\phi}\to (Q_2,\Theta_2)$ is a $\wt{G}$-equivariant map $Q_1 \overset{\phi}\to Q_2$ which is an isomorphism of $K$-bundles with connection.
\end{defn}

\begin{rem}For $\wt{G} = G \times K$, the condition $\Theta \in \big(\O^1(Q) \otimes \fk\big)^{G\times K}$ can be rewritten as $k^*\Theta = \Ad_{k^{-1}}\Theta$ and $g^*\Theta = \Theta.$  Also, in this case we may refer to $\wt{G}$-equivariant $K$-bundles as $G$-equivariant $K$-bundles.  This is a standard convention, and we hope it does not cause any confusion.  
\end{rem}

Given an element $(Q,\Theta) \in \wt{G}\GBunc[K](M)$, we want to define a natural map $\Theta_G$
\[ \xymatrix{ \En \wt{G} \times_{\wt{G}} Q \ar[r]_-{q_{/K}} & \En G \times_G M \ar@/_1pc/[l]_-{\Theta_G} 
} \]
such that the composition $q_K \o \Theta_G$ is naturally isomorphic to the identity.  We first describe the construction, hopefully emphasizing the geometric nature and naturalness.  The details, which are relatively straightforward, are then checked in Proposition  \ref{prop:Reduction}.  While the induced maps in cohomology and equivariant forms that we recover are well-known, and the construction of $\Theta_G$ appears implicitly in \cite{MR1850463}, we do not know any references where this construction is done in general at the level of principal bundles with connection.

We now give a functor between the two groupoids produced when the stacks are evaluated on a test manifold $X$.  Denoting $\Theta=\Theta_Q$ for added clarity, the map is given by the following construction.
\begin{align*}  &(\En G \times_G M ) (X) &&\xrightarrow{\Theta_G(X)} &&(\En \wt{G} \times_{\wt{G}} Q ) (X) \\
&\vcenter{ \xymatrix{& (Q,\Theta_Q) \ar[d]^{\pi_M} \\ (P, \Theta_P) \ar[r]^{f} \ar[d]^{\pi_X} & M \\X }} &&\xmapsto{\quad \quad}
&& \vcenter{ \xymatrix{\Big(f^*Q, \, \pi_P^*\Theta_P \oplus (\wt{f}^*\Theta_Q - \iota_{\pi_P^* \Theta_P}\wt{f}^*\Theta_Q) \Big) \ar[r]^-{\wt f} \ar@{..>}[d]^-{\pi_P} \ar@/_3pc/[dd]_{\pi'_X} &(Q,\Theta_Q) \ar@{..>}[d]^{\pi_M} \\ (P, \Theta_P) \ar@{..>}[r]^{f}  \ar@{..>}[d] & M \\X} }
 \end{align*}
Our new $\wt{G}$-bundle is given by pulling back $Q$.  The connection on $f^*Q$ is given by the pulling back the connections on $P$ and $Q$ and subtracting a correction term.  This extra term is defined to be the image 
\begin{align*}
\O^1(\wt{f}^*Q) \otimes \fg \otimes \O^1(\wt{f}^*Q) \otimes \fk & \longrightarrow \O^1(\wt{f}^*Q) \otimes \fk \\
\pi^*_P \Theta_P \otimes \wt{f}^*\Theta_Q &\longmapsto \iota_{\pi_P^* \Theta_P}\wt{f}^*\Theta_Q
\end{align*}
under the natural contraction $\fg \otimes \O^1(\wt{f}^*Q) \xrightarrow{\iota} \O^0(\wt{f}^*Q)$.  While this term may initially seem obscure, it is necessary to ensure that we produce a connection on $f^*Q$, and it has a simple description in the Weil model.

Define the equivariant extension of the connection and curvature forms by 
\begin{align}
\label{eq:equivConn} \Theta_G &\= \Theta - \iota_{\theta_\fg}\Theta &&\in \big( \W(\fg)\otimes \O(Q) \big)^1\otimes \fk, \\
\label{eq:equivCurv} \Omega_G &\= d_G \Theta_G + \tfrac{1}{2} [\Theta_G \wedge \Theta_G] &&\in  \big( \W(\fg)\otimes \O(Q) \big)^2 \otimes \fk.
\end{align}
To clarify, $\iota_{\theta_\fg} \Theta \in \L^1 \fg^* \otimes \O^0(Q) \otimes \fk$ and it is evaluated on elements $\xi \in \fg$ by contracting the connection along the vector field in $Q$ generated by the $G$-action; i.e.
\[ \langle \iota_{\theta_\fg} \Theta, \xi\rangle \= \iota_{\xi} \Theta \in \O^0(Q;\fk).\]
The $\fk$ portion of the connection on $\wt{f}^*Q$ is simply the image of the equivariant connection $\Theta_G$ under the Weil homomorphism 
\begin{align*}
\W(\fg)\otimes \O(Q) \otimes \fk &\xrightarrow{(\pi^*_P\Theta_P)^*\otimes \wt{f}^* \otimes 1} \O(\wt{f}^*Q) \otimes \fk \\
\Theta_G = \Theta - \iota_{\theta_\fg} \Theta &\xmapsto{\quad\quad\quad} \wt{f}^*\Theta_Q - \iota_{\pi_P^* \Theta_P}\wt{f}^*\Theta_Q.
\end{align*}

Note that the additional term $\iota_{\pi_P^* \Theta_P}\wt{f}^*\Theta_Q$ uses the connection on the $G$-bundle $P$ to detect the $G$-action on $Q$.  This is key to understanding the equivariant Chern--Weil homomorphism in Section \ref{sec:EquivCW}.
%\[\left[ \vcenter{ \xymatrix{(f^*Q, \pi_P^*\Theta_P \oplus \wt{f}^*\Theta_Q ) \ar[r]^-{\wt f} \ar[d]^-{\pi_P} \ar@/_4pc/@{-->}[dd]_{\pi_X} &(Q,\Theta_Q) \ar@{..>}[d]^{\pi_M} \\ (P, \Theta_P) \ar@{..>}[r]^{f}  \ar[d] & M \\X} }\right] \longmapsfrom \left[ \vcenter{ \xymatrix{& (Q,\Theta_Q) \ar[d]^{\pi_M} \\ (P, \Theta_P) \ar[r]^{f} \ar[d]^{\pi_X} & M \\X }} \right]  \]


\begin{rem}\label{Rem:CartanCurv}In the Cartan model for equivariant forms (see Section \ref{subsec:CartanMQ}), the equivariant connection and curvature simplify to
\begin{align*}
\Theta_G &\longleftrightarrow \Theta &&\in S^0 \fg^* \otimes \O^1(Q)\otimes \fk, \\
\Omega_G &\longleftrightarrow \Omega - \iota_{\Omega_\fg} \Theta &&\in \Big(S^0 \fg^* \otimes \O^1(Q) + S^2\fg^* \otimes \O^0(Q)\Big) \otimes \fk,
\end{align*}
with $\Omega_G$ becoming the more familiar equivariant curvature defined in \cite[\textsection 2]{MR705039}.
%We view $\varOmega_\fg \in S^1 \fg^* \otimes \fg$ as the Koszul derivative of the Maurer--Cartan form, and $\langle \iota_{\varOmega_\fg} \Theta,  \xi \rangle := \iota_{\xi} \Theta$ for $\xi \in  S^1 \fg$.  While the Cartan model is far more convenient when performing calculations, we find it easier to explain the underlying geometric constructions when using the Weil model.
\end{rem}

We proceed to check that $\Theta_G$ and $\Omega_G$ are equivariant forms and the above construction satisfies the desired properties.


\begin{lemma}\label{lem:equivConn}For $(Q,\Theta) \in \wt{G}\GBunc[K](M)$, the forms $\Theta_G$ and $\Omega_G$ are $\wt{G}$-invariant and $\fg$-horizontal; i.e. they live in the subcomplex $\left(\W(\fg)\otimes \O(Q) \otimes \fk\right)^{\wt{G}}_{\fg \text{-}\hor}$.  When $\wt{G} = G\times K$, this can be written
\[ \Theta_G \in \O^1_G(Q;\fk), \quad \Omega_G \in \O^2(Q;\fk).\]
\end{lemma}
\begin{proof}
To see that $\Theta_G$ is $\fg$-horizontal, let $\xi \in \fg$.  Then
\[ \iota_\xi \Theta_G = \iota_\xi \Theta - \iota_{\theta_\fg(\xi)} \Theta = \iota_\xi \Theta - \iota_\xi \Theta = 0.\]
We now show both terms in $\Theta_G$ are $\wt{G}$-invariant.  First note that $\Theta \in \left( \O^1(Q) \otimes \fk\right)^{\wt{G}}.$  To check the term $\iota_{\theta_{\fg}}\Theta$, observe that $\theta_{\fg} \in \left( \L^1 \fg^* \otimes \fg \right)^G = \left( \L^1 \fg^* \otimes \fg \right)^{\wt{G}}$, and the contraction map 
\begin{align*} \fg \otimes \O^1(Q) &\overset{\iota}\longrightarrow \O^0(Q) \\
\xi \otimes \omega &\longmapsto  \iota_{\xi} \omega 
\end{align*}
is $\wt{G}$-equivariant.  Hence,
\[ \iota_{\theta_{\fg}}\Theta \in \left( \L^1 \fg^* \otimes \O^0(Q) \otimes \fk\right)^{\wt{G}},\]
so $\Theta_G$ is $\fg$-horizontal and $\wt{G}$-invariant.

From the Leibniz rule for $\iota_\xi$ and the definition of tensor product of representations, it immediately follows that $[\Theta_G \wedge \Theta_G]$ is also $\fg$-horizontal and $\wt{G}$-invariant.  The standard  identities \eqref{eq:gstar1} and \eqref{eq:gstar2} imply that $d_G \Theta_G$ is also $\wt{G}$-invariant and $\fg$-horizontal.  Consequently, $\Omega_G$ is invariant and horizontal.
%The term $[\Theta_G \wedge \Theta_G]$ is also $\fg$-basic since the Leibniz rule implies
%\[ \iota_{\xi}[\Theta_G \wedge \Theta_G] = [\iota_{\xi}\Theta_G \wedge \Theta_G] - [\Theta_G \wedge \iota_{\xi}\Theta_G] = 0-0 = 0.\] 
\end{proof}


%\begin{prop}Via the construction described above, an object $(Q,\Theta) \in \wt{G}\GBunc[K](M)$ induces a natural map $\Theta_G$
%\[ \xymatrix{ \En \wt{G} \times_{\wt{G}} Q \ar[r]_-{q_K} & \En G \times_G M \ar@/_1pc/[l]_-{\Theta_G} 
%} \]
%such that the composition $q_K \o \Theta_G$ is naturally isomorphic to the identity.
%\end{prop}


\begin{prop}\label{prop:Reduction}Assume $\wt{G}/K \iso G$ satisfying \eqref{eq:KGassumption}.  Let $(Q,\Theta) \in \wt{G}\GBunc[K](M)$.
\begin{enumerate} 
\item\label{Reduction1} The above construction defines a natural morphism $\Theta_G$
\[ \xymatrix{ \En \wt{G} \times_{\wt{G}} Q \ar[r]_-{q_{/K}} & \En G \times_G M \ar@/_1pc/[l]_-{\Theta_G} 
} \]
such that the composition $q_{/K} \o \Theta_G$ is naturally isomorphic to the identity.
\item\label{Reduction2} The induced map 
\[ \O\big( \En \wt{G} \times_{\wt{G}} Q  \big) \xrightarrow{\Theta_G^*} \O\big( \En G \times_G M \big) \]
is equivalent to the homomorphism $\Theta_G^*\colon  \O_{\wt{G}}(Q) \to \O_G(M)$ induced from
\begin{align*}
\W(\fg)\otimes \W(\fk) \otimes \O(Q) &\overset{\Theta_G^*}\longrightarrow \W(\fg) \otimes \O(Q) \\
\alpha \cdot \beta \cdot \gamma &\longmapsto \alpha \cdot \beta(\Theta_G) \cdot \gamma.
\end{align*}

\item \label{Reduction3} The induced map in cohomology is the standard isomorphism
\[ H^*_{\wt{G}} (Q;-) \overset{\iso}\longrightarrow H^*_{G}(M;-). \]
\end{enumerate}
\end{prop}
\begin{proof}
For  part \eqref{Reduction1}, we must check that our construction defines a principal $\wt{G}$-bundle with connection.  Since $P \xrightarrow{f} M \xleftarrow{\pi_M} Q$ are equivariant maps between $\wt{G}$-manifolds, the pullback $f^*Q$ is naturally a $\wt{G}$-manifold.  The freeness of the $\wt{G}$-action follows easily by the following argument.  Suppose $\wt{g}\in \wt{G}$ satisfies $\wt{g} (p,q) = (p,q)$ for some point in $f^*Q \subset P \times Q$; then $gp =p \in P$ implies $g=1$ and hence $\wt{g} \in K$.  Since $K$ acts freely on $Q$, then $\wt{g}=1$.  Hence, $f^*Q \to X$ is a principal $\wt{G}$-bundle.  To show that 
\begin{equation}\label{eq:newconnection}  \Theta_{f^*Q} = \pi_P^*\Theta_P \oplus \big(\wt{f}^*\Theta_Q - \iota_{\pi_P^* \Theta_P}\wt{f}^*\Theta_Q \big) \in \O^1(f^*Q)\otimes (\fg \oplus \fk)\end{equation}
is a connection, we must check it is $\wt{G}$-equivariant and restricts fiberwise to the Maurer--Cartan 1-form on $\wt{G}$.  By Lemma \ref{lem:equivConn}, we see that $\wt{f}^*\Theta_Q - \iota_{\pi_P^* \Theta_P}\wt{f}^*\Theta_Q$ must be $\wt{G}$-invariant and $\fg$-horizontal, since it is the image of $\Theta_G$ under the map
\[ \big(\W(\fg)\otimes \O(Q) \otimes \fk \big)^{\wt{G}}_{\fg \text{-}\hor} \longrightarrow \big( \O(f^*Q) \otimes \fk \big)^{\wt{G}}_{\fg\text{-}\hor}.  \]
This, combined with the fact that $\Theta_P \in \big(\O^1(P) \otimes \fg \big)^G$, implies that \eqref{eq:newconnection} is an element of $\big( \O^1(f^*Q) \otimes(\fg \oplus \fk)\big)^{\wt{G}}.$  Similarly, if $\xi_1 \oplus \xi_2 \in \fg \oplus \fk$, then
\begin{align*}
\iota_{\xi_1 \oplus \xi_2} &\big( \pi_P^*\Theta_P \oplus (\wt{f}^*\Theta_Q - \iota_{\pi_P^* \Theta_P}\wt{f}^*\Theta_Q) \big)  \\
&= \big( \iota_{\xi_1} \Theta_P \big) \oplus \big( \iota_{\xi_2} \Theta_Q \big) + 0\oplus \iota_{\xi_1}\big( \wt{f}^*\Theta_Q - \iota_{\pi_P^* \Theta_P}\wt{f}^*\Theta_Q\big) \\
&= \xi_1 \oplus \xi_2.
\end{align*}
Therefore, the morphism of stacks $\Theta_G$ is well-defined.  The map $\pi_P$ gives a natural isomorphism $(f^*Q)/K \iso P$ compatible with the connections.  Hence, $q_{/K} \o \Theta_G$ is naturally isomorphic to the identity on $\En G \times_G M$.  

The proof of part \eqref{Reduction2} is given by tracing through the Freed--Hopkins isomorphism, discussed in Section \ref{subsec:FreedHopkins}, and the definition of our map.  Essentially, we must show that the diagram 
\[ \xymatrix{ \O_{G\times K}(Q) \ar[d]^{\Theta_G^*} \ar[r]^-{\iso}& \O\left(\En (G\times K) \times_{G\times K} Q \right) \ar[d]^{\Theta_G^*} \\
\O_G(M) \ar[r]^-{\iso} & \O(\En G \times_G M)
}\]
commutes by evaluating on a test manifold $X$.  Consider a homogeneous element $\alpha  \beta  \gamma \in \O_{G\times K}(Q) \subset \W(\fg)\otimes \W(\fk) \otimes \O(Q)$.  We first check the clockwise direction.  Given $X \to \En G \times_G M$, we have 
\[ \vcenter{\xymatrix{ (P,\Theta_P) \ar[d] \ar[r]^f &M \\ X
} }  \quad \mapsto \quad \vcenter{ \xymatrix{ (f^*Q, \Theta_{f^*Q}) \ar[r]^{\wt f} \ar[d]^{\pi_P} & (Q,\Theta) \ar[d]\\ (P,\Theta_P) \ar[r]^{f} \ar[d] & M \\ X
} }\]
where 
\[ \Theta_{f^*Q} =  \pi_P^* \Theta_P \oplus \left( \wt f^*\Theta - \iota_{\pi_P^*\Theta_P} \wt f^*\Theta \right).\]
Therefore, $\alpha \beta \gamma \in \O_{G\times K}(Q)$ gets mapped to
\[ \alpha(\pi_P^*\Theta_P) \wedge \beta( \wt f^*\Theta - \iota_{\pi_P^*\Omega_P} \wt f^*\Theta) \wedge \wt f^* \gamma \in \O(f^*Q)_{G\times K \text{-} \basic} \iso \O(X).\]
Going counter-clockwise, we first have that $\alpha \beta\gamma \mapsto \alpha \beta(\Theta_G)\gamma$, which lives in \[\left( \W(\fg) \otimes \O(Q) \right)_{G\times K \text{-} \basic } \iso \left(\W(\fg) \otimes \O(M) \right)_{G \text{-} \basic}.\]
%In particular, we know that $\beta(\Theta_G)\gamma$ descends to $\W(\fg) \otimes \O(M)$.  
When we evaluate on $X \leftarrow (P,\Theta) \xrightarrow{f} M$, 
\[ \alpha \beta(\Theta_G) \gamma \quad \longmapsto \quad \alpha(\Theta_P) \beta(f^*\Theta - \iota_{\Theta_P} f^*\Theta) f^*\gamma \, \in \,  \O(P)_{G\text{-}\basic} \iso \O(X).\]
Pulling this back up to $f^*Q$ via $\pi_P^*$ gives us the form from the other direction.  Hence, they restrict to the same form in $\O(X)$.

Finally, part \eqref{Reduction1} implies that $\Theta_G^* \, \o\,  q_K^* = \Id$, and we know from \eqref{eq:QuotientFree} that $q_{/K}^*$ is an isomorphism in ordinary cohomology.  Therefore, part \eqref{Reduction3} follows immediately.
\end{proof}

%In its most basic case, this takes the form.
%
%\begin{cor}If $(P,\Theta) \in \Bunc[G](M)$, there are natural homomorphisms
%\[ \HG[G]^k(P) \overset{\Theta^*}\longrightarrow \HG[]^k(M)\]
%that recover the usual $H_G^*(P;\Z) \iso H^*(M;\Z)$ and $\O_G(P) \overset{\Theta^*}\to \O(M)$.
%\end{cor}


%\begin{prop}\label{prop:GxK-Reduction}An object $(Q,\Theta) \in G\GBunc[K](M)$ induces a natural homomorphism
%\[ \HG[G\times K](Q) \xrightarrow{\Theta_G^*} \HG[G](M).\]
%In the short exact sequences, this gives
%\[ \xymatrix{ 0 \ar[r]& \frac{\O^{*-1}_{G\times K}(Q)}{\O^{*-1}_{G\times K}(Q)_\Z} \ar[r] \ar@/_1pc/[d]_{\Theta_G^*}&\HG[G\times K]^*(Q) \ar@/_1pc/[d]_{\Theta_G^*}\ar[r] & H^*_{G\times K}(Q;\Z) \ar[r] \ar@{<->}[d]^{\iso} &0 \\
%0 \ar[r]& \frac{\O^{*-1}_G(M)}{\O_G^{*-1}(M)_\Z} \ar[r] \ar@/_1pc/[u]_{q_K^*}&\HG[G]^*(M) \ar[r] \ar@/_1pc/[u]_{q_K^*} & H_G^*(M;\Z) \ar[r] &0 
%}\]
%\[ \xymatrix{ 0 \ar[r] & H_{G\times K}^{*-1}(Q;\R/\Z) \ar@{<->}[d]^{\iso} \ar[r] & \HG[G\times K]^*(Q) \ar@/_1pc/[d]_{\Theta_G^*} \ar[r] & \O^*_{G\times K}(Q)_\Z \ar@/_1pc/[d]_{\Theta_G^*} \ar[r] &0 \\
%0 \ar[r] & H_G^{*-1}(M;\R/\Z) \ar[r] & \HG[G]^*(M) \ar@/_1pc/[u]_{q_K^*} \ar[r] & \O_G^*(M)_\Z \ar@/_1pc/[u]_{q_K^*} \ar[r] &0 
%} \]
%where $\Theta_G^*: \O_{G\times K}(Q) \to \O_G(M)$ is the homomorphism defined in Proposition \ref{prop:ReductionForms}.
%\end{prop}
%\begin{proof}
%\end{proof}

%\reminder{Possibly rewrite the above for $G \times K$ for simplicity, and then do additional proposition for $\wt{G}$ noting the slight change in details?}

%Extend Proposition \ref{prop:GxK-Reduction} to the case of $K \into \wt{G} \to G$.  We must assume that the principal $K$-bundle $\wt{G} \to G$ has a $\wt{G}$-invariant connection; i.e. ....   This is equivalent to a Lie algebra splitting 
%\[\wt{g} \iso \fg \oplus \fk.\]
%\begin{equation}\label{eq:GroupCondition}
%\end{equation}
%Analogously, and in a slight abuse of notation, define $\wt{G}\GBunc[K](M)$.
%
%\begin{prop}Suppose that $K \triangleleft \wt{G}$ satisfies \eqref{eq:GroupCondition}.  An object $(Q,\Theta) \in \wt{G}\GBunc[K](M)$ induces a natural map
%\[ \En G \times_G M \xrightarrow{\Theta_{\wt{G}} } \En \wt{G} \times_{\wt{G}} Q.\]
%The results of Proposition \ref{prop:GxK-Reduction} hold, with $G\times K$ replaced by $\wt G$.
%\end{prop}


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\subsection{Uniqueness of $\HG^*$}\label{subsec:Uniqueness}

A natural question is whether one can give an axiomatic characterization of the functor $\HG^*$.  We now show that the character diagram, combined  with the maps $\HG^*(P) \xrightarrow{\Theta^*} \HG[]^*(M)$ for principal $G$-bundles with connection, uniquely characterize our differential extension of Borel equivariant cohomology.  The idea is to use $\Theta^*$ to regard $\HG^*(M)$ a subgroup of $\HG[]^*(EG\times_G M)$ and invoke the Simons--Sullivan axiomatic characterization of differential cohomology.  It is unknown to the author if the uniqueness result holds without assuming the additional structure maps $\Theta^*$.


\begin{prop}\label{prop:Uniqueness}If $\bH^*_G$ is a functor from $G$-manifolds to graded abelian groups satisfying parts \ref{thm1diag} and \ref{thm1conn} of Theorem \ref{thm:Properties}, then there is a natural equivalence $\bH^*_G \to \HG^*$ that commutes with the identity map on all other functors in the character diagram.
\end{prop}
\begin{proof}
Assume that $\bH_G$ satisfies Theorem \ref{thm:Properties} parts \ref{thm1diag} and \ref{thm1conn}.  As described in Section \ref{subsec:deRhamModel}, let $(EG,\Theta_{EG}) \to BG$ be the universal bundle with connection obtained as a direct limit of smooth manifolds.  The projection map $EG \times M \to M$ is $G$-equivariant and can be combined with the universal connection to produce
\[ \bH_G^*(M) \to \bH_G^*(EG\times M) \xrightarrow{\Theta_{EG}^*} \bH^*(EG \times_G M). \]
On ordinary cohomology, this map is an isomorphism.  Since the Weil homomorphism  $\O_G(M) \to \O(EG\times_G M)$ is injective (Lemma \ref{lem:WeilEGinj}), it follows from the short exact sequence \eqref{ses2} that $\bH_G^*(M)$ is naturally an abelian subgroup of $\bH^*(EG\times_G M)$.  The same argument shows $\HG^*(M)$ is a subgroup of $\HG[]^*(EG\times_G M)$.  

In \cite{SS08a}, Simons--Sullivan show the character diagram uniquely characterizes ordinary differential cohomology, and there exists a unique natural transformation $\bH^* \xrightarrow{\iso} \HG[]^*$.  This isomorphism combines with \eqref{ses2} to give the following.
\[ \xymatrix{ 0 \ar[r] &\frac{\O^{*-1}_G(M)}{\O^{*-1}_G(M)_\Z} \ar[r] \ar@{^(->}[d] &\bH_G^*(M) \ar[r] \ar@{^(->}[d]& H^*_G(M;\Z) \ar[r] \ar[d]^{\iso}&0 \\
0 \ar[r] & \frac{\O^{*-1} (EG\times_G M)}{\O^{*-1} (EG\times_G M)_\Z} \ar[r]  & \HG[]^*(EG\times_G M) \ar[r] & H^*(EG\times_G M;\Z) \ar[r]&0 \\
0 \ar[r] & \frac{\O^{*-1}_G(M)}{\O^{*-1}_G(M)_\Z} \ar[r] \ar@{_(->}[u]& \HG^*(M) \ar[r] \ar@{_(->}[u] & H^*_G(M;\Z) \ar[r] \ar[u]_{\iso}&0
} \]
It immediately follows that $\bH_G^*(M)$ and $\HG^*(M)$ can be naturally considered as the same abelian subgroup of $\HG[]^*(EG\times_G M)$, and this gives us our desired natural isomorphism.
\end{proof}






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\section{Equivariant Chern--Weil theory}\label{sec:EquivCW}  
The constructions in  Propositions  \ref{prop:GroupChange} and \ref{prop:Reduction} combine to give a geometric interpretation of equivariant Chern--Weil theory.  For simplicity, let $\wt{G}=G\times K$ in the following preliminary discussion.

Suppose that $M$ is a $G$-manifold and $(Q,\Theta) \to M$ is a $G$-equivariant principal $K$-bundle with $G$-invariant connection; i.e.  $(Q,\Theta)\in G\GBunc[K](M)$.  We briefly explain two classical ways to construct equivariant characteristic classes.  The first construction is purely topological and does not use the connection.  Consider the principal $K$-bundle $EG \times_G Q \to EG \times_G M$.  The choice of classifying map 
\begin{equation}\label{eq:ClassMapEquivBundle} EG \times_G M \longrightarrow BK.\end{equation}
induces a homomorphism
\[ H^*(BK;A) \longrightarrow H^*_G(M;A) \]
for any abelian group $A$.
The second method, due originally to Berline--Vergne  \cite{MR705039}, uses the connection to construct an equivariant differential form.  Given $\omega \in (S^n \fk^*)^K \iso H^{2n}(BK;\R)$, define the equivariant Chern--Weil form
\[ \omega(\Theta_G) \= \omega(\Omega_G^{\wedge n}) \in \O_G^{2n}(M). \]
As noted in Remark \ref{Rem:CartanCurv}, this becomes
\[ \omega\big( (\Omega - \iota_{\varOmega_\fg } \Theta)^{\wedge n} \big) \in \left( S \fg^* \otimes \O(M) \right)^G\]
in the more commonly used Cartan model.  While the form $\omega(\Theta_G)$ depend on the connection $\Theta$, its class in $H_G^{2n}(M;\R)$ does not.  These give equivariant Chern classes and forms when $K=U(n)$, and equivariant Pontryagin classes and forms when $K=SO(n)$.  In \cite{MR1850463}, Bott--Tu use the universal connection on $EG$ to show these two constructions give the same classes in $H^*_G(M;\R)$.  The following construction is similar to theirs,  but we replace $EG$ with $\En G$ and view everything in the world of stacks.  

Suppose that $(Q,\Theta) \in G\GBunc[K](M)$.  The following maps compose to give a refinement of \eqref{eq:ClassMapEquivBundle} to stacks.
\begin{equation}\label{eq:ChernWeil} \En G \times_G M \xrightarrow{\Theta_G} \En (G\times K) \times_{G\times K} Q \longrightarrow \Bn (G \times K)  \xrightarrow{q_{/G}} \Bn K.\end{equation}
The first map is given by the bundle pullback construction of Section \ref{subsec:Reduction}, the second is induced by the equivariant map $Q\to \pt$, and the third map is induced by the quotient $G\times K \xrightarrow{/G} K$.  When applied to a test manifold $X$, the composition performs the following construction.
\[  \vcenter{ \xymatrix{& (Q,\Theta_Q) \ar[d]^{\pi_M} \\ (P, \Theta_P) \ar[r]^{f} \ar[d]^{\pi_X} & M \\X }} \longmapsto  \vcenter{ \xymatrix{ \big(f^*Q, \pi_P^* \Theta_P \oplus (\wt f^* \Theta_Q - \iota_{\Theta_P} \wt{f}^* \Theta_Q) \big) \big/ G \ar[d] \\ X} } \]
It pulls back $Q$, modifies the connection to create a $G\times K$-bundle with connection, and then quotients by $G$.


\begin{thm}\label{thm:equivCCS}Associated to $(Q,\Theta) \in G\GBunc[K](M)$ is 
a natural morphism
\[ \En G \times_G M \xrightarrow{(Q_G,\Theta_G)} \Bn K, \]
such that the induced map 
\[ \HG[K]^* \xrightarrow{(Q_G,\Theta_G)^*} \HG^*(M)\] 
refines the traditional equivariant characteristic classes and forms as described above.
%, and it is given by the composition 
%\[ \HG[K]^* \to \HG[G\times K]^* \to \HG[G\times K]^*(Q) \xrightarrow{\Theta_G^*} \HG[G]^*(M)\]
\end{thm}
\begin{proof}We know the morphism $\En G \times_G M \to \Bn K$ is well-defined, since it is defined in \eqref{eq:ChernWeil} as the composition of three morphisms.

We now must show that at the level of cohomology, $(Q_G,\Theta_G)^*$ is the homomorphism $\psi^*$, where $EG\times_G M \xrightarrow{\psi} BK$ is a classifying map for $EG\times_G Q \to EG\times_G M$.  By Proposition \ref{prop:StackCohIso}, the maps $EG\times_G M \to \En G \times_G M \to \E G \times_G M$ and $BK \to \Bn K \to \B K$  induce isomorphisms in cohomology.  Therefore, we must check that the diagram
\begin{equation}\label{eq:equivCCSdiag} \vcenter{ \xymatrix{ EG\times_G M \ar[d]_{\psi} \ar[r]  & \E G \times_G M \ar[d]^{Q_G} \\ BK \ar[r]  & \B K,} } \end{equation}
where the two horizontal morphisms given by viewing $EG\to BG$ and $EK \to BK$ as principal bundles, is commutative up to isomorphism.

We evaluate on a test manifold $X$ and trace through where $X\xrightarrow{s} EG\times_G M$ is sent, using the following diagram to identify maps.
\[ \xymatrix{ \wt{s}^*(\Pi_M^* Q) \ar[r]\ar[d] & \Pi_M^*Q \ar[r] \ar[d] & Q \ar[d] \\
s^*(EG\times M) \ar[d] \ar[r]^{\wt{s}} & EG \times M \ar[d] \ar[r]^-{\Pi_M} & M \\
X \ar[r]^{s} &EG\times_G M 
}\]
Going clockwise in \eqref{eq:equivCCSdiag}, we have
\[  X\xrightarrow{s} EG\times_G M  \;  \mapsto \;  \vcenter{ \xymatrix{s^*(EG\times M) \ar[r]^(.65){\Pi_M \o \wt{s} } \ar[d] & M \\ X} } \; \mapsto \; \vcenter{ \xymatrix{ \big((\Pi_M \o \wt{s})^*Q\big)/G \ar[d] \\ X } }, \]
and the evident isomorphism $\Pi_M^*Q \iso EG\times Q$ gives 
\[ \big( (\Pi_M \o \wt{s})^*Q \big)/G \iso \big( \wt{s}^*(EG\times Q) \big)/G  \iso s^* (EG\times_G Q). \]
Going counter-clockwise in \eqref{eq:equivCCSdiag}, we have
\[  X \xrightarrow{s} EG\times_G M   \; \mapsto \;   X \xrightarrow{\psi \o s} BK   \; \mapsto \;   \vcenter{ \xymatrix{ (\psi \o s)^* EK \ar[d] \\ X }}. \]
Since
\[ (\psi \o s)^* EK \iso s^*(\psi^*EK) \iso s^*(EG\times_G Q),\]
the diagram  \eqref{eq:equivCCSdiag} commutes up to isomorphism.

For differential forms, we trace through the composition \eqref{eq:ChernWeil}, which is given by the restriction of the following to the basic sub-complexes
\[ \W(\fk) \to \W(\fg) \otimes \W(\fk) \to \W(\fg)\otimes \W(\fk) \otimes \O(Q) \xrightarrow{\Theta^*_G} \W(\fg)\otimes \O(M).  \]
The first two maps are given by the mapping $\W(\fk)$ into its factor in the tensor product, so the composition is simply
\begin{align*}
\left( S \fk^*\right)^K = \W(\fk)_{\basic}  &\longrightarrow \left( \W(\fg) \otimes \O(M) \right)_{\basic} = \O_G(M) \\
\omega &\longmapsto \omega(\Theta_G),
\end{align*}
which is precisely the equivariant Chern--Weil form in the Weil model.
\end{proof}



One can also construct equivariant extensions of the Chern--Simons forms.  As explained in \eqref{eq:ChernSimonsWeil}, to $\omega \in (S^k\fk^*)^K$ is naturally associated a Chern--Simons form $\CS_\omega \in \W^{2k-1}(\fk)^K$ satisfying $d_\W \CS_\omega = \omega$.  Given $(Q,\Theta) \in G\GBunc[K](M)$, we define the equivariant Chern--Simons form to be the image of $\CS_\omega$ under the equivariant equivariant Weil homomorphism 
\begin{align}\label{eq:EquivChernSimonsForm}\begin{split}
\W^{2k-1} &\xrightarrow{\Theta_G^*} \O^{2k-1}_G(Q) \\
\CS_\omega &\longmapsto \CS_\omega(\Theta_G).  \end{split}
\end{align}
As in the case of differential characters, the Chern--Simons forms are closely related to the differential refinements of characteristic classes.  

\begin{prop}\label{prop:EquivChernSimons}Let $\check{\omega} \in \HG[]^{2k}(\Bn K) \iso H^{2k}(BK;\Z)$ be a differential refinement of $\omega \in (S^k \fk^*)^K$.  Suppose that $(Q,\Theta) \in G\GBunc[K](M)$ admits a $G$-equivariant section $s:M \to Q$.  Then 
\[ [ s_G^* \CS_\omega(\Theta_G) ] = \check{\omega}(\Theta_G) \in \HG^{2k}(M) .\]
\end{prop}
\begin{proof}
First note that because $EG$ is contractible, the character diagram for $\En G$ gives isomorphisms
\[ \xymatrix@R=3mm@C=10mm{& \HG[]^{2k}(\En G) \ar[dr]^{\iso} \\
{\displaystyle \frac{\W^{2k-1}(\fk)}{d_{\W}  \W^{2k-2}(\fk) }} \ar[ur]^{\iso} \ar_{d_{\W}}[rr]&& \W^{2k}(\fk)_0.
}\]
This gives the universal property $\pi^*\check{\omega} = [\CS_\omega] \in \HG[]^{2k}(\En K)$.

The construction \eqref{eq:ChernWeil} readily generalizes to the following diagram,
\[ \xymatrix@C=1.5cm{ \En G \times_G Q \ar[r] \ar[d] &\En K \ar[d]^{\pi} \\
\En G \times_G M \ar[r]^-{(Q_G,\Theta_G)} \ar@/_1pc/[u]_{s_G}  & \Bn K }\]
giving us two ways of writing the morphism $\En G \times_G M \to \Bn K$.  The bottom morphism induces $\check{\omega}(\Theta_G)$; the other one gives
\[\xymatrix@R=0mm{ \HG[](\Bn K) \ar[r] & \HG[](\En G) \ar[r] & \HG[](\En G\times_G Q) \ar[r] &  \HG[](\En G \times_G M) \\
\check{\omega} \ar@{|->}[r] & [\CS_\omega]  \ar@{|->}[r] & [\CS_\omega(\Theta_G)]  \ar@{|->}[r] & [s_G^*\CS_\omega(\Theta_G)] .
} \]
\end{proof}

%\[ \xymatrix@R=3mm@C=6mm{ 0 \ar[dr]&&&&0 \\
%&0 \ar[dr] \ar[rr] && 0 \ar[ur]  \ar[dr]\\
%0 \ar[ur] \ar[dr]&& \HG[]^{2k}(\En G) \ar_{\cc}[ur] \ar^{\curv}[dr] \ar@{}_>>>>>{\dR}[rr]&& 0 \\
%& {\displaystyle \frac{\W^{2k-1}(\fk)}{d_{\W}  \W^{2k-2}(\fk) }} \ar[ur] \ar_{d_{\W}}[rr]&& \W^{2k}(\fk)_0 \ar[ur] \ar[dr] \\
%0 \ar[ur]&&&&0
%} \]




\begin{rem}In the statement of Theorem \ref{thm:equivCCS}, we assumed that the total space of the bundle $Q$ had an action of $G\times K$.  Suppose instead that $(Q,\Theta) \in G\GBunc[K](M)$, where $\wt{G}/K\iso G$ with a fixed Lie algebra splitting \eqref{eq:KGassumption}.  It no longer makes sense to quotient by $G$ at the end, but we still have the composition
\[ \En G \times_G M \xrightarrow{\Theta_G} \En \wt{G} \times_{\wt{G}} Q \longrightarrow \Bn \wt{G}.\]
This induces differential equivariant characteristic classes via the homomorphism
\[ \HG[\wt{G}]^* \longrightarrow \HG^*(M).\]
Note that Proposition \ref{prop:EquivChernSimons} would not be applicable unless there exists an isomorphism $\wt{G} \iso G \times K$.  
%This fact is not surprising, as the universal characteristic classes for Borel equivariant cohomology in this situation would be in $H^*(B\wt{G})$ (Cor 2 from May's \cite{May87}).
\end{rem}




%In ordinary differential cohomology, the secondary invariants of bundles with connection can often be computed via Chern--Simons forms.  The same thing happens here in the equivariant situation, as we now explain.  
%
%Let $\check{\omega} \in \HG[]^{2k}(\Bn G) \iso H^{2k}(BK;\Z)$ be a differential refinement of $\omega \in (S^k \fk^*)^K$.  As explained in Example \ref{ex:ChernSimons}, the universal Chern--Simons form lives in
%\[ \CS_\omega \in \W^{2k-1}(\fk)^K \iso \O^{2k-1}(\En G).\]
%This gives a natural way to define equivariant Chern--Simons forms.
%
%The class $\check{\omega}(\Theta_G)$ is given by the equivariant Chern--Simons form.  Specifically, 
%in cases where the bundle $Q$ admits an equivariant section, then there is an equality.  More generally, one can often compute the ``equivariant holonomies" by using the equivariant Chern--Simons form.
%
%
%
%
%Given $(Q,\Theta) \in G\GBunc[K](M)$, we construct equivariant extensions of the usual Chern--Simons forms.  To do so, we mimic the construction of the original Chern--Simons forms.  
%
%
%Define the following formal 1-parameter family of path of connection forms on the total space of $\En K$ by
%\begin{align*}
%\theta_t &\= t\theta & \in \Big( \O^0[0,1] \otimes \L^1\fk^* \Big)\otimes \fk \\
%\varOmega_t &\= dt\tfrac{\partial}{\partial t} (\theta_t) + d_{\W} \theta_t + \tfrac 12 [\theta_t \wedge \theta_t] &\in \Big( \O[0,1] \otimes \W(\fk) \Big)^2 \otimes \fk 
%\end{align*}
%$\varOmega_t^{\wedge k} \in \Big( \O[0,1] \otimes \W(\fk)  \Big)^{2k} \otimes S^k \fg$ and hence 
%\[ \omega(\varOmega_t^{\wedge k} ) \in \Big( \O[0,1] \otimes \W(\fk)  \Big).\]
%
%$\varOmega_t =  dt\,\theta + t \varOmega + \tfrac{1}{2}(t^2-t)[\theta \wedge \theta]$.
%
%For $\omega \in (S^k \fg^*)^K$, the universal Chern--Simons form is defined
%\begin{equation}
%\CS_\omega \= \int_{t=0}^{t=1} \omega \big( \varOmega_t^{\wedge k} \big) \in \W^{2k-1}(\fk)^K,
%\end{equation}
%where 
%
%Satisfies $d_\W \CS_\omega = \omega(\varOmega^{\wedge k}) = \omega,$ and restricted to the completely vertical part, gives element in $(\L^{2i-1} \fk^*)^K$ representing transgression in 
%\[ \xymatrix{ H^{2k}(BK;\R) \ar[r]^{\tau} & H^{2k-1}(K;\R) \\
%(S^k \fk^*)^K \ar[u]^{\iso} \ar[r]^{\CS} & (\L^{2k-1} \fg^*)^K \ar[u]^{\iso}
%} \]
%
%\begin{rem}This is one way to show that $\W(\fk)$ is acyclic.  Alternatively, you can use a change of coordinates and recover the Koszul complex.
%\end{rem}
%
%Note that the space of $G$-invariant $K$-connections is affine over $\O^1(M;\fk)^G$.
%
%Given $(Q, \Theta) \in \Bunc[K](M)$, this gives (if we start with $\omega$ $K$-invariant, then everything will be $K$-invariant.)
%\[ \xymatrix{ \W^{2k-1}(\fk) \ar[r]^{\Theta^*} \ar[d]^{d_\W} & \O^{2k-1}(Q) \ar[d]^{d} \\
%\W^{2k}(\fk) \ar[r]^{\Theta^*} & \O^{2k}(Q) }  \quad \quad \xymatrix{ \CS_\omega \ar@{|->}[r] \ar@{|->}[d] & \CS_\omega(\Theta) \ar@{|->}[d] \\ \omega \ar@{|->}[r] & \omega(\Theta)  } \]
%
%
%This construction immediately extends to the equivariant case.
%
%\begin{defn}Fix $\omega \in (S^k \fk^*)^K$.  Associated to any $(Q,\Theta) \in G\GBunc[K](M)$ is the \textit{equivariant Chern--Simons form}
%\[ \CS^G_\omega(\Theta) \= \CS_\omega(\Theta_G) \in \O_G^{2k-1}(Q). \]
%\end{defn}
%
%\begin{prop}The equivariant Chern--Simons form is functorial.
%\[ d_G \CS^G_\omega(\Theta) = \omega^G(\Theta) \in \pi^* \O_G(M).\]
%\end{prop}
%
%\begin{cor}
%\[ \pi_G^* \lambda^G(\Theta) = \CS^G_\lambda (\Theta) \in \HG(P).\]
%\end{cor}
%Let $(Q,\Theta) \in G\GBunc[K](M)$ with equivariant connection $\Theta_G \in \O^1_G(Q;\fk)$.  The generalization of the Weil homomorphism, described in the previous section, gives
%\[ \xymatrix{ \W(\fk) \ar[r]^-{\Theta_G^*} \ar[d]^{d_\W} & \W(\fg) \otimes\O(Q) \ar[d]^{d_G} \\
%\W(\fk) \ar[r]^-{\Theta_G^*} & \W(\fg) \otimes \O(Q) }  \quad \quad \xymatrix{ \CS_\omega \ar@{|->}[r] \ar@{|->}[d] & \CS_\omega(\Theta_G) \ar@{|->}[d] \\ \omega \ar@{|->}[r] & \omega(\Theta_G)  } \]
%Since $\Theta_G^*$ is a morphism of $(G\times K)$-da's, then 
%\[ \W(\fk) \into \left( \W(\fg)\otimes \W(\fk) \right)_{\basic G} \to \left( \W(\fg) \otimes \O(Q) \right)_{\basic G},\]
%\[ S(\fk)^K \to \left( \W(\fg) \otimes \O(Q) \right)_{\basic G\times K} = \left( \W(\fg) \otimes \O(Q)_{\basic K} \right)_{\basic G},\]
%hence \marginnote{In fact, $(S\fk^*)^K \into \W(\fk)^K \to \O_G(Q)^K$.}
%\[ \xymatrix{ \W(\fk) \ar[r] & \O_G(Q) \\
%(S\fk^*)^K \ar@{^(->}[u] \ar[r] & \O_G(M) \ar@{^(->}[u]}.\]
%Therefore, 
%\begin{align*}
%\CS_\omega( \Omega_G) &\in \O^{2k-1}_G(Q) \\
%d_G \CS_\omega(\Omega_G) = \omega(\Omega_G) &\in \O^{2k}_G(Q)_{\basic K} \iso \O^{2k}_G(M)
%\end{align*}
%
%We can also use the notation \marginnote{e.g.  $c_i^G(\Omega) = c_i(\Omega_G)$.}
%\[ \CS^G_\omega(\Theta) = \CS_\omega(\Theta_G), \quad \omega^G(\Theta) = \omega(\Theta_G).\]
%Given two invariant connections $\Theta_0, \Theta_1$ for $(Q,\Theta) \in G\GBunc[K](M)$, we can define 
%\[ \CS^G_\omega(\Theta_0, \Theta_1) = \int_0^1 \omega^G(\Omega_t). \]
%
%\begin{prop}\[ d_G \CS^G_\omega(\Theta_0, \Theta_1) = \omega^G(\Theta_1) - \omega^G(\Theta_0)\]
%\end{prop}
%\begin{proof}
%\begin{align*}
%d_G \CS^G_\omega(\Theta_0, \Theta_1) &= (d + d_\fg) \CS^G_\omega(\Theta_0, \Theta_1) \\
%&= (d + d_\fg) \int_0^1 \omega^G(\Omega_t) \\
%&= \left( \int_0^1 d\omega^G(\Omega_t) + \omega^G(\Omega_t) \big|^{t=1}_{t=0} \right) + d_\fg \int_0^1 \omega^G(\Omega_t) \\
%&= \left( \int_0^1 d\omega^G(\Omega_t) + \omega^G(\Omega_t) \big|^{t=1}_{t=0} \right) + \int_0^1 d_\fg \omega^G(\Omega_t) \\
%&=\int_0^1 d_G \omega^G(\Omega_t) + \omega^G(\Omega_t) \big|^{t=1}_{t=0} \\
%&= \omega^G(\Omega_1) - \omega^G(\Omega_0).
%\end{align*}
%They key identity, that $d_\fg \int_0^1 \omega^G(\Omega_t)  =  \int_0^1 d_\fg \omega^G(\Omega_t)$, comes from the fact that on $\W(\fg) \otimes \O(M)$, integration acts as $1 \otimes \int$ and $d_\fg$ acts as $d_\fg \otimes 1$.  We also use the fact that $d_G\omega_G(\Theta)=0$.
%\end{proof}
%
%\begin{lemma}The space of $G$-invariant $K$-connections is affine over $\O^1(M;\fk)^G$.
%\end{lemma}
%\begin{proof}If $\Theta_0, \Theta_1$ are $K$-connections, then $\Theta_1 - \Theta_2 \in \pi^* \Omega^1(M;\fk)$.  If they are also $G$-invariant, then $g^*(\Theta_1-\Theta_0) = \Theta_1-\Theta_0$, so $\Theta_1 - \Theta_0 \in \pi^* \Omega^1(M)^G \otimes \fk$.  Finally, it is easy to show that $\Theta + A$ is a connection for any $A \in \O^1(M;\fk)^G$.\end{proof}

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%\newpage

%
%\section{Additional properties}

\subsection{Equivariant differential characters}\label{subsec:equivDC}
One convenient description of differential cohomology is via differential characters \cite{CS85}.  An element $\x \in \HG[]^n(M)$, with curvature $\curv(\x) = \omega$, associates to every smooth singular $(n-1)$-cycle $X_{n-1} \xrightarrow{z} M$ an element  $\langle \x, z \rangle \in \R/\Z$; if $z=\partial \wt{z}$ for some smooth $n$-chain $Y_n \xrightarrow{\wt{z}} M$, then
\[ \langle \x, z \rangle = \int_{Y_n} \wt{z}^{^*} \omega \mod \Z.\]
In degree 2, an $\R/\Z$-bundle with connection on $M$ naturally defines an element of $\HG[]^2(M)$ via its holonomy.  More generally, evaluating a differential character on a $(n-1)$-cycle can be thought of as the holonomy for some higher gerbe or abelian gauge field.  

This idea generalizes to the equivariant setting, with classes in $\HG^n(M)$ giving ``differential characters'' on the stack $\En G \times_G M$.  This type of structure naturally appears in the physics literature as WZW terms for gauged sigma models \cite{MR1311654}.  We state the main idea here, but we do not give enough details on singular cycles to consider this a definition of $\HG^*(M)$.

Assume that $X_{n-1}$ is a closed smooth manifold.  To any map 
\[ X_{n-1} \xrightarrow{z} \En G \times_G M,\]
a class $\x \in \HG^n(M)$ associates an ``equivariant holonomy'' in $\R/\Z$, as seen in the following diagram.  
\begin{align*}
\vcenter{ \xymatrix{(P,\Theta) \ar[r]^f \ar[d] & M \\ X_{n-1}} }  \quad &\Longrightarrow \quad \vcenter{\xymatrix{ \HG^n(P) \ar[d]^{\Theta^*} & \HG^n(M) \ar[l]_{f_G^*} \\ \HG[]^n(X_{n-1}) \iso \R/\Z}  }\end{align*}
If the cycle is a boundary, with $[(P,\Theta) \to X_{n-1}] = \partial\left[ (P',\Theta') \to Y_n \right]$ and $f$ extending to a map $P' \xrightarrow{\wt{f}} M$, then the holonomy can be computed by integrating the equivariant curvature $\omega = \curv(\x) \in \O^n_G(M)$
\[ \langle \x, \; X_{n-1} \leftarrow (P,\Theta) \xrightarrow{f} M \rangle = \int_{Y_n} \left( \Theta^* \otimes \wt{f}^* \right) \omega \mod \Z.\]
Furthermore, these equivariant holonomies are gauge-invariant; if two objects in the groupoid $\Map(X_{n-1}, \En G\times_G M)$ are isomorphic, they associate to $\x \in \HG^n(M)$ equal elements in $\R/\Z$.

\begin{rem}The entire above discussion concerning equivariant characters immediately generalizes from $\Z \subset \R$ to $\L \subset V$, where elements of $\HG^*(M;\L)$ give characters valued in $V/\L$.
\end{rem}

\subsection{Geometric models in low degrees}

\begin{prop}There are natural isomorphisms
\begin{align*}
 \HG^0(M) &\iso H^0(M/G;\Z), \\
 \HG^1(M) &\iso \HG[]^1(M)^G \iso C^\infty(M, \R/\Z)^G.
\end{align*}
\end{prop}
\begin{proof}
The short exact sequence \eqref{ses1} implies that $\HG^0(M) \iso  \O^0_G (M)_\Z.$  Unraveling this, we obtain the following chain of equalities and isomorphisms,
\[ \HG^0(M) \iso \O_G^0(M)_\Z = C^\infty(M,\Z)^G \iso C^0(M/G,\Z) \iso H^0(M/G;\Z).\]
%(where $\cC^0$ are continuous functions and $\cC^0(X;A) \iso Z^0(X;A) \iso \Map(\pi_0(X),A)$ for any topological space $X$ and discrete group $A$.

In degree one, consider the homomorphism $\HG^1(M) \to C^\infty(M,\R/\Z)^G$ defined as follows using the equivariant differential character construction of Section \ref{subsec:equivDC}.  Given $\x \in \HG^1(M)$ and a point $m\in M$, define the element $F_{\x}(M) \in \R/\Z$ by evaluating on the $G$-orbit of $m$,
\[ F_{\x}(m) \=  \left\langle  \x, \; \pt \leftarrow G \xrightarrow{G\cdot m} M \right\rangle.\]
The gauge invariance of the equivariant character implies that the function $F_{\x}$ is $G$-invariant.  Furthermore, $\O^1_G(M)_\Z \iso \O^1(M)^G_\Z$, and integration shows that $F_{\x}$ is a smooth function with derivative $\curv(\x)$.  Hence, $F_{\x} \in C^\infty(M,\R/\Z)^G$.

Note that the kernel of $C^\infty(M,\R/\Z)^G \xrightarrow{d} \O^1(M)_\Z^G$ is given by the constant functions
\[ \Ker(d) = \O^0(M;\R/\Z)^G_{\cl} \iso H^0(M/G;\R/\Z) \iso H^0_G(M;\R/\Z).\]
Therefore, the homomorphism of short exact sequences 
 \[ \xymatrix{
0 \ar[r] & H^0_G(M;\R/\Z) \ar[r] \ar[d]^{\iso} &\HG^1(M) \ar[r] \ar[d] & \O_G^1(M)_\Z \ar[d]^{\iso} \ar[r] & 0 \\
0\ar[r] & \Ker(d) \ar[r]  & C^\infty(M, \R/\Z)^G \ar[r] &\O^1(M)^G_\Z  \ar[r] &0 \\
}  \]
gives our desired isomorphism via the Five Lemma.
\end{proof}

\begin{prop}In degree two, there is a natural isomorphism
\begin{align*}
 \HG^2(M) &\iso \pi_0\big(G\GBunc[\R/\Z](M)\big);
\end{align*}
i.e. $G$-equivariant $\R/\Z$-bundles with connection on $M$, modulo equivariant connection-preserving isomorphisms, are in bijection with elements of $\HG^2(M)$.
\end{prop}

\begin{proof}
%
%The isomorphism is induced by the natural map
%\[ G\GBunc[\R/\Z](M) \longrightarrow \Map(\EnG \times_G M, \Bn \R/\Z) \]
%from Theorem \ref{thm:equivCCS}.  The equivariant holonomy gives a homomorphism 
%\[ \Map(\EnG \times_G M, \Bn \R/\Z )  \overset{\hol}\longrightarrow \HG[]^2(\EnG \times_G M).\]
%Composing these induces a homomorphism
%\[G\GBunc[\R/\Z](M) \longrightarrow \HG^2(M). \]
%
%... or say ...

Let $\D c \in \HG[\R/\Z]^2 \iso H^2(B \R/\Z;\Z) \iso \Z$ be the standard generator.  The equivariant Chern--Weil construction of Theorem \ref{thm:equivCCS} gives us a map
\begin{align} \label{eq:EquivLineBundles} \begin{split} G\GBunc[\R/\Z](M) &\longrightarrow \HG^2(M) \\
(Q, \Theta) &\longmapsto \D c(Q_G, \Theta_G). \end{split}
\end{align}
Multiplication in $\R/\Z$ makes $G\GBunc[\R/\Z](M)$ into a Picard groupoid (this symmetric monoidal structure corresponds to the tensor product of line bundles), and \eqref{eq:EquivLineBundles} is symmetric monoidal.  Furthermore, the map only depends on the isomorphism class of $(Q,\Theta)$ in $G\GBunc[\R/\Z](M)$.

Hence, \eqref{eq:EquivLineBundles} descends to an abelian group homomorphism $\pi_0 \left( G\GBunc[\R/\Z](M) \right) \to \HG^2(M).$  Furthermore, the characteristic class $c(Q_G) \in H^2_G(M;\Z)$ does not depend on the connection $\Theta$.  This gives the following homomorphism of short exact sequences.
\[ \xymatrix{ 0 \ar[r]& \ar[r] \ar[d] \Ker \Pi \ar[r] \ar[d] & \pi_0 \left[ G\GBunc[\R/\Z](M) \right]\ar[r]^-{\Pi} \ar[d]& \pi_0 \left[ G\GBun[\R/\Z](M) \right] \ar[r] \ar[d]& 0 \\
0 \ar[r] & \ar[r] \frac{\O^1_G(M)}{\O^1_G(M)_\Z} \ar[r]& \HG^2(M) \ar[r]& H^2_G(M;\Z) \ar[r]& 0} \]
We now show that each of the above vertical maps is an isomorphism.

It is a classical result \cite{MR711050} that equivariant bundles with abelian structure group are classified topologically by Borel cohomology, i.e. $\pi_0 \big( G\GBun[A](M) \big) \iso [ EG\times_G M, BA]$ when $A$ is abelian.  This implies the right vertical map is an isomorphism.

The $G$-invariant connections on the trivial bundle $M \times \R/\Z$ are in bijection with $\O^1(M)^G$.  Such a connection form is gauge equivalent to the trivial connection if and only if it is the derivative of a gauge transformation $C^\infty(M,\R/\Z)^G \iso \HG^1(M)$; i.e. if it lives in $\O^1(M)^G_\Z$.  Hence, 
\[ \Ker \Pi \iso  \frac{\O^1(M)^G}{\O^1(M)^G_\Z} \xrightarrow{\iso} \frac{\O^1_G(M)}{\O^1_G(M)_\Z},\]
giving that the left vertical arrow is an isomorphism.  The proposition now follows by the Five Lemma.
\end{proof}


%\reminder{Description of equivariant holonomy and how moment map appears?}

%
%
%\begin{rem}It also agrees with the classification of Gomi \cite{MR2147734}, though this is due somewhat to the fact his group equals ours in the degree 2 case.
%\end{rem}




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%
%\subsection{Associated Bundles}
%Very generally, if $G_1 \overset{\varphi}\to G_2$ is a Lie group homomorphism, the associated bundle construction gives morphisms
%\[ \xymatrix{ \EnG[G_1] \ar[d] \ar[r]^{E\varphi} &\EnG[G_2] \ar[d]\\
%\BnG[G_1] \ar[r]^{B\varphi} & \BnG[G_2].}\]
%To a principal $G_1$-bundle $P_1 \to M$, we define the principal $G_2$-bundle as
%\[ P_1 \times_{G_1} G_2 \to M.\]
%If $\Theta_1 \in \O^1(P;\fg_1)$ is a connection, the form $\varphi_*(\Theta_1) + \theta_{G_2} \in \O^1(P\times G_2;\fg_2)$ is invariant and basic, and hence is the pullback of a connection form $\Theta_2 \in \O^1(P \times_{G_1} G_2; \fg_2)$.  In the case where $1 \to K \to H \to G \to 1$, we can also describe this as quotienting by $K$, as seen by the map (which is a well-defined, equivariant map, between principal $G$-bundles.
%\begin{align*}
%P\times_H G &\longrightarrow P/K \\
%[p, g] &\mapsto [pg]
%\end{align*}
%We are using that since $K$ is a normal subgroup of $H$, and $K$ acts freely on $P$, then $P/K$ still has a natural free $H/K \iso G$ action given by $[p]g = [p\wt{g}]$ for $\wt{g} \in H$ mapping to $G$.
%%
%%Suppose that $(P,\Theta) \overset{\pi}\to M$ is a principal $G$-bundle with connection.  Then, we have two maps 
%%\[ \EnG\times_G P \longleftrightarrow M. \]
%%Given map $(P_\Sigma,\Theta_{\Sigma}) \to P$, you can take the induced quotient $\Sigma \to M$.  Given any $\Sigma \overset{\sigma}\to M$, you have $\sigma^*(P,\Theta) \to \Sigma$, which is a map $\Sigma \to \EnG \times_G P$.  One of the compositions is the identity, and the other has a fiber given by the maps $\Sigma \to \EnG$.
%
%%The above is actually part of a more general construction.  Suppose that $K$ is a normal subgroup of $H$ with quotient $H/K \iso G$,
%%\[ 1 \to K \to H \to G \to 1.\]
%Suppose that $M$ and $P$ are $H$-equivariant, and $(P,\Theta) \to M$ is a $H$-equivariant map, and $(P,\Theta)$ is a principal $K$-bundle with connection.  Then, there are natural morphisms
%\[ \EnG[H] \times_{H} P  \longleftrightarrow \EnG \times_G (P/K)\]
%generalizing the construction in Section \ref{subsec:Principal}.  This is given by the following.  Let $\Sigma \to \EnG[H] \times_{H} P$ be
%\[ \xymatrix{ (Q, \Theta_Q) \ar[r]\ar[d] &P \\
%\Sigma }\]
%Associated to $(Q,\Theta_Q)$ is the principal $H/K$-bundle $(Q/K,\Theta_{Q/K})$, given by the quotient of $K$.  Since the $K$-action on $P$ is free, then
%\[ \xymatrix{ (Q, \Theta_Q) \ar[r]\ar[d] &P \ar[d] \\
%(Q/K, \Theta_{Q/K}) \ar[r]\ar[d] &P/K \\
%\Sigma }.\]
%

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

\section{Details of simplicial sheaf construction}\label{Section:DefnProof}


This section contains a more detailed account of Section \ref{subsec:sPreDefn}.  The results essentially follow by combining the work of \cite{MR3049871} and \cite{1311.3188}; the reader may refer to these works, along with \cite[Chapter 5]{MR2522659} for further details.  Let us briefly explain our notation.  

We will use simplicial sets in order to deal with groupoids, non-negatively graded chain complexes, and topological spaces simultaneously.  Let $\Gpdinfty$ be the $(\infty,1)$-category of ``$\infty$-groupoids,'' taken here to be the full simplicially enriched subcategory of simplicial sets spanned by Kan complexes.  The category $\Gpdinfty$ is naturally equivalent, as an $(\infty,1)$-category, to the topologically enriched category $\Top$ of topological spaces with the homotopy type of a CW complex.  Though it is not strictly necessary, we use this equivalence $\Gpdinfty \simeq \Top$ to make certain statements easier to read.
Any groupoid is naturally a Kan complex via the nerve construction $\N$, and the Dold--Kan correspondence $\DK$ makes a non-negatively graded chain complex into a simplicial abelian group.  These structures, together with the singular functor $\S$ and geometric realization $|\,|$, fit into the following picture.
\[ \xymatrix{  & \Top  \ar@<1.2ex>[d]^{\S}_{  \simeq }\\
\Gpd \ar@{^`->}[r]^{\N} &\Gpdinfty \ar@<1.2ex>[u]^{| \,|}& \Ch_{\geq 0} \ar@{_`->}[l]_{\DK} } \]
We implicitly consider any ordinary category as an $(\infty, 1)$-category via the nerve construction, though we do not explicitly write it.

Let $\PShv_{\Gpdinfty} = \Fun(\Man^{\op}, \Gpdinfty)$ be the $(\infty, 1)$-category of $\Gpdinfty$-valued presheaves on manifolds.  A functor $\cF \in \PShv_{\Gpdinfty}$ is a sheaf if it satisfies the following descent condition: for any covering $\cU \to X$, the canonical map $\cF(X) \xrightarrow{\simeq} \lim_{\Delta} \cF(\cU^\bullet)$ is an equivalence.  The full subcategory of simplicial sheaves is denoted $\Shv_{\Gpdinfty}$, and there is a sheafification functor $L$ forming the adjunction
\[ L: \PShv_{\Gpdinfty} \rightleftarrows \Shv_{\Gpdinfty} : \text{inclusion}. \] 
%See \cite[Chapter 5]{MR2522659} for further details.

We continue to suppress the embedding of $\Set \into \Gpd$, but we use the nerve symbol to denote $\N\colon  \Shv_{\Gpd}\to \Shv_{\Gpdinfty}$.  As in previous sections, we will use $M$ to denote both a manifold $M$ and the stack it represents, but we will begin using the notation $\uline{M}$ within proofs for added clarity.  For $Y \in \Top$, we define $\Sing Y \in \Shv_{\Gpdinfty}$ by $\Sing Y(X) = \S(Y^X)$, the singular complex of the space of continuous maps from $X\to Y$.  Let $\cK(A,n)\in \Shv_{\Gpdinfty}$ denote any simplicial sheaf equivalent equivalent to $\Sing K(A,n)$.  



%We let $\Shv_{\Gpdinfty}$ be the $(\infty,1)$-category of $\Gpdinfty$-valued sheaves on manifolds.  This is the full subcategory of sheaves in $\Fun(\Man^\op, \Gpdinfty)$ satisfying descent, and we will commonly refer to them as simplicial sheaves.  





We now recount the results we need from the theory of homotopy-invariant sheaves developed by Bunke--Nikolaus--V\"olkl in \cite{1311.3188}.  Their results hold for general $\cC$-valued sheaves on manifolds, where $\cC$ is an $(\infty, 1)$-category, but we have specialized them to the case of $\cC = \Gpdinfty \simeq \Top$.  A sheaf $\cF$ is homotopy-invariant if, for all manifolds $X$, the projection $X \times I \to X$ induces an equivalence
\[ \cF(X) \xrightarrow{\simeq} \cF(X \times I).\]
Let  $\Shv^{\h}_{\Gpdinfty}$ denote the full subcategory of homotopy-invariant sheaves.  The following results are found in Proposition 2.6, equation (63), and Lemma 7.13 of \cite{1311.3188}.
\begin{itemize}
\item The functor $\Sing$ gives an equivalence $\Top \xrightarrow{\simeq} \Shv_{\Gpdinfty}^{\h}$;  the inverse is given by evaluating at a point and taking the geometric realization.  There is also a homotopification functor $\cH$, and these form the following adjunctions.
\[ \xymatrix{ \Shv_{\Gpdinfty} \ar@<.75ex>[r]^{\cH} &\Shv_{\Gpdinfty}^{\h}  \ar@<.75ex>[r]^-{\text{ev}(\pt)}_-{\sim}  \ar@<.75ex>@{_`->}[l]& \Top \ar@<.75ex>[l]^-{\Sing}} \]
\item These adjunctions combine to give the following adjunction:
\begin{equation}\label{eq:HtpyAdjunction} \h: \Shv_{\Gpdinfty} \rightleftarrows \Top : \Sing .
\end{equation}
Because $\h$ is a left adjoint, it automatically commutes with homotopy colimits.  The functor $\h$ also preserves finite products.
\item Let $M$ be a manifold and $\uline{M}\in \Shv_{\Gpd}$ the stack it represents.  Then, 
\begin{equation}\label{eq:HtpyYoneda} \h( \N(\uline{M}) ) \simeq  M \in \Top.\end{equation}
\item If $\cA \in \Shv_{\Ab} \into \Shv_{\Gpd}$ is a sheaf of abelian groups that are $C^\infty$-modules, then  \begin{equation}\label{eq:FineSheaf} \h( \N(\cA)) \simeq \pt \in \Top. \end{equation}
\end{itemize}



We now make the following definition/abbreviation.
\begin{defn}\label{defn:StackCoh}Let $A$ be a fixed abelian group.  For a stack $\cM \in \Shv_{\Gpd}$,
\[ H^n(\cM; A) \=  \ho \Shv_{\Gpdinfty}(\N(\cM), \cK(A,n) ).\]
\end{defn}

\begin{example}For $M \in \Man$, consider $\N(\uline{M}) \in \Shv_{\Gpdinfty}$, which is defined by considering the set $C^\infty(X, M)$ as a discrete simplicial set.  The Yoneda lemma shows that Definition \ref{defn:StackCoh} agrees with our ordinary definition of cohomology,
\[ H^n( \uline{M}, A) = \ho \Shv_{\Gpdinfty}(\N(\uline{M}), \Sing K(A,n)) \iso \ho \Top(M,K(A,n)) \iso H^n(M;A).\]
\end{example}




%In fact, BNV actually proves that $\h (\N(\Bn G)) \simeq BG$, and their proof immediately generalizes to $\h(\N(\En G \times_G M)) \simeq EG\times_G M$. 

%\reminder{I don't think my desired sequence is actually a cofiber sequence.  It's a fiber sequence.  I'm not in a stable category.}
%\begin{lemma}There is a free action of $\O^1 \otimes \fg$ on $\En G \times_G M$, and the quotient is $\E G \times_G M$.
%\end{lemma}
%\begin{proof}Note that, for each $X$, the space of connections on a $G$-bundle is affine over $\O^1(X)\otimes \fg$.  The map $\O^1 \otimes \fg \times \En G \times_G M \to \En G \times_G$ is given by modifying the connection.  The action is clearly free, for the usual action of $\O^1(X)\otimes \fg$ on the space of connections is free and transitive.  We analyze the quotient.  For objects in $(\En G \times_G M)(X)$, this equates $(X \leftarrow (P,\Theta)\xrightarrow{f} M)$ with any other object $(X \leftarrow (P,\Theta')\xrightarrow{f} M)$ that only differs by the connection.  Similarly, the 1-morphisms from $[(X \leftarrow (P,\Theta)\xrightarrow{f} M)] \to [(X \leftarrow (P',\Theta')\xrightarrow{f'} M)]$ is the set of bundle morphisms $P\xrightarrow{\varphi} P'$ such that $f' \o \varphi = f$, and $\varphi^*\Theta' = \Theta + A$ for some $A \in \O^1(X)\otimes \fg$.  This is simply the set of morphisms $(X \leftarrow P \xrightarrow{f} M) \xrightarrow{\varphi} (X \leftarrow P' \xrightarrow{f'} M)$ in $(\E G \times_G M)(X)$.  Therefore, the quotient is naturally equivalent to $\E G \times_G M$. 
%\end{proof}

%\begin{prop}\label{prop:HtpyConns}The natural map $\En G \times_G M \to \E G \times_G M$ induces an equivalence on the homotopifications
%\[ \h (\N(\En G \times_G M)) \xrightarrow{\simeq} \h(\N(\E G \times_G M)). \]
%\end{prop}
%\begin{proof}
%We first show that $\N(\E G \times_G M)$ is the homotopy cofiber of 
%\[ \N(\O^1 \otimes \fg) \into \N(\En G \times_G M). \] 
%Note that, for each $X$, the space of connections on a $G$-bundle is affine over $\O^1(X)\otimes \fg$.  The map $\O^1 \otimes \fg \into \En G \times_G M$ is given by .......
%
%This map follows by noting that, for each $X$, the space of connections on a $G$-bundle is naturally affine over $(\O^1\otimes \fg)(X)$.  The map is is a monomorphism of simplicial sets when evaluated on each $X$, so it is a cofibration in the local injective model structure on simplicial presheaves.  Hence, the homotopy cofiber is equivalent to the strict cofiber, which we can compute explicitly.  At the level of 0-simplices, for objects in $(\En G \times_G M)(X)$, this equates $(X \leftarrow (P,\Theta)\xrightarrow{f} M)$ with any other object $(X \leftarrow (P,\Theta')\xrightarrow{f} M)$ that only differs by the connection.  Similarly, the induced 1-morphisms from $[(X \leftarrow (P,\Theta)\xrightarrow{f} M)] \to [(X \leftarrow (P',\Theta')\xrightarrow{f'} M)]$ is the set of bundle morphisms $P\xrightarrow{\varphi} P'$ such that $f' \o \varphi = f$, and $\varphi^*\Theta' = \Theta + A$ for some $A \in \O^1(X)\otimes \fg$.  This is simply the set of morphisms $(X \leftarrow P \xrightarrow{f} M) \xrightarrow{\varphi} (X \leftarrow P' \xrightarrow{f'} M)$ in $(\E G \times_G M)(X)$.  Therefore, the colimit is equivalent to $\E G \times_G M$.  Consequently, we have that
%\[\hocolim  \left(\N(\O^1 \otimes \fg) \into \N(\En G \times_G M)  \right) \simeq \N (\E G \times_G M).\]
%
%Since $\O^1 \otimes \fg$ is a sheaf of $C^\infty$-modules, this implies $\h( \N(\O^1 \otimes \fg)) \simeq \pt$.  We also know that $\h$ is a left adjoint, so it automatically commutes with homotopy colimits.  The induced cofiber sequence
%\[ \pt \simeq \h(\N(\O^1 \otimes \fg)) \to \h (\N( \En G \times_G M)) \xrightarrow{\simeq} \h(\N(\E G \times_G M))\]
%gives us our first desired equivalence.
%\end{proof}

%Let us use the following notation for an action groupoid.  

The following proposition and proof are taken almost directly from Lemma 5.2 in \cite{1311.3188}. 


\begin{prop}\label{prop:EnGHtpy}The natural map $EG\times_G M \to \En G \times_G M$ induces equivalences
\[ EG \times_G M \xrightarrow{\simeq} \h(\N(\En G \times_G M)) \xrightarrow{\simeq} \h(\N(\E G \times_G M)). \]
\end{prop}
\begin{proof}
We first show $\h(\N(\En G \times_G M)) \xrightarrow{\simeq} \h(\N( \E G \times_G M))$.  Consider the following commutative diagram of prestacks
\[ \xymatrix{  \En G\times_G M \ar[r] & \E G \times_G M \\
 \left(\O^1 \otimes \fg \times M\right)\sslash G  \ar[u] \ar[r] & M \sslash G,  \ar[u] 
}\]
where we use the notation $\bullet \sslash G$ to denote the action groupoid $[\bullet \leftleftarrows G \times \bullet]$ associated to a $G$-action.  The group $G$ acts on $\O^1 \otimes \fg$ via the usual change of connection formula, $\omega \cdot g = \Ad_{g^{-1}} \omega + g^*\theta_{\fg} $.  The right vertical maps is given by associating to $X \to (M\sslash G)$ the trivial bundle $X \leftarrow X\times G$, together with the map $X\times G \to M$ determined by the map $X\to M$.  Likewise, the left vertical map is given by the same construction, but with the connection on $X\times G$ determined by the map $X\to \O^1\otimes \fg$.  

Since any bundle is locally trivializable, these vertical maps induce equivalences of groupoids when evaluated on stalks.  Hence, upon taking the nerve, the vertical maps induce equivalences in $\Shv_{\Gpdinfty}$ (see \cite[Lemma 5.2]{1311.3188} and \cite[(7.23)]{MR3049871}).
\[ \xymatrix{  \N(\En G\times_G M )\ar[r] & \N(\E G \times_G M) \\
L( \N\left( \left(\O^1 \otimes \fg \times M\right)\sslash G \right))  \ar[u]^{\simeq} \ar[r] & L(\N \left(M \sslash G\right))  \ar[u]^{\simeq} 
}\]


We now use the general fact that for a simplicial set $B$, the induced simplicial object in simplicial sets, given by the composition $\Delta^{\op} \xrightarrow{B} \Set \xrightarrow{\N} \sSet$, gives a natural equivalence
\[ \hocolim_{\Delta^{\op}} \N(B) \simeq B.\] 
Rewriting the bottom entries as colimits, and applying the properties of $\h$ described above, gives the following sequence of equivalences. 
\begin{align*}
\h \big( L( \N &((\O^1 \otimes \fg \times M ) \sslash G  ))\big) \simeq 
\h \left(L( \O^1 \otimes \fg \times \uline{M} \leftleftarrows \uline{G}\times \O^1 \otimes \fg \times \uline{M} \tripleleftarrow  \cdots )\right) \\
&\simeq \h\Big(L \Big( \hocolim_{\Delta^{\op}}\big( \N(\O^1 \otimes \fg\times \uline{M}) \leftleftarrows \N(\uline{G}\times \O^1 \otimes \fg\times \uline{M}) \tripleleftarrow  \cdots \big)\Big)\Big) \\
&\simeq \hocolim_{\Delta^{\op}} \left( \h(\N(\O^1\otimes \fg) \times \h(\N(\uline{M})) \leftleftarrows \h(\N(\uline{G})) \times \h(\N(\O^1\otimes \fg)) \times \h(\N(\uline{M})) \tripleleftarrow   \cdots )\right) \\
&\xrightarrow{\simeq} \hocolim_{\Delta^{\op}} \left( M \leftleftarrows G \times M \tripleleftarrow G \times G \times M \cdots \right) \simeq \h \big(L \big( \N(M\sslash G)\big)\big).
\end{align*}
The equivalence $\h(\N(\En G \times_G M)) \xrightarrow{\simeq} \h(\N(\E G\times_G M))$ follows immediately.  In the above argument, the third equivalence was given by fact that $\h$ and $L$ commute with colimits,  $\N(\uline{G}^{\times k} \times \O^1 \otimes \fg \times \uline{M})$ is a sheaf, and $\h$ preserves finite products.  The fourth equivalence was given by the fact that $\h(\N(\uline{X})) \simeq X$, and $\h(\N(\cA)) \simeq \pt$ for $\cA$ a sheaf of $C^\infty$-modules.  Note that the middle entries were given as the homotopy colimit of a simplicial space (simplicial object in $\Top$).  For proper simplicial spaces, the homotopy colimit is equivalent to the geometric realization; this is a common construction of $BG$ when $M=\pt$.

%Therefore, the map $\En G \times_G M \to \E G \times_G M$ induces an equivalence
%\[ \h(\N(\En G \times_G M)) \xrightarrow{\simeq} \h(\N(\E G\times_G M)).\]

Now consider also the following commutative diagram of prestacks
\[ \xymatrix{  EG\times_G M \ar[r] & \E G \times_G M \\
(EG \times M) \sslash G  \ar[u] & M \sslash G, \ar[u] \ar[l]
}\]
where the bottom left map is given by the homotopy equivalence $M \xrightarrow{\simeq} EG\times M$.  The left vertical map is given by quotienting, and since $G$ acts freely, it leads to an equivalence of groupoids when evaluated on any manifold $X$.  Hence, both vertical maps give equivalences in $\Shv_{\Gpdinfty}$.  
\[ \xymatrix{ \N( EG\times_G M) \ar[r] & \N(\E G \times_G M) \\
L(\N[EG \times M \leftleftarrows G \times (EG \times M) ]) \ar[u]^{\simeq} &L(\N [M \leftleftarrows G \times M]) \ar[u]^{\simeq} \ar[l]
}\]
By the same calculation  as above
\begin{align*} \h(L(\N(M\sslash G))) &\simeq \hocolim_{\Delta^{\op}} \left( M \leftleftarrows G \times M \tripleleftarrow G \times G \times M \cdots \right) , \\
 \h(L(\N((EG \times M)\sslash G)))  &\simeq \hocolim_{\Delta^{\op}} \left( EG \times M \leftleftarrows G \times EG \times M \tripleleftarrow G \times G \times M \cdots \right), 
\end{align*}
and the homotopy equivalence $\pt \xrightarrow{\simeq} EG$ induces a homotopy equivalence of the relevant simplicial spaces.  This implies $\h(L(\N(M\sslash G))) \xrightarrow{\simeq}\h(L(\N((EG \times M)\sslash G)))$, which in turn gives the equivalence
\[ EG \times_G M \simeq \h(L(\N( EG\times_G M))) \xrightarrow{\simeq} \h(\N(\E G \times_G M)).\]
\end{proof}

%Now, we proceed to show that the bottom map induces an equivalence after homotopification.
%We now use the general fact that for a simplicial set $B$, the induced simplicial object in simplicial sets, given by the composition $\Delta^{\op} \xrightarrow{B} \Set \xrightarrow{\N} \sSet$, gives a natural equivalence
%\[ \hocolim_{\Delta^{\op}} \N(B) \simeq B.\] 
%Rewriting $\N[ M \leftleftarrows G \times M]$ as a colimit, and applying the properties of $\h$ described above, gives the following sequence of equivalences 
%\begin{align*}
%\h \left( \N[ M \leftleftarrows G \times M ]  \right) &\simeq 
%\h \left( \uline{M} \leftleftarrows \uline{G \times M} \tripleleftarrow \uline{G\times G  \times M}  \cdots \right) \\
%&\simeq \h \left( \hocolim_{\Delta^{\op}}( \N(\uline{M}) \leftleftarrows \N(\uline{G\times M}) \tripleleftarrow \N(\uline{G \times G \times M})  \cdots )\right) \\
%&\simeq \hocolim_{\Delta^{\op}} \left( \h(\N(\uline{M})) \leftleftarrows \h(\N(\uline{G \times M})) \tripleleftarrow \h(\N(\uline{G \times G \times M}))  \cdots )\right) \\
%&\simeq \hocolim_{\Delta^{\op}} \left( M \leftleftarrows G \times M \tripleleftarrow G \times G \times M \cdots \right) ,
%\end{align*}
%where the last line is the homotopy colimit of a simplicial space (simplicial object in $\Top$).  It is worth noting that, for proper simplicial spaces, the homotopy colimit is equivalent to the geometric realization; for $M=\pt$, this is a common construction of $BG$.  By the same argument, we  calculate
%\[ \h \left( \N[ EG\times M \leftleftarrows  G \times (EG\times M) ]  \right) \simeq \hocolim_{\Delta^\op} \left( EG \times M \leftleftarrows G \times EG \times M \tripleleftarrow  \cdots \right). \]
%The map between the simplicial sheaves in question is therefore induced by the homotopy equivalences   
%\[ \h \h \left( \N[ M \leftleftarrows G \times M ]  \right) \xrightarrow{interrupted} \]
%Therefore, 
%he homotopy equivalences
%thus 
%
%
%Now, we need to show $\h(\N(\E G \times_G  M)) \simeq EG\times_G M$.  As noted in many places, including \cite{MR3049871}, the stack $\E G \times_G M$ is equivalent to
%\[ \left( \uline{M} \leftleftarrows \uline{G\times M} \right) \in \Shv_{\Gpd},\]
%the more commonly used version of the quotient stack.  
%
%The intermediate step $\hocolim_{\Delta^{\op}}( M \leftleftarrows G \times M \cdots)$ is the homotopy colimit of a simplicial space.  It is equivalent to the geometric realization of this simplicial space via the Bousfield--Kan map, and 
%\[ \big| M \leftleftarrows G \times M \tripleleftarrow G \times G \times M \cdots \big| \]
%is a common construction of $EG\times_G M$.
%\end{proof}



\begin{prop}\label{prop:StackCohIso}The map $EG\times_G M \to \En G \times_G M$, defined by the connection on $EG$, induces an isomorphism in cohomology
\[ H^*(\En G \times_G M; A) \xrightarrow{\iso} H^*(EG\times_G M; A).\]
\end{prop}
\begin{proof}
This follows almost immediately from Definition \ref{defn:StackCoh}, the adjunction \ref{eq:HtpyAdjunction}, and the calculation of $\h(\N(\En G \times_G M))$ in Proposition \ref{prop:EnGHtpy}.  Together, these give the following sequence of isomorphisms:
\begin{align*} H^n( \En G \times_G M; A) &= \ho \Shv_{\Gpdinfty} ( \N( \En G \times_G M), \cK(A,n) ) \\
&\iso \ho \Top( \h( \N( \En G \times_G M)), K(A,n)) \\
& \xrightarrow{\iso} \ho \Top( EG \times_G M, K(A,n)) \\
&\iso H^n(EG\times_G M;A).
\end{align*}
\end{proof}

%\reminder{Insert justification for fact that Freed--Hopkins implies the result in our infinity-categories.}


Finally, we repeat an explicit construction of a simplicial sheaf representing differential cohomology.  This construction, and minor variations, have already appeared in several places, including \cite{HS05} implicitly and \cite{1311.3188,1208.3961, MR3019405, MR3335251, 1310.7930} explicitly.  In order to obtain simplicial sets from cochain complexes, we use the standard trick of reversing the grading and shifting, and then using the Dold--Kan construction.  The inclusions of
 \[ \Z \into \R \into \O^0 \]
induce the following maps of presheaves of chain complexes, where degree 0 is written on the left.
\[ ( \O^n_\cl \leftarrow 0 \leftarrow \cdots )  \into ( \O^n_{\cl} \xleftarrow{d} \O^{n-1} \xleftarrow{d} \cdots \xleftarrow{d} \O^0 )  \hookleftarrow (0 \leftarrow \cdots \leftarrow \Z \leftarrow \cdots )\]
Applying Dold--Kan (and sheafifying where necessary), we then take the homotopy pullback in $\Shv_{\Gpdinfty}$, obtaining
\begin{equation}\label{eq:DiffK(Z,n)} \vcenter{ \xymatrix{ \K(\Z,n) \ar[d] \ar[r] & L(\DK(0 \leftarrow \cdots \leftarrow \Z \leftarrow \cdots )) \ar[d] \\
\DK ( \O^n_\cl \leftarrow 0 \leftarrow \cdots )   \ar[r] & \DK( \O^n_{\cl} \xleftarrow{d} \O^{n-1} \xleftarrow{d} \cdots \xleftarrow{d} \O^0 ) } }\end{equation}
The top right corner is equivalent to $\cK(\Z,n)$, and the bottom right corner is equivalent to $\cK(\R,n)$, so the construction of $\K(\Z,n)$ fits the description from \eqref{eq:SheafSquare}.

%\begin{rem}Following the convention that a cochain complex $(C^*, d)$ defines a chain complex $(C_*, \delta)$ by $C_* = C^{-*}$ and $\delta = (-1)^i d$, and the corresponding shift for a chain complex given by $C[n]_k = C_{n+k}$, then we obtain
%defined as a homotopy pullback in simplicial presheaves.
%\end{rem}


%This can also be rewritten explicitly by
%\[ \xymatrix{ \KZ \ar[d] \ar[r] & \DK \left( \Z \to 0 \to \cdots \to 0 \right) \ar[d] \\
%\DK \left( \cdots \to 0 \to \Omega^n \right)  \ar[r] & \DK \left( \Omega^0 \xrightarrow{d} \Omega^1 \xrightarrow{d} \cdots \Omega^n  \right) }\]
%where the right-most term of the chain complex is in degree $0$.
%
%\begin{rem}It is important to note here that $\O^n \in \sPre$ is discrete (no non-trivial morphisms).  The Dold--Kan construction $\DK(\O^n)$ is equivalent to the nerve $\N(\O^n)$, where $\O^n$ is considered as a discrete category.  On the other hand, $\O^*$ has non-trivial morphisms given by coboundaries.  It is not given by the set of differential forms.
%\end{rem}

We now have two potential definitions of $\HG^*(M)$.  The first was given in Section \ref{sec:HS} using the cochain complex $\CG(n)(M;\Z)$, and the second is given in this section as 
\[ \ho \Shv_{\Gpdinfty}\left(\N(\En G \times_G M), \K(\Z,n)\right).\]
We now prove these two possible definitions of $\HG^*(M)$ agree.

\begin{proof}[Proof of Theorem \ref{thm:TwoDefnsAgree}]
We use Dold--Kan to consider $\CG(n)^*(M)$ as a homotopy pullback of simplicial abelian groups via the following.
\begin{equation}\label{eq:DKofHS} \vcenter{\xymatrix{\DK( \ZG(n)^n(M) \xleftarrow{d} \CG(n)^{n-1}(M) \xleftarrow{d}  ) \ar[r] \ar[d] &\DK(Z^n(EG\times_G M;Z) \xleftarrow{\delta} \cdots) \ar[d] \\
\DK(\O^n_G(M)_\cl \xleftarrow{}0) \ar[r]  & \DK(Z^n(EG\times_G M;\R) \xleftarrow{\delta} \cdots) 
} }\end{equation}

The Freed--Hopkins isomorphism \cite[Propositions 6.22 and Theorem 7.28]{MR3049871} gives us 
\[ \ho \Shv_{\Gpdinfty}(\N(\En G \times_G M), \N(\O^n_{\cl})) \iso \Shv_{\Gpd}( \En G \times_G M, \O^n_{\cl}) \iso \O^n_G(M)_\cl.\]
Likewise Proposition \ref{prop:EnGHtpy} shows us that
\begin{align*}
\Shv_{\Gpdinfty}&(\N(\En G \times_G M), L(\DK(0 \leftarrow \cdots \leftarrow \Z ) ) \simeq \Shv_{\Gpdinfty} ( \N(\En G \times_G M), \cK(\Z, n)) \simeq \\
\simeq &\Shv_{\Gpdinfty}^{\h}\left( \h(\N(\En G \times_G M)), \cK(\Z,n)   \right)  \simeq \S\left( K(\Z,n) ^{EG\times_G M } \right),  \\
\Shv_{\Gpdinfty}&(\N(\En G \times_G M), \DK( \O^n_{\cl} \xleftarrow{d} \O^{n-1} \xleftarrow{d} \cdots ) ) \simeq \Shv_{\Gpdinfty} ( \N(\En G \times_G M), \cK(\R, n)) \\ 
\simeq &\Shv_{\Gpdinfty}^{\h}\left( \h(\N(\En G \times_G M)), \cK(\R,n)   \right)  \simeq \S\left( K(\R,n) ^{EG\times_G M } \right). 
\end{align*}
There is a natural homotopy equivalence 
\[ \S(K(A,n)^X ) \xrightarrow{\simeq} \DK( Z^n(X;A) \xleftarrow{\delta} C^{n-1}(X;A) \xleftarrow{\delta} \cdots ) \]
induced by pulling back a fundamental cocycle in $Z^n(K(A,n);A)$ and integrating over the simplex \cite[Proposition A.12]{HS05}.  Therefore, we have equivalences from three corners of \eqref{eq:DiffK(Z,n)}  evaluated on $\N(\En G \times_G M)$, and the corresponding three corners of \eqref{eq:DKofHS}.  Therefore, the induced map between homotopy pullbacks 
\[ \Shv_{\Gpdinfty}(\N(\En G \times_G M), \K(\Z,n)) \xrightarrow{\sim} \DK( \ZG(n)^n \xleftarrow{d} \CG(n)^{n-1} \xleftarrow{d} \cdots ) \]
is also an equivalence in $\Gpdinfty$, which gives us the desired isomorphism in the homotopy category.
\end{proof}


%Theorem 18.7.4 from \cite{MR1944041} states that for $X$ a Reedy cofibrant simplicial object in a simplicial model category, then the Bousfield--Kan map $\hocolim X \longrightarrow |X|$ is a natural weak equivalence.  

%%Look at the following to see if there's anything I forgot to include.
\begin{comment}
Pullback diagram
\[ \xymatrix{ \K(\Z, n) \ar[d] \ar[r] & \cK(\Z,n) \ar[d] \\
\O^n_{\cl}   \ar[r] & \cK(\R,n) }\]
The easiest way to construct such a diagram is as follows:
\[ \xymatrix{ \KZ \ar[d] \ar[r] & \DK \left( \Z \to 0 \to \cdots \to 0 \right) \ar[d] \\
\DK \left( \cdots \to 0 \to \Omega^n \right)  \ar[r] & \DK \left( \Omega^0 \xrightarrow{d} \Omega^1 \xrightarrow{d} \cdots \Omega^n  \right) }\]
where the right-most term of the chain complex is in degree $0$.
\[ \xymatrix{ K(\Z,n)(q) \ar[d] \ar[r] & \Gamma(\Z[-n]) \ar[d] \\
\DK \left( \Omega^{*\geq q}[-n] \right)   \ar[r] & \DK \left( \Omega^*[-n]  \right) ,}\]
defined as a homotopy pullback in simplicial presheaves.

\begin{thm}$\HG[]^*(\En G\times_G M) \iso \HG^*(M)$.
\end{thm}
\begin{proof}
The universal connection gives morphism $EG \times_G M \to \En G\times_G M$.  As shown in \ref{}, since $K(\Z,n)$ and $\DK(\O^{*})$ are homotopy-invariant, this gives weak equivalences

Since $\En G \times_G M$ is a simplicial sheaf (satisfies descent), then
\begin{align*} \sPre\left(\N(\En G\times_G M) , \, K(\Z,n) \right) &\xrightarrow{\sim} \sPre\left( EG \times_G M, \,K(\Z,n) \right), \\
\sPre\left(\En G\times_G M, \, \DK(\Omega^{*}[n]) \right) &\xrightarrow{\sim} \sPre\left( EG \times_G M, \,\DK(\Omega^{*}[n]) \right).
\end{align*}
The inclusion $\Z \into C^*(\Z)$ and $\Omega^* \into C^*(\R)$ give weak equivalences 
\begin{align*}  \sPre\left( EG \times_G M, \,K(\Z,n) \right) &\xrightarrow{\sim} \sPre( EG\times_G M, \DK (C^*(\Z)[n])) = \DK  (C^*(EG\times_G M, \Z)[n]), \\
\sPre\left( EG \times_G M, \,\DK(\Omega^{*}[n]) \right)  &\xrightarrow{\sim} \sPre( EG\times_G M, \DK (C^*(\R)[n]))  = \DK (C^*(EG\times_G M, \R)[n]).
\end{align*}

Furthermore, the Freed--Hopkins Theorem \ref{} implies that 
\[ \sPre\left( \En G \times_G M,\, \Omega^n \right) \xrightarrow{\iso} \O_G^n(M)\]
is a bijection.

These combine to give weak equivalences at all corners of the pullback square
\[ \xymatrix{  \sPre(\En G\times_G M, \K(\Z,n)) \ar[d]\ar[r] & \sPre(\En G \times_G M, K(\Z,n)) \ar[d] \\
\sPre( \En G \times_G M, \DK( \O^n)) \ar[r] &\sPre( \En G \times_G M, \DK( \O^*[n]))}  \]
to
\end{proof}

We denote the simplicial presheaf representing degree $n$ differential cohomology by $\KZ \in \sPre$, and it is defined as the homotopy fiber (in $\sPre$) in the following diagram.
\[ \xymatrix{ \KZ \ar[d] \ar[r] & \DK \left( \Z \to 0 \to \cdots \to 0 \right) \ar[d] \\
\DK \left( \cdots \to 0 \to \Omega^n \right)  \ar[r] & \DK \left( \Omega^0 \xrightarrow{d} \Omega^1 \xrightarrow{d} \cdots \Omega^n  \right) }\]
Here, the chain complexes are written with the right-most term being placed in degree $0$.  More generally, 
More generally, we can modify the cut-off of forms, and using the same notation as \cite{MR3335251}, we have 
\[ \xymatrix{ K(\Z,n)(q) \ar[d] \ar[r] & \Gamma(\Z[-n]) \ar[d] \\
\DK \left( \Omega^{*\geq q}[-n] \right)   \ar[r] & \DK \left( \Omega^*[-n]  \right) .}\]
If $M$ is an ordinary manifold, then under the Yoneda embedding, we recover the usual differential cohomology groups by
\[ \ho \sPre \left(M, \KZ \right) \iso \HG[]^n(M).\]


%Preliminary: \[ \xymatrix{ \K(\Z,n) \ar[d] \ar[r] & K(\Z,n) \ar[d] \\
%\DK (\Omega^n)   \ar[r] & \DK \left( \Omega^*[n]  \right) .}\] 

Use this as the more convenient definition:
\[ \xymatrix{ \KZ \ar[d] \ar[r] & \DK \left( \Z \to 0 \to \cdots \to 0 \right) \ar[d] \\
\DK \left( \cdots \to 0 \to \Omega^n_\cl \right)  \ar[r] & \DK \left( \Omega^0 \xrightarrow{d} \Omega^1 \xrightarrow{d} \cdots \Omega^n_\cl  \right) }\]

Then, use the convention that $\DK (C^*[n]) = \DK \left( C^0 \xrightarrow{d} C^1 \xrightarrow{d} \cdots Z^n  \right) \in \sSet$, and hence 
\[ \pi_k\big( \DK (C^*[n]) \big) \iso H^{n-k}(C^*),\]
so in particular $\pi_0\big( \DK (C^*[n]) \big) \iso H^{n}(C^*)$.  
\end{comment}
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\appendix
\renewcommand\K{\operatorname{K}}

\section{Equivariant de Rham theory}\label{sec:Background}
We briefly recall some standard facts about equivariant cohomology, principal bundles with connection, and equivariant differential forms.  Our goal is to describe the Weil algebra $\W(\fg)$ from the perspective of differential forms on principal bundles.  We hope this makes clear why having connections is so essential to using $\O_G(M)$.  Among the many wonderful references, our treatment is most heavily influenced by: the introduction to principal bundles in Section 1 of \cite{Fre95}, the summary of equivariant cohomology \cite{Meinrenken-EquivCoh}, and the standard textbook for $G^\star$-algebras and equivariant de Rham cohomology \cite{GuilleminSternberg99}.

%introduction to stacks on manifolds  \cite{MR2206877}.

%Equivariant cohomology is defined using a universal principal $G$-bundle $EG$, and de Rham models are defined using an algebraic object that should be thought of as the differential forms on $EG$.  So, we will proceed to discuss differential forms on principal bundles (with connection) and see how $G^\star$-algebras (with connection) provide a natural algebraic generalization.  Finally, we describe how the universal $G$-bundle with connection is most naturally defined using the language of stacks, or presheaves of groupoids.

%\reminder{We wish to state things in a more coordinate-free manner.}

%\reminder{Brief paragraph stating:  equivariant de Rham cohomology is essentially based on algebraic structures seen in differential forms on principal bundles with connection.  $G^\star$-algebras are a useful algebraic generalization of this.}

%\reminder{Insert some of these references into the text.} References: Section 1 of \cite{Fre95} for principal bundles, \cite{KN63,KN69} for principal bundles, \cite{MR0116341} for the foundational work on Borel's equivariant cohomology, \cite{GuilleminSternberg99} for the standard text on the de Rham theory of equivariant forms, \cite{Meinrenken-EquivCoh} for an excellent summary of \cite{GuilleminSternberg99}, and \cite{MR3049871} for an account of the Weil algebra from the viewpoint of stacks (simplicial presheaves).  Also, \cite{MathaiQuillen} has a nice account of the Weil and Cartan models and proves their equivalence (the Matthai--Quillen isomorphism).  As expected, anything written by Atiyah and/or Bott will be amazing.  See Atiyah--Bott's \cite{MR721448} for an explanation of localization phenomena.  Bott--Tu's paper \cite{MR1850463} explains the Chern--Weil construction for equivariant bundles, specifically why the integral classes constructed via the Borel quotient and the equivariant forms in $\O^*_G(M)$ are equal in $H^*_G(M;\R)$.  A good introduction to stacks, for a geometer, is Heinloth's notes \cite{MR2206877}.


\subsection{Equivariant cohomology}\label{subsec:BorelCoh}
Let $G$ be a reasonably nice topological group.  Then there exists a contractible space $EG$ on which $G$ acts freely from the right, giving us a universal $G$-bundle $EG \to BG$.  If $M$ has a continuous left $G$-action, the \textit{homotopy quotient} (or \textit{Borel construction}) of $M$ is defined
\[ EG \times_G M \= \left( EG \times M\right)\big/ \big((eg,x) \sim (e,gx)\big). \]
Borel's construction of the \textit{equivariant cohomology} of $M$, with coefficients in an abelian group $A$, is simply the ordinary cohomology of the homotopy quotient
\[ H_G^*(M;A) \= H^*(EG \times_G M; A).\]
A $G$-equivariant map $M \overset{f}\to N$ induces a map on the Borel constructions, giving the desired homomorphisms
\[ H^*_G(N;A) \overset{f_G^*}\longrightarrow H^*_G(M;A).\]

There are important, though subtle, relationships between equivariant and ordinary cohomology.  In particular, $H^*_G(\pt;A) = H^*(BG;A)$, and the natural map $M \to \pt$ makes $H^*_G(M;R)$ into a $H^*(BG;R)$-module when $R$ is a ring.  When $G$ acts freely on $M$, there is a natural isomorphism $H^*_G(M;A) \iso H^*(M/G;A)$.  This follows from the fact that the projection $EG\times_G M \to M/G$ is a locally trivial fiber bundle with fiber $EG$, and hence it is a homotopy equivalence.

\subsection{Principal $G$-bundles with connection}\label{subsec:GBun}Let $G$ be a Lie group, and let $\fg$ be the Lie algebra of left-invariant vector fields on $G$.

A \textit{principal $G$-bundle} on a manifold $X$ is a manifold $P$ equipped with a free right $G$-action and map to $X$ such that
\[ P \overset{\pi}\longrightarrow X\iso P/G.\]
Note that on differential forms, $\pi^*\colon  \O(X) \to \O(P)$ is an injective map, and the subspace $\pi^*\O(X)$ is naturally identified as the \textit{basic} (i.e. invariant horizontal) forms
\begin{equation}\label{eq:HorizontalFormsP} \O(X) \iso  \pi^*\O(X) = \O(X)_{\hor}^G \subset \O(P).\end{equation}
Here, the horizontal forms are defined by: $\omega \in \O(P)_{\hor}$ if  $\iota_X \omega = 0$ for all $X\in \fg$. 

A \textit{connection} on $P$ is an $\Ad$-equivariant $\fg$-valued 1-form $\Theta \in \O^1(P;\fg)$ that restricts fiberwise to the Maurer--Cartan form $\theta_\fg$.  More explicitly,
\begin{itemize}
\item $\iota_X \Theta = X \in \O^0(P;\fg)$ for any vector field $X \in \fg$,
\item $g^*(\Theta) = \Ad_{g^{-1}} \Theta$, where $g\colon  P\to P$ is right multiplication by $g$.
\end{itemize}
The $\Ad$-equivariance can be rewritten as $\Theta \in \big( \O^1(P) \otimes \fg \big)^G$, where the second factor $\fg$ is the adjoint representation of $G$.  The set of all connections on $P$ is an affine space, with $\O^1(X;\fg)$ acting freely and transitively.  The \textit{curvature} is defined 
\[ \Omega \= d\Theta + \tfrac{1}{2} [\Theta \wedge \Theta] \in \O^2(P;\fg).  \]
%\[ \xymatrix{ TP  \ar[r]^{=} &T^H P \oplus T^V P  \ar[r]^{\iso} &T^H P \oplus \fg \ar[r] & \pi^* T\S \oplus \fg \\
%X \ar@{|->}[rr]&& (X - \Theta(X)) \oplus \Theta(X) \ar[r]& \pi_* X \oplus \Theta(X) .
%}\]

There is also an equivalent, but more geometric, interpretation of a connection.  The $G$-action defines the distribution $\TV P$ of vertical tangent vectors in $P$ by
\[ \fg \overset{\iso}\longrightarrow \Ker \pi_* =: \TV P  \subset TP.\]
A connection on $P$ is the choice of an equivariant horizontal distribution $\TH P$; i.e. a connection is equivalent to an equivariant splitting of the tangent bundle
\[ TP = \TH P \oplus \TV P \iso \pi^*TM \oplus \fg. \]
The connection 1-form $\Theta$ gives a projection from $TP$ onto $\TV P\iso \fg$, thus defining the horizontal distribution by 
\[ \TH P \= \Ker \Theta \subset TP.\]

The splitting of $TP$ into horizontal and vertical subspaces induces a bi-grading on the differential forms
\[ \O^{i,j}(P) \= C^\infty(P, \L^i \TH P^* \otimes \L^j \TV P^*) \iso C^\infty(P, \pi^* \L^i TX^* \otimes \L^j \fg^* ).\]
In this bi-grading, the exterior derivative decomposes as 
\[ d = d^{0,1} + d^{1,0} + d^{2,-1} \iso  (-1)^i d_{\fg} + d_{\nabla} + (-1)^i \iota_\Omega.\]
Here, $d_{\fg}\colon  \L^j \fg^* \otimes C^\infty(P) \to \L^{j+1}\fg^* \otimes C^\infty(P)$ is the Lie algebra (or Chevalley--Eilenberg) differential for the $G$-module $C^\infty(P)$ \cite{CE48}.  If we restrict to $\O(P)^G$, then $(\L \fg^*, d_\fg)$ is naturally the de Rham complex of left-invariant forms on $G$.  The connection $\Theta$ induces the covariant derivative $d_\nabla$, and $\iota_\Theta$ is the derivation induced by contracting along the vector-valued 2-form $\Omega$.  See Section 3 of \cite{MR3004281} for more details.


\subsection{$G^\star$-algebras}   Let $(\cA, d)$ be a commutative differential graded algebra (DGA), where $d$ is a derivation of degree $+1$ and commutative means $ab = (-1)^{|a||b|}ba$ for homogeneous elements.  A $\operatorname{DGA}$-automorphism of degree 0 is an algebra automorphism $\phi\colon  \cA \to \cA$ that preserves grading and commutes with $d$.

%\marginnote{In \cite{MR1813265} it is an operation, in \cite{Meinrenken-EquivCoh} a $\fg$-da.}
\begin{defn}[Section 2.3 of \cite{GuilleminSternberg99}]\label{defn:G*alg}A $G^\star$-algebra is a commutative DGA $(\cA, d)$ equipped with representations
\[ G \overset{\rho}\longrightarrow \Aut_{\operatorname{DGA}}(\cA) \quad \text{ and }\quad \fg \overset{\iota}\longrightarrow \Der(\cA) \]
of degree 0 and -1, respectively, such that $\iota$ is $G$-equivariant with respect to $\rho$ and satisfies the Cartan equation; i.e. for all $X\in \fg$
\begin{align}
\label{eq:gstar1} \rho_g \iota_X \rho_{g^{-1}} &= \iota_{\Ad_g X}, \\
\label{eq:gstar2} \iota_X d + d\iota_X &= L_X.
\end{align}
Here, $L\colon  \fg \to \Der(\cA)$ is the Lie algebra representation induced by $\rho$.

A map $\phi\colon  \cA_1 \to \cA_2$ is a morphism of $G^\star$-algebras if $\phi$ commutes with multiplication, $\rho$, $d$, and $\iota$.
\end{defn}

\begin{example}\label{ex:G*Forms}Suppose that a manifold $M$ has a left $G$-action.  Then the de Rham complex $\big(\O(M), d \big)$ is naturally a $G^\star$-algebra.  The $G$-action on $\O(M)$ is defined by
\[ \rho_g \omega \= (g^{-1})^*\omega,\]
and $\iota$ is defined by composing the usual interior derivative with the action of $\fg$ on vector fields
\[ \fg \to \mathfrak{X}(M) \overset{\iota}\to \Der( \O(M) ).\]
If $f\colon  M_1 \to M_2$ is $G$-equivariant, then $f^*\colon  \O(M_2) \to \O(M_1)$ is a morphism of $G^\star$-algebras.
\end{example}

%
%\begin{rem}Usually this is phrased in terms of properties of the Lie derivative $L$, but these can all be deduced by using the fact that $\cA$ is a Lie group representation, and hence there is an induced Lie algebra representation $\fg \overset{L}\to \End\left( \cA \right)$.
%\end{rem}

\begin{rem}\label{rem:LeftVsRight}We use the convention that manifolds $M$ have a left $G$-action, while principal bundles $P$ have a right $G$-action.  However, we sometimes implicitly use the natural switch between left and right actions.  Given a left $G$-action on a set $Y$, define the right $G$-action via the formula $y \cdot g \= g^{-1} \cdot y$; similarly, a right $G$-action induces a left $G$-action.
\end{rem}

%\begin{example}
%A \textit{principal $G$-bundle} on a manifold $\S$ is a manifold $P$ equipped with a free right $G$-action and map to $\S$ such that
%\[ P \overset{\pi}\longrightarrow \S \iso P/G.\]
%The above example shows that $\O(P)$ is naturally a $G^\star$-algebra, though we write $\rho_g \omega = g^*\omega$ due to the fact that $G$ acts on the right.
%
%In this case, $\pi^*: \O(\S) \to \O(P)$ is an injective map, and there is an easy way to identify the subspace $\pi^*\O(\S)$ as the invariant horizontal forms
%\[ \O(\S) \iso  \pi^*\O(\S) = \O_{\hor}^G \subset \O(P),\]
%where the horizontal forms are defined by: $\omega \in \O(P)_{\hor}$ if  $\iota_X \omega = 0$ for all $X\in \fg$.
%\end{example}

\begin{defn}Let $\cA$ a $G^\star$-algebra.  An element $a$ is \textit{invariant} if $\rho_g a = a$ for all $g\in G$, and it is \textit{horizontal} if $\iota_X a = 0$ for all $X \in \fg$.  The \textit{basic} sub-algebra is the intersection of the invariant and horizontal elements:
\[ \cA_{\basic} \= \cA^G \cap \cA_{\hor} = \cA_{\hor}^G.\]
\end{defn}

The definition of a $G^\star$-algebra implies that $(\cA_{\basic}, d)$ is a sub-$\DGA$ of $(\cA,d)$.  Furthermore, if $\phi\colon  \cA \to \cB$ is a morphism of $G^\star$-algebras, then $\phi$ restricts to a $\DGA$-morphism on the basic subcomplexes $\phi\colon  \cA_{\bas} \to \cB_{\bas}$.

%\subsection{Connections}
%Let $P \overset{\pi}\to \S$ be a principal $G$-bundle.  Geometrically, the $G$-action defines a vertical distribution $\TV P$ by
%\[ \fg \overset{\iso}\longrightarrow \Ker \pi_* =: \TV P  \subset TP.\]
%A connection on $P$ is the choice of an equivariant horizontal distribution $\TH P$, giving an equivariant splitting of the tangent bundle
%\[ TP = \TH P \oplus \TV P \iso \pi^*TM \oplus \fg. \]
%%\[ \xymatrix{ TP  \ar[r]^{=} &T^H P \oplus T^V P  \ar[r]^{\iso} &T^H P \oplus \fg \ar[r] & \pi^* T\S \oplus \fg \\
%%X \ar@{|->}[rr]&& (X - \Theta(X)) \oplus \Theta(X) \ar[r]& \pi_* X \oplus \Theta(X) .
%%}\]
%
%Algebraically, a \textit{connection} on $P$ can be defined as $\Ad$-equivariant $\fg$-valued 1-form $\Theta \in \O^1(P;\fg)$ that restricts fiberwise to the Maurer--Cartan form $\theta_\fg$.  More explicitly,
%\begin{itemize}
%\item $\iota_X \Theta = X \in \O^0(P;\fg)$ for any vector field $X \in \fg$,
%\item $g^*(\Theta) = \Ad_{g^{-1}} \Theta$, where $g: P\to P$ is right multiplication by $g$.
%\end{itemize}
%The \textit{curvature} is defined $\Omega \= d\Theta + \frac{1}{2} [\Theta \wedge \Theta] \in \O^2(P;\fg)$.  The form $\Theta$ gives a projection onto $\TV P\iso \fg$, thus defining a horizontal distribution by 
%\[ \TH P \= \Ker \Theta \subset TP.\]
%
%The $\Ad$-equivariance can be rewritten as $\Theta \in \big( \O^1(P) \otimes \fg \big)^G$, where the second factor $\fg$ is the adjoint representation of $G$.  This leads to the following algebraic generalization.
%

\begin{defn}A connection on a $G^\star$-algebra $\cA$ is an element $\Theta \in \big(\cA^1 \otimes \fg\big)^G$ such that $\iota_X \Theta = X$ for all $X\in \fg$.  The curvature of $\Theta$ is defined \[ \Omega \= d\Theta + \tfrac{1}{2}[\Theta, \Theta] \in \big( \cA^2 \otimes \fg \big)^G.\]
\end{defn}


\begin{example}Suppose $P\overset{\pi}\to X$ is a principal $G$-bundle.  Example \ref{ex:G*Forms} and Remark \ref{rem:LeftVsRight} show that $\O(P)$ is naturally a $G^\star$-algebra, though we write $\rho_g \omega = g^*\omega$ due to the fact that $G$ acts on the right.  As noted in \eqref{eq:HorizontalFormsP}, the basic subcomplex $\O(P)_{\basic}$ is naturally isomorphic to $\O(X)$.  Furthermore, connections for the principal bundle $P$ are equivalent to connections for the $G^\star$-algebra $\O(P)$.
\end{example}


%
%A \textit{connection} on the principal $G$-bundle $P \overset{\pi}\to \S$ is an $\Ad$-equivariant $\fg$-valued 1-form $\Theta \in \O^1(P;\fg)$ that restricts fiberwise to the Maurer--Cartan form $\theta_\fg$.  More explicitly,
%\begin{itemize}
%\item $\iota_X \Theta = X \in \O^0(P;\fg)$ for any vector field $X \in \fg$,
%\item $g^*(\Theta) = \Ad_{g^{-1}} \Theta$, where $g: P\to P$ is right multiplication by $g$.
%\end{itemize}
%The \textit{curvature} is defined $\Omega \= d\Theta + \frac{1}{2} [\Theta \wedge \Theta] \in \O^2(P;\fg)$.
%
%Geometrically, the $G$-action defines a vertical subspace $\TV P$ by
%\[ \fg \overset{\iso}\longrightarrow \Ker \pi_* =: \TV P  \subset TP.\]
%The connection $\Theta$ gives a projection onto the vertical tangent space, thus defining a horizontal subspace $\TH P \= \Ker \Theta \subset TP$.  In fact, a connection is equivalent to an equivariant horizontal distribution in $TP$.  The splitting of the tangent bundle
%\[ TP = \TH P \oplus \TV P \iso \pi^*TM \oplus \fg \]
%%\[ \xymatrix{ TP  \ar[r]^{=} &T^H P \oplus T^V P  \ar[r]^{\iso} &T^H P \oplus \fg \ar[r] & \pi^* T\S \oplus \fg \\
%%X \ar@{|->}[rr]&& (X - \Theta(X)) \oplus \Theta(X) \ar[r]& \pi_* X \oplus \Theta(X) .
%%}\]
%induces a bi-grading on the differential forms
%\[ \O^{i,j}(P) \= C^\infty(P, \L^i \TH P^* \otimes \L^j \TV P^*) \iso C^\infty(P, \pi^* \L^i T\S^* \otimes \L^j \fg^* ),\]
%under which the exterior derivative decomposes as 
%\[ d = d^{0,1} + d^{1,0} + d^{2,-1} \iso  (-1)^i d_{\fg} + d_\nabla + (-1)^i \iota_\Omega.\]
%Here, $d_{\fg}: \L^j \fg^* \otimes C^\infty(P) \to \L^{j+1}\fg^* \otimes C^\infty(P)$ is the Lie algebra (or Chevalley--Eilenberg) differential for the $G$-module $C^\infty(P)$ \cite{CE48}.  When restricted to $\O(P)^G$, then $(\L \fg^*, d_\fg)$ is naturally the de Rham complex for left-invariant forms on $G$.  The connection $\Theta$ induces the covariant derivative $d_\nabla$, and $\iota_\Theta$ is the derivation induced by contracting along the vector-valued 2-form $\Omega$.


\begin{example}\label{WeilAlg}
The \textit{Weil algebra} $\W(\fg)$ is a $G^\star$-algebra with connection, and it is constructed so that it canonically maps to any other $G^\star$-algebra with connection.  Explicitly, 
\begin{gather*}\W(\fg) \= S \fg^* \otimes \Lambda \fg^*, \\
\deg S^1\fg^* = 2, \quad \deg \Lambda^1\fg^* = 1, \\
d_{\W} \= d_{\fg} + d_{\K}.
\end{gather*}
Here $S$ and $\Lambda$ are the total symmetric and exterior powers, so the coadjoint representation $\fg^*$ naturally makes $\W(\fg)$ into a $G$-representation.  The differential $d_{\fg}$ is the Chevalley--Eilenberg differential for Lie algebra cohomology with values in the $\fg$-module $S\fg^*$
\[ S^i \fg^*  \otimes \Lambda^j \fg^* \overset{d_{\fg}}\longrightarrow S^i \fg^*  \otimes \Lambda^{j+1} \fg^*;\]
it has degree $(0,1)$ under the bi-grading $\W^{2i,j}(\fg) = S^i \fg^* \otimes \L^j \fg^*$.  The Koszul differential $d_{\K}$ has degree $(2,-1)$ and is defined by extending the natural isomorphism
\[ \Lambda^1 \fg^* \overset{d_{\K}}{\underset{\iso}\longrightarrow} S^1(\fg^*)\]
to a derivation 
\[ S^i(\fg^*) \otimes \Lambda^j(\fg^*) \overset{d_{\K}}\longrightarrow S^{i+1}(\fg^*) \otimes \Lambda^{j-1}(\fg^*).\]
The derivation $\iota$ has degree $(0,-1)$ and is induced by the usual contraction 
\[ \fg \overset{\iota}\longrightarrow \End(\L \fg^*).\]


The Weil algebra $\W(\fg)$ is acyclic; i.e. $H^0(\W(\fg),d_{\W}) = \R$ and $H^i(\W(\fg), d_{\W}) = 0$ for $i>0$.  It has a natural connection
\[ \theta_\fg \in \L^1 \fg^* \otimes \fg = \W^{0,1}(\fg)\otimes \fg \]
given by the identity map $\fg \to \fg$; i.e. $\theta_\fg(X) = X$ for $X\in \fg$.  When there is no risk of confusion, we will drop the subscript and write $\theta$ for $\theta_\fg$.  Using the fact that $d_{\fg} \theta = - \tfrac{1}{2}[\theta \wedge \theta]$, we see that the curvature $\varOmega = \varOmega_{\fg}$ equals $d_{\K} \theta_\fg$.  In light of this, we rewrite the Koszul derivative $d_{\K}$ as $\iota_{\varOmega_\fg}$.
%  Using the convention that the $\fg$-valued outputs of $\theta$ and $\varOmega$ can be By thinking of the inputs of $\L \fg^*$ and $S \fg^*$ as accepting vectors fromWe will therefore use the notation $\iota_{\varOmega}$ for the Koszul derivative $d_{\K}$, since for any $\eta \in \L^1 \fg^*$
%\[ (d_{\K} \eta)(X') = \eta(X) = \iota_X \eta = (\iota_\theta \eta)(X) = \big(\iota_{\varOmega}\eta\big)(X).\]
%
%The curvature is defined analogously,
%\[ (Curvature) \quad \varOmega = \varOmega_{\fg} \= d_{\W} \theta_{\fg} + \tfrac{1}{2} [\theta_\fg \wedge \theta_\fg].\]
%A quick shows that $\varOmega_{\fg} = d_{\K} \theta_\fg$, and we will use the notation $\iota_{\varOmega}$ for the Koszul derivative $d_{\K}$.
\end{example}

\begin{prop}[Weil homomorphism]\label{prop:WeilToConn}Let $\cA$ be a $G^\star$-algebra with connection $\Theta$.  Then, there is natural morphism of $G^\star$-algebras with connection $\W(\fg) \overset{\Theta^*}\longrightarrow \cA$ induced by $\theta \mapsto \Theta$, $\varOmega \mapsto \Omega$.
\end{prop}

Therefore, we see that $\W(\fg)$ serves as a natural algebraic model for differential forms on $EG$.  It is acyclic, and to any $G$-bundle with connection $(P,\Theta) \to X$, the Weil homomorphism is a natural map 
\begin{align}\label{eq:WeilHomo} \begin{split} \W^{2i,j}(\fg) &\overset{\Theta^*}{\longrightarrow} \O^{2i,j}(P) \\
\omega\otimes \eta &\longmapsto \omega( \Omega^{\wedge i}) \wedge \eta(\Theta^{\wedge j}) \end{split}
 \end{align}
that is compatible with the bi-grading, multiplication, $G$-action, derivative $d$, and contraction $\iota$.  Below is a diagram showing this.  For generic $\alpha \in \W(\fg)$, we use the notation $\alpha(\Theta)$ for $\Theta^*(\alpha)$.
{\scriptsize
\[ \hskip -.8in \vcenter{ \xymatrix@C=3mm{ & \vdots& & \vdots& &\vdots \\
&\Lambda^2\fg \ar[u]_{d_\fg} \ar[drr]_<<<<<<<<{\iota_{\varOmega}} & 0
 & S^1 \fg^* \otimes \Lambda^2\fg^* \ar[u]_{d_{\fg}} \ar[drr]_<<<<<<<<{\iota_{\varOmega}}& 0 & S^2 \fg^* \otimes \Lambda^2 \fg^* \ar[u]_{d_{\fg}}   \\
& \fg^*  \ar[u]_{d_\fg} \ar[drr]_<<<<<<<<{\iota_{\varOmega}}^{\iso}& 0 & S^1 \fg^* \otimes \Lambda^1 \fg^* \ar[u]_{d_{\fg}} \ar[drr]_<<<<<<<<{\iota_{\varOmega}}& 0 & S^2 \fg^* \otimes \Lambda^1 \fg^* \ar[u]_{d_{\fg}}  \\
&\R \ar[u]_{0} & 0 & S^1 \fg^* \ar[u]_{d_{\fg}} & 0 & S^2\fg^* \ar[u]_{d_{\fg}} 
}  } \Longrightarrow 
\vcenter{ \xymatrix{ \vdots &\vdots &\vdots &\vdots &\\
\O^{0,2}(P) \ar[u]_{d_\fg} \ar@{.>}[r]^{\nabla} \ar[drr]_<<<<<<<<{\iota_\Omega} & \O^{1,2}(P) \ar@{.>}[u]_{-d_\fg} \ar@{.>}[r]^{d_\nabla} \ar@{.>}[drr]_<<<<<<<<{-\iota_\Omega}& \Omega^{2,2}(P) \ar[u]_{d_\fg} \ar@{.>}[r]^{d_\nabla} \ar[drr]_<<<<<<<<{\iota_\Omega}& \O^{3,2}(P) \ar@{.>}[u]_{-d_\fg} \ar@{.>}[r]^{d_\nabla}&\cdots\\
\O^{0,1}(P)  \ar[u]_{d_\fg} \ar@{.>}[r]^{\nabla} \ar[drr]_<<<<<<<<{\iota_\Omega}& \O^{1,1}(P)\ar@{.>}[u]_{-d_\fg} \ar@{.>}[r]^{d_\nabla} \ar@{.>}[drr]_<<<<<<<<{-\iota_\Omega}& \O^{2,1}(P) \ar[u]_{d_\fg} \ar@{.>}[r]^{d_\nabla} \ar[drr]_<<<<<<<<{\iota_\Omega}& \O^{3,1}(P) \ar@{.>}[u]_{-d_\fg} \ar@{.>}[r]^{d_\nabla}&\cdots\\
\Omega^{0,0}(P) \ar[u]_{d_\fg} \ar@{.>}[r]^{d_\nabla} & \Omega^{1,0}(P) \ar@{.>}[u]_{-d_\fg} \ar@{.>}[r]^{d_\nabla}& \Omega^{2,0}(P) \ar[u]_{d_\fg} \ar@{.>}[r]^{d_\nabla}& \Omega^{3,0}(P) \ar@{.>}[u]_{-d_\fg} \ar@{.>}[r]^{d_\nabla}&\cdots
} } \] }


%\begin{example}\label{ex:ChernSimons}  

The universal Chern--Weil and Chern--Simons forms are naturally described via the Weil model.  The usual Chern--Weil forms are given by simply restricting \eqref{eq:WeilHomo} to the basic subcomplex
\begin{align*} (S^k \fg^*)^G = \W^{2k}(\fg)_{\basic} &\xrightarrow{\Theta^*} \O^{2k}(P)_{\basic} \iso \O^{2k}(M) \\
\omega &\longmapsto \omega(\Theta^{\wedge n}).
\end{align*}
That $\W(\fg)$ is acyclic implies any such $\omega \in (S\fg^*)^G$ is exact in $\W(\fg)$, and there is a standard way to pick out such a coboundary.  Using the trivial $G^\star$-algebra $\O([0,1])$ with natural coordinate $t$, define 
%the following formal 1-parameter family of path of connection forms on the $G^\star$-algebra $\left( \O([0,1]) \otimes \W(\fg), d \right)$ by 
\begin{align*}
\theta_t &\= t\theta & &\in \big( \O([0,1]) \otimes \W(\fg)  \big)^1\otimes \fg, \\
\varOmega_t &\= d(\theta_t) + \tfrac{1}{2} [ \theta_t \wedge \theta_t]
& &\in \big( \O([0,1]) \otimes \W(\fg) \big)^2 \otimes \fg, 
\end{align*}
where one can rewrite the curvature to give $\varOmega_t =  dt\,\theta + t \varOmega + \tfrac{1}{2}(t^2-t)[\theta \wedge \theta]$.
%$dt\tfrac{\partial}{\partial t} (\theta_t) + d_{\W} \theta_t + \tfrac 12 [\theta_t \wedge \theta_t]$

For $\omega \in (S^k \fg^*)^K$, then $\omega(\varOmega_t^{\wedge k}) \in \big( \O([0,1]) \otimes \W(\fg)  \big)_{\basic}$, and the Chern--Simons form is defined using integration over the interval $[0,1]$ by 
\begin{equation}\label{eq:ChernSimonsWeil}
\CS_\omega \= \int_{[0,1]} \omega \big( \varOmega_t^{\wedge k} \big) \in \W^{2k-1}(\fg)^G.
\end{equation}
Stokes Theorem implies that $d\CS_\omega = \omega \in \W(\fg)$, and the Weil homomorphism sends this to the standard Chern--Simons forms on the total space of principal bundles
\begin{align*} \W(\fg)^G &\xrightarrow{\Theta^*} \O(P)^G \\
\CS_\omega &\longmapsto \CS_\omega(\Theta).
\end{align*}
%Satisfies $d_\W \CS_\omega = \omega(\varOmega^{\wedge k}) = \omega,$ and restricted to the completely vertical part, gives element in $(\L^{2i-1} \fk^*)^K$ representing transgression in 
%\[ \xymatrix{ H^{2k}(BK;\R) \ar[r]^{\tau} & H^{2k-1}(K;\R) \\
%(S^k \fk^*)^K \ar[u]^{\iso} \ar[r]^{\CS} & (\L^{2k-1} \fg^*)^K \ar[u]^{\iso}
%} \]

%\end{example}

%
%{\scriptsize
%\[ \hskip -.45in \xymatrix@C=5mm{ & \vdots& & \vdots& &\vdots \\
%&\Lambda^2\fg \ar[u]_{d_\fg} \ar[drr]_<<<<<<<<{\iota_{\varOmega}} & 0
% & S^1 \fg^* \otimes \Lambda^2\fg^* \ar[u]_{d_{\fg}} \ar[drr]_<<<<<<<<{\iota_{\varOmega}}& 0 & S^2 \fg^* \otimes \Lambda^2 \fg^* \ar[u]_{d_{\fg}}  &\cdots \\
%& \fg^*  \ar[u]_{d_\fg} \ar[drr]_<<<<<<<<{\iota_{\varOmega}}^{\iso}& 0 & S^1 \fg^* \otimes \Lambda^1 \fg^* \ar[u]_{d_{\fg}} \ar[drr]_<<<<<<<<{\iota_{\varOmega}}& 0 & S^2 \fg^* \otimes \Lambda^1 \fg^* \ar[u]_{d_{\fg}} &\cdots  \\
%&\R \ar[u]_{0} & 0 & S^1 \fg^* \ar[u]_{d_{\fg}} & 0 & S^2\fg^* \ar[u]_{d_{\fg}} &\cdots 
%} \] }
%
%{\scriptsize
%\[ \hskip -.45in \xymatrix{ &\vdots &\vdots &\vdots &\vdots &\\
%&\O^{0,2}(P) \ar[u]_{d_\fg} \ar@{.>}[r]^{\nabla} \ar[drr]_<<<<<<<<{\iota_\Omega} & \O^{1,2}(P) \ar@{.>}[u]_{-d_\fg} \ar@{.>}[r]^{d_\nabla} \ar@{.>}[drr]_<<<<<<<<{-\iota_\Omega}& \Omega^{2,2}(P) \ar[u]_{d_\fg} \ar@{.>}[r]^{d_\nabla} \ar[drr]_<<<<<<<<{\iota_\Omega}& \O^{3,2}(P) \ar@{.>}[u]_{-d_\fg} \ar@{.>}[r]^{d_\nabla}&\cdots\\
%&\O^{0,1}(P)  \ar[u]_{d_\fg} \ar@{.>}[r]^{\nabla} \ar[drr]_<<<<<<<<{\iota_\Omega}& \O^{1,1}(P)\ar@{.>}[u]_{-d_\fg} \ar@{.>}[r]^{d_\nabla} \ar@{.>}[drr]_<<<<<<<<{-\iota_\Omega}& \O^{2,1}(P) \ar[u]_{d_\fg} \ar@{.>}[r]^{d_\nabla} \ar[drr]_<<<<<<<<{\iota_\Omega}& \O^{3,1}(P) \ar@{.>}[u]_{-d_\fg} \ar@{.>}[r]^{d_\nabla}&\cdots\\
%&\Omega^{0,0}(P) \ar[u]_{d_\fg} \ar@{.>}[r]^{d_\nabla} & \Omega^{1,0}(P) \ar@{.>}[u]_{-d_\fg} \ar@{.>}[r]^{d_\nabla}& \Omega^{2,0}(P) \ar[u]_{d_\fg} \ar@{.>}[r]^{d_\nabla}& \Omega^{3,0}(P) \ar@{.>}[u]_{-d_\fg} \ar@{.>}[r]^{d_\nabla}&\cdots
%} \] }
%


\subsection{Equivariant Forms}\label{subsec:deRhamModel}
We now describe an explicit de Rham model $\O_G(M)$, often referred to as the Weil model, for $H^*_G(M;\R)$.  While the definition of $\O_G(M)$ does not require $G$ to be compact, the cohomology of $\O_G(M)$ is not necessarily isomorphic to $H^*_G(M;\R)$ for non-compact $G$.

Suppose $M$ is a manifold equipped with a smooth left $G$-action.  The tensor product $\W(\fg) \otimes \O(M)$ is a $G^\star$-algebra with derivative $d_G = d_{\W}\otimes 1 + 1 \otimes d$.  

\begin{defn}The complex of \textit{equivariant forms} is the basic sub-complex
\begin{equation*} \left(  \O_G^*(M), d_G \right) \= \big( \left( \W(\fg)\otimes \O(M) \right)^G_{\hor}, \> d_{\W} \otimes 1 + 1 \otimes d \big). \end{equation*}
\end{defn}

Note that if $f\colon  M \to N$ is $G$-equivariant map, then 
\[\W(\fg)\otimes \O(N) \xrightarrow{1 \otimes f^*} \W(\fg) \otimes \O(M)\] restricts to the basic complex, giving a naturally induced DGA-morphism we denote
\[ \O_G(N) \overset{f^*_G}\longrightarrow \O_G(M).\]


To relate $\O_G(M)$ to the cohomology of $EG \times_G M$, we use the following geometric fact.  There exist finite-dimensional smooth $n$-classifying bundles with connection $(E^{(n)}G, \Theta_{E^{(n)}G}) \to B^{(n)}G$ (\cite{MR0133772}).  This means that if $\dim(X) \leq n$, any bundle with connection $(P,\Theta)\to X$ is isomorphic to the pullback $f^*(E^{(n)}G, \Theta_{E^{(n)}G})$ for some smooth map $X \overset{f}\to B^{(n)}G$.  Furthermore, any two maps classifying $(P,\Theta)$ are homotopic.  The universal bundle $EG\to BG$ can then be constructed as a direct limit of finite-dimensional manifolds.
\[ \vcenter{ \xymatrix{  (EG,\Theta_{EG}) \ar[d]^{\pi} \\ BG } }  \quad \= \quad \varinjlim
\left(\vcenter{\xymatrix{ (E^{(n)}G, \Theta_{E^{(n)}G}) \ar[d]^{\pi} \\ B^{(n)}G } } \right) \]
When discussing differential forms and cochains, we use the notations
\begin{align*} \O^*(EG) &= \varprojlim \O^*(E^{(n)}G), & C^*(EG;A) &= \varprojlim C^*(E^{(n)}G;A),\\\O^*(BG) &= \varprojlim \O^*(B^{(n)}G), & C^*(BG;A) &= \varprojlim C^*(B^{(n)}G;A).\end{align*}

The connection $\Theta_{EG}$ and the Weil homomorphism give a $G^\star$-algebra homomorphism 
\begin{align*}
\W(\fg) \xrightarrow{\Theta_{EG}^*} \O(EG),
\end{align*}
which in turn gives
\begin{equation}\label{eq:EquivToBorel} \O_G(M) = \big(\W(\fg) \otimes \O(M) \big)_{\basic} \xrightarrow{\Theta_{EG}^* \otimes 1}\O(EG \times M)_{\basic} \iso \O(EG\times_G M).  \end{equation}

\begin{thm}[Equivariant de Rham Theorem]For $G$ compact, \eqref{eq:EquivToBorel} induces an isomorphism in cohomology
\[ H^*\big( \O_G(M), d_G \big) \overset{\iso}\longrightarrow H^*\big( \O(EG\times_G M),d \big) \iso H^*_G(M;\R).\]
\end{thm}

A proof of the above theorem can be found in Theorems 2.5.1 (and Theorems 4.3.1 and 6.7.1) of \cite{GuilleminSternberg99}.  The following lemma is used in the first construction of $\HG^*(M)$ in Section \ref{sec:HS}, along with the proof of \ref{prop:Uniqueness}.  Though it is certainly well-known, we are unaware of a specific reference, and we prove it directly so that the first construction of $\HG^*(M)$ does not rely on results from \cite{MR3049871}.

\begin{lemma}\label{lem:WeilEGinj}The homomorphism $\W(\fg) \xrightarrow{\Theta_{EG}^*} \O(EG)$ is injective, as are the induced homomorphisms $\O_G(M) \to \O(EG\times_G M)$.
\end{lemma}
\begin{proof}The second homomorphism is given by restricting 
\[ \W(\fg) \otimes \O(M) \longrightarrow \O(EG)\otimes \O(M) \iso \O(EG\times M)\]
to the basic subcomplex.  Showing this map is injective is equivalent to showing $\W(\fg) \to \O(EG)$ is injective.

Any bundle with connection is isomorphic to the pullback of $(EG,\Theta_{EG})$, so it suffices to show that for any element $\alpha \in \W(\fg)$, there exists some $(P,\Theta) \xrightarrow{\pi} X$ such that $\alpha(\Theta) \neq 0 \in \O(P)$.  At any point $p$, $\Lambda T_p^*P$ is an algebra freely generated by the horizontal and vertical cotangent spaces.  On the vertical part, $\W^{0,*}(\fg) \to \L T_p^V P^* \iso \L \fg^*$  is an isomorphism, so it suffices to show the Weil homomorphism is injective on $\W^{*,0}(\fg)$. 

Let $\omega \in S^n \fg^*$ be any non-zero element.  Then $\omega( \xi_{i_1}\cdots \xi_{i_n}) \neq 0$ for some $(i_1, \ldots, i_n)$, where $\{\xi_i\}$ be a basis of $\fg$.  Let $\R^{2n} \times G \to \R^{2n}$ be the trivial bundle.  In a neighborhood of $0$, use the canonical frame $p$ to define a connection $\Theta$ by 
\[ p^*\Theta = x^1 dx^2 \xi_{i_1} + \cdots + x^{2n-1} dx^{2n} \xi_{i_n} \in \O^1 (\R^{2n};\fg).\]
The local curvature is given by 
\begin{align*} p^*\Omega &= dx^1 dx^2 \xi_{i_1} + \cdots + dx^{2n-1} dx^{2n} \xi_{i_n} +  \tfrac{1}{2}[p^*\Theta \wedge p^*\Theta],
\end{align*}
and the terms involving $p^*\Theta$ vanish when all $x^i=0$.  Evaluating $p^*\omega(\Theta^{\wedge n})$ at the origin gives 
\begin{align*}
p^*\omega(\Theta^{\wedge n})_{(0)}\big( \partial_1, \ldots, \partial_{2n}  \big) = n! \omega( \xi_{i_1} \cdots \xi_{i_n} ) \neq 0. 
\end{align*} 
\qedhere
\end{proof}



\subsection{Cartan model and Matthai--Quillen}\label{subsec:CartanMQ}There is another $G^\star$-algebra structure on $\W(\fg)\otimes \O(M)$ that leads to the Cartan model for $H^*_G(M;\R)$.  It is obtained by modifying the $G^\star$-structure so that the interior derivative $\iota$ only acts on the factor $\W(\fg)$.

The $G$-equivariant homomorphism
\[ \O^1(M) \overset{\iota_{\theta}}\longrightarrow \L^1 \fg^*, \]
defined by $(\iota_\theta \psi) (X) = \iota_{\theta(X)} \psi = \iota_X \psi$,  induces a derivation on associative algebra $\W(\fg)\otimes \O(M)$.  It exponentiates to a $G$-equivariant automorphism of associative algebras
\[ \W(\fg)\otimes \O(M) \xrightarrow{\exp(\iota_\theta)} \W(\fg)\otimes \O(M) \]
known as the Mathai--Quillen isomorphism \cite{MathaiQuillen}.  Conjugating by $e^{\iota_\theta}$ gives the new operators 
\[ d_C \= (\exp{\iota_{\theta}})\, d_G\, (\exp{- \iota_\theta}), \quad \iota^C \= (\exp{\iota_{\theta}} )\, \iota \, (\exp - \iota_\theta) \]
where $\iota^C_X (\alpha \otimes \psi) = (\iota_X \alpha) \otimes \psi$.  The isomorphism between $G^\star$-algebras induces an isomorphism of the basic subcomplexes, one of which is the previously discussed Weil model.  The other, known as the Cartan model, is 
\[ \big( \W(\fg) \otimes \O(M)\big)^G_{\Ker \iota^C} =  \big( S\fg^* \otimes \O(M) \big)^G,\]
and the derivative takes the form  $d_C = d - \iota_{\varOmega_\fg}$,
where $\iota_{\varOmega_\fg}$ is the degree 1 derivation induced by 
\[ \O^k(M) \xrightarrow{\varOmega_\fg \otimes 1} S^1 \fg^* \otimes \fg \otimes \O^k(M) \xrightarrow{1 \otimes\iota} S^1 \fg^* \otimes \O^{k-1}(M).\]
The Mathai--Quillen isomorphism may be used to interpret $\O_G(M)$ as the Cartan model throughout the entire paper.
%\[ \xymatrix{ &\left(S\fg^* \otimes \O(M) \right)^G \ar@{<->}[dd]^{\iso} \ar[dl] \\
%\O(X) \\ & \left( \W(\fg) \otimes \O(M) \right)_{\basic} \ar[ul]
%}\]

%As noted in \cite{MR721448}, proving that $\O_G(M) \to \O(EG\times_G M)$ is a quasi-isomorphism relies on the fact that:
%$\L\fg^* \into \O(G)$ induces an isomorphism in cohomology, there exists the map, and spectral sequence.

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

%\subsection{Trivializations}\label{subsec:Trivializations}The definitions of trivializations and the results from \cite{Redden11a} carry over immediately as well.  We briefly phrase them as follows, but they will make more sense after the differential character description in Section \ref{subsec:characters}.
%
%We can discuss equivariant trivializations, 

%\subsection{Generalized cohomology theories}


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



%\appendix

\begin{comment}
\section{Equivariant singular chains}\label{app:StackChains}

Let $\cC$ be a groupoid.  We define $\Z \cC$ as the the free (strict) Picard groupoid generated by $\cC$.  The objects/morphisms of $\Z\cC$ are finite formal sums of objects/morphisms in $\cC$ with integer coefficients.  More precisely the objects $\Obj(\Z\cC)$ are defined as the free abelian group generated by $\Obj(\cC)$.  They are finite formal sums $\sum_{i=1}^N n_i x_i$, where $n_i \in \Z$ and $x_i \in \Obj(\cC)$, subject to the usual relations:
\begin{itemize}
\item $0x_1 = 0x_2$ for all $x_i\in \Obj(\cC)$, 
\item $n_1 x + n_2 x = (n_1+n_2)x$,
\item addition is associative and commutative.
\end{itemize}
The morphisms of $\Z\cC$ are similarly defined as the free abelian group generated by the morphisms in $\cC$.  If $f_i \in \Hom_{\cC}(x_i, y_i)$, then 
\[ \sum n_i f_i \in \Hom_{\Z\cC}\left( \sum n_i x_i, \sum n_i y_i  \right).\]
The sums are subject to the same relations used in defining $\Obj(\Z\cC)$.  The source and target maps $s, t$ are group homomorphisms.  Composition is defined as follows: assume that $f = \sum m_i f_i$,  $g = \sum n_j g_j$, and $s(f) = t(g)$.  Then, we define the composition
\[ f \o g \= f+ g - s(f) = \sum_i m_i f_i + \sum_j n_j g_j - s(f) .\]
This makes $\Z\cC$ into a strict Picard category, and we have a natural identification
\[ \Hom_{\Z\cC} \left( \sum m_i x_i, \sum n_i y_i \right) \iso \Hom_{\Z\cC} \left( 0, \sum n_i y_i - m_i x_i\right)\]
induced by the natural map
\[ \Hom_{\cC}(x,y) \overset{\bullet -\Id_x}\longrightarrow \Hom_{\Z\cC}(0, y-x),\]
where $f:x\to y$ gets mapped to $f-\Id_x$.  Note that the definition of composition gives rise to several natural looking equalities, including $f+g = (g\o f) + \Id_y$, and $f + f^{-1} = \Id_x + \Id_y$. \reminder{Insert picture}

Let $\cX$ be a presheaf of groupoids on manifolds (a pre-stack).  Evaluating $\cX$ on the smooth $k$-simplex $\Delta_k$ gives the category $\cX(\Delta_k) = \Map(\Delta_k, \cX)$.  The face maps $d^i:\Delta_k \to \Delta_{k+1}$ induce functors $\Map(\Delta_{k+1}, \cX) \overset{d_i}\to \Map(\Delta_k, \cX)$.  We define $C_k(\cX)$ as the free Picard category $\Z \Map(\Delta^k, \cX)$.  The alternating sum of the face maps $\partial = \sum (-1)^i d_i$ is then a Picard functor 
\[ C_{k+1}(\cX) \overset{\partial}\longrightarrow C_k(\cX) \]
satisfying $\partial^2 = 0$.  This makes $C_*(\cX)$ into a categorical analog of the singular chain complex.

Let $A$ be an abelian group $A$, which we consider as a Picard category with $\Obj(A) = A$ and no non-trivial morphisms.  We define the $A$-valued cochains on $\cX$ as the Picard functors 
\[ C^k(\cX;A) \= \Hom_{Pic}\left( C_k(\cX), A\right).\]
The functor $\partial$ induces a functor $\delta: C^k(\cX;A) \to C^{k+1}(\cX;A)$ satisfying $\delta^2=0$.  In this way, $C^*(\cX;A)$ is a categorical analog of singular cochain complex with coefficients in $A$.

We can then define $Z_k(\cX)$ and $Z^k(\cX;A)$ as the subcategories defined by $\Ker \partial$ and $\Ker \delta$, respectively.  In other words, an object in $Z_k(\cX)$ is a formal sum of elements in $\cX(\Delta^k)$ such that, when restricted to the boundary $\Delta_{k-1}$, their alternating sum is the identity element $0$ (they ``cancel out").  Similarly, a morphism in $Z_k(\cX)$ is a formal sum of morphisms that equals the identity morphism $0$ under the boundary operator $\partial$.  We define $Z^k(\cX;A)$ analogously.  

%Note that $\Hom_{Pic}(\bullet, \bullet)$ is naturally an abelian group. 

%\reminder{Size considerations.}
%\reminder{To have any nice functoriality properties, we will need $\cX$ to be a stack (probably).  To generalize to $A$ being a more general Picard category, we will need to be more careful about morphisms, gluings.}


\newpage
\section{Leftovers}


%\subsection{Integration}
%The integration (or pushforward) map in differential cohomology, defined in \cite{HS05} using a differential Thom class for the stable normal bundle, extends to our equivariant theory without any additional difficulty (beyond the original definition; the only difficulties are the ones already taken care of by Hopkins--Singer).  Suppose that $M_1 \overset{f}\to M_2$ is an equivariant between $G$-manifolds.
%
%\begin{defn}An $\HG[G]$-oriented map.
%\end{defn}
%\reminder{For $M$ compact, there exists an equivariant Euclidean embedding.  But, is the space of embeddings contractible?  The existence of an equivariant embedding is often referred to as the Mostow--Palais Theorem.  Look at Wasserman's ``Equivariant Differential Topology" for some details on Thom spaces, equivariant embeddings.}
%
%\reminder{Pushforward maps using Mathai--Quillen form}
%\subsection{A calculation}Let $G_1$ and $G_2$ be compact Lie groups, and let $H_1$ and $K_1$ be subgroups of $G_1$.  We will consider the trivial $H_1$-equivariant bundle $G_1/K_1 \times G_2 \to G_1/K_1$.

%\section{Relationship to other work}[If I have time.]
%\subsection{Orbifolds} Reference Bunke et al, Uribe, others?
%\subsection{Gomi's construction}
%\subsection{Equivariant gerbes}
%In \cite{MR2817778}, Behrend--Xu discuss (at least in literature) equivariant gerbes.  Gomi's papers \cite{MR2147734,MR2140550} discuss equivariant gerbes with connection, but this is a different notion of connection than I use.  

\end{comment}

%\newpage

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\end{document}


\newpage
Denote
\begin{align*} 
\Omega^\bullet(EG) &\= W^\bullet(\fg) = S^\bullet(\fg^*) \otimes \Lambda^\bullet(\fg^*), \\
\Omega^\bullet (EG \times M) &= W^\bullet(\fg) \otimes \Omega^\bullet(M), \\
\Omega_G^\bullet(M)  &\= \Omega^\bullet(EG \times_G M) = \big( W(\fg) \otimes \Omega(M) \big)_\bas \iso \big(S(\fg^*) \otimes \Omega(M) \big)^G.
\end{align*}
See Appendix \ref{sec:EquivForms} for more details.

Let $(EG, \Theta_{EG})\to BG$ be a classifying space with universal connection.  The specific model should not really matter, but we require that $G$ acts freely on $EG$, $EG$ is ``contractible," and there is a natural injective map $\w: W(\fg) \into C^*(EG;\R)$ given by the Weil homomorphism.  For $G$ compact, we can simply take $EG$ to be embedded in the Stiefel manifolds classifying $U(N)$ for some $N$.

EXPLICITLY: we assume $EG$ is an object in some (model) category that classifies $G$-bundles, there is a notion of singular cochains, $EG$ is acyclic (all the homotopy groups vanish, and $C^*(EG;A)$ is an acyclic complex), and $\Omega(EG) \into C^*(EG;\R)$ is an inclusion (as sets) and a weak equivalence.

Let $\L \subset V$ be a completely disconnected subgroup of a vector space (over $\R$ or $\C$).  Then, we use the Hopkins--Singer construction to define $\HG^\bullet(M;\L) \= \HG[]^\bullet(EG\times_G M;\L)$, where the right-hand side is interpreted using the above definition of $\Omega(EG\times_G M)$.  More concretely,

\begin{defn}For a left $G$-manifold $M$, define the differential equivariant cochain complex $\CG^\bullet(M;\L) \= \CG[]^\bullet(EG\times_G M;\L)$ by 
\begin{align*}
\CG^k(M;\L) &\= C^k(EG \times_G M;\L) \times C^{k-1}(EG\times_G M;V) \times \Omega_G^k(M;V) \\
d(c,h,\omega) &\= (\delta c, \w(\omega) - c - \delta h, d_G \omega).
\end{align*}
\end{defn}

Natural cochain maps
\[ \xymatrix{ & C_G^\bullet(\L) \\
\CG^\bullet(\L) \ar[ur]^{\cc} \ar[dr]_{\curv} \\
& \Omega_G^\bullet (V)
} \]

There is a natural descending filtration of cochain complexes $\CG(M;\L)$
\[ \CG^\bullet = F^0 \CG \supset F^1 \CG \supset \cdots \]
given by requiring
\[ \curv F^s \CG^k = 0 \text{ when } k < s. \]

\begin{equation}
\HG^k (M;\L) \= H^k \big(F^k\CG^\bullet(M;\L) \big) = \frac{\ZG^k(M;\L)}{d \CG^{k-1}(M;\L)_{\flat} }
\end{equation}

\begin{prop}Commutative square
\begin{equation}\label{eq:FundSquare} \xymatrix{ \HG^\bullet(M;\L) \ar[r]^{\cc} \ar[d]_{\curv}& H^\bullet_G(M;\L) \ar[d]\\
\Omega^k_G(M;V)_\L \ar[r]^{\dR}  & H^\bullet_G(M;V)_\L
}\end{equation}
and short exact sequences
\begin{align*}
0 \longrightarrow H^{\bullet - 1}_G(M; V/\L) \longrightarrow &\HG^\bullet(M;\L) \longrightarrow \Omega^\bullet_G(M;V)_\L \longrightarrow 0 \\
0 \longrightarrow \frac{\Omega^{\bullet -1}_G(M;V)}{\Omega^{\bullet -1}_G(M;V)_\L} \longrightarrow &\HG^\bullet(M;\L) \longrightarrow H^\bullet_G(M;\L) \longrightarrow 0 \\
0 \longrightarrow \frac{H^{\bullet - 1}_G(M;V)}{H^{\bullet - 1}_G(M;V)_\L} \longrightarrow &\HG^\bullet(M;\L) \longrightarrow A^\bullet_G(M;\L) \longrightarrow 0
\end{align*}
where $A^\bullet_G(M;\L)$ is the pullback (in sets) of the commutative square \eqref{eq:FundSquare}.
\begin{proof}Follows same proof as \cite{HS05}, and more formally, the outline in Appendix \ref{}.
\end{proof}\end{prop}

Character diagram:
\[ \xymatrix{ 0 \ar[dr]&&&&0 \\
&H_G^{\bullet - 1}(M;V/\L) \ar[dr] \ar@/^1pc/^{-B}[rr]&& H_G^{\bullet}(M;\L) \ar[ur]  \ar@/^1pc/[dr]\\
H_G^{\bullet -1}(M;V) \ar@/^1pc/[ur] \ar@/_1pc/[dr]&& \HG^{\bullet}(M;\L) \ar_{\cc}[ur] \ar^{\curv}[dr] \ar@{}^>>>>>{\dR}[rr]&& H_G^{\bullet}(M;V) \\
& {\displaystyle \frac{\Omega^{\bullet-1}_G(M;V)}{\Omega^{\bullet-1}_G(M;V)_\L}} \ar[ur] \ar@/_1pc/_{d_G}[rr]&& \Omega^{\bullet}_G(M;V)_\L \ar@/_1pc/[ur] \ar[dr]\\
0 \ar[ur]&&&&0
} \]


\begin{example}When $G=1$ is trivial, we can take $EG= \pt$, and we recover 
\[ \CG[1]^\bullet(M;\L) \iso \CG[]^\bullet(M;\L), \quad \HG[1]^\bullet(M;\L) \iso \HG[]^\bullet(M;\L).\]
\end{example}

\begin{example}
When $M=\pt$ and $G$ is compact, then $H^{2k}(BG;\R) \iso S^k(\fg^*)^G$ and $H^{2k+1}(BG;\R) = 0$.
\begin{align*}
0 \longrightarrow H^{2k-1}(BG;\R/\Z) \longrightarrow &\HG^{2k}(\pt;\Z) \longrightarrow H^{2k}(BG;\Z) \longrightarrow 0 \\
0 \longrightarrow \frac{S^k(\fg^*)^G}{S^k(\fg^*)^G_\Z} \longrightarrow &\HG^{2k+1}(\pt;\Z) \longrightarrow H^{2k+1}(BG;\Z) \longrightarrow 0
\end{align*}
%\[ \xymatrix{ 0 \ar[dr]&&&&0 \\
%&H^{2k - 1}(BG;\R/\Z) \ar[dr] \ar@/^1pc/[rr]&& H^{2k}(BG;\Z) \ar[ur]  \ar@/^1pc/[dr]\\
%0 \ar@/^1pc/[ur] \ar@/_1pc/[dr]&& \HG^{2k}(\pt;\Z) \ar[ur] \ar[dr]&& H^{2k}(BG;\R)_\Z \\
%& 0 \ar[ur] \ar@/_1pc/[rr]&& S^{k}(\fg^*)^G_\Z \ar@/_1pc/[ur]^{\iso} \ar[dr]\\
%0 \ar[ur]&&&&0
%} \]
%\[ \xymatrix{ &&H^{2k}(BG;\R/\Z) && H^{2k+1}(BG;\Z) \\
%&\frac{H^{2k}(BG;\R)}{H^{2k}(BG;\R)_\Z} && \HG^{2k+1}(\pt;\Z) && 0 \\
%&& \frac{S^{k}(\fg^*)}{S^{k}(\fg^*)_\Z} && 0
%} \]
\end{example}

\bibliographystyle{alphanum}
\bibliography{MyBibDesk}

\end{document}

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A more concise way to say that $(EG, \Theta_{EG}) \to BG$ is a universal bundle is that the natural map
\[ \Map(\S, BG) \longrightarrow \pi_0\big( \Bunc[G](\S) \big) \]
is surjective, and homotopy classes of maps classify isomorphism classes of bundles without connection:
\[ \pi_0 \big( \Map(\S,BG) \big) \overset{\iso} \longrightarrow \pi_0 \big( \Bun[G](\S) \big).\]
Viewing bundles as maps $\S\to BG$ is an extremely useful tool in topology, for it allows one to perform universal calculations on $BG$.  One must be careful, however, because a classifying map for an abstract bundle $P\to \S$ is an extra choice, and the map
\[ \Map(\S,BG) \longrightarrow \Bun[G](\S)\] is not naturally an equivalence of categories.  

To emphasize the intuitive geometric ideas of certain constructions, and to avoid problems that accompany choosing classifying maps, we make use of the language of stacks in this paper.  However, none of our results or arguments will make use of the assorted technical tools that have been developed for stacks.

The language of stacks gives us a ``classifying space" $\BnG$ that is more functorial than $BG$, but at the expense of leaving the category of topological spaces.  Luckily, this will not pose a serious problem for anyone uncomfortable with the language of sheaves on manifolds.  Since $(EG, \Theta_{EG}) \to BG$ is a principal $G$-bundle with connection, there are natural maps
\[ \xymatrix{ EG \ar[r]\ar[d] &\En G \ar[d]\\
BG \ar[r] &\Bn G . } \]
The fact that $EG\to BG$ is a universal bundle implies that any map $M \to \Bn G$ factors through $BG$.  While our constructions are often more natural and geometric when using stacks, it will always be possible to make choices and use $EG \to BG$.

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%\subsection{Weil algebra}
%
%
%When $G$ is a compact Lie group, there is a particularly nice de Rham model for $H^*_G(M;\R)$, which we now explain.  The \textit{Weil algebra} $\W(\fg)$ is a commutative differential graded algebra (dga), and it is defined
%\begin{gather*}\W(\fg) \= S (\fg^*) \otimes \Lambda (\fg^*), \\
%\deg S^1\fg^* = 2, \quad \deg \Lambda^1\fg^* = 1.
%\end{gather*}
%Here, $S$ and $\Lambda$ are the total symmetric and exterior powers, and they make $\W(\fg)$ into a $\fg$-representation via the coadjoint representation $\fg^*$.  The differential, which we write
%\[ d_W \= d_{CE} + d_K,\]
%is the sum of the Chevalley--Eilenberg differential $d_{CE}$ and the Koszul differential $d_K$.  The differential $d_{CE}$ is defined using Lie algebra cohomology with values in the $\fg$-module $S(\fg^*)$,
%\[ S^i(\fg^*) \otimes \Lambda^j \fg^* \overset{d_{CE}}\longrightarrow S^i(\fg^*) \otimes \Lambda^{j+1}(\fg^*),\]
%and 
%\[ S^i(\fg^*) \otimes \Lambda^j(\fg^*) \overset{d_K}\longrightarrow S^{i+1}(\fg^*) \otimes \Lambda^{j-1}(\fg^*)\]
%is defined by extending the natural isomorphism
%\[ \Lambda^1 \fg^* \overset{\iso}\longrightarrow S^1(\fg^*)\]
%to a derivation.  Connection $\Theta_{\W}$ on $\W(\fg)$.  Hence, $\W(\fg)$ is simultaneously a commutative differential graded algebra (dga) and a $\fg$-representation, and it is equipped with an algebraic notion of a connection.  (This data is often packaged together and called a $\fg$-differential algebra ($\fg$-da) with connection \cite{Meinrenken-EquivCoh}, or a $G^{\star}$ algebra with connection \cite{GuilleminSternberg99}.)  
%
%Given a principal $G$-bundle with connection $(P,\Theta_P) \to M$, there is a natural homomorphism
%\begin{align*} \W^*(\fg) &\overset{w(P,\Theta_P)}{\longrightarrow} \O^*(P) \\
%\omega\otimes \eta &\longmapsto \omega( \Omega_P^{\wedge i}) \wedge \eta(\Theta_P^{\wedge j})
% \end{align*}
%and this is compatible with the connections, $\fg$-action, the derivative, and the bi-grading.  In other words, it is a morphism of $\fg$-da's.  We call this the Weil homomorphism.
%

%ill define constructions using the abstract groupoidsThis will allow us to describe constructions using the Furthermore, any two choices of classifying map are homotopic (though not necessarily a flat homotopy).  In this paper, we will work with quantities that are invariant under gauge transformations (bundle isomorphisms).  Practically speaking, this means there will be no real difference between $BG$ and $\BnG$.  However, we find that many constructions are more natural using the language of stacks.  And, in general one cannot simply say that $BG$ and $\BnG$ are equivalent since, viewing them as categories, the morphisms can be different (even the homotopy type).

%
%are of type $(0,1)$ and $(2,-1)$ in the bi-grading on $\W(\fg)$; the first The Koszul operator 
%


%
%simultaneously a commutative differential graded algebra (dga) and a $\fg$-representation, and it is equipped with an algebraic notion of a connection.  (This data is often packaged together and called a $\fg$-differential algebra ($\fg$-da) with connection \cite{Meinrenken-EquivCoh}, or a $G^{\star}$ algebra with connection \cite{GuilleminSternberg99}.)  It is defined
% There is a natural bi-grading, which we denote by $S^i(\fg^*) \otimes \Lambda^j(\fg^*) = \W^{(2i,j)}(\fg)$.  The $G$-action is induced by the coadjoint representation $\fg \to \End(\fg^*)$, and the differential $d_W$ is defined 
%\[ d_W \= d_{CE} + d_K,\]
%as the sum of the Chevalley--Eilenberg differential $d_{CE}$ \cite{CE48} and the Koszul differential $d_K$.  The differentials $d_{CE}$ and $d_K$ are of type $(0,1)$ and $(2,-1)$ in the bi-grading on $\W(\fg)$; the first is defined using Lie algebra cohomology with values in the $\fg$-module $S(\fg^*)$,
%\[ S^i(\fg^*) \otimes \Lambda^j \fg^* \overset{d_{CE}}\longrightarrow S^i(\fg^*) \otimes \Lambda^{j+1}(\fg^*).\]
%The Koszul operator 
%\[ S^i(\fg^*) \otimes \Lambda^j(\fg^*) \overset{d_K}\longrightarrow S^{i+1}(\fg^*) \otimes \Lambda^{j-1}(\fg^*)\]
%is defined by extending the natural isomorphism
%\[ \Lambda^1 \fg^* \overset{\iso}\longrightarrow S^1(\fg^*)\]
%to a derivation.
%
%Has an algebraic version of a connection.  $\Theta_{\W}$. Weil homomorphism.

%\subsection{Differential forms on principal bundles}
%
%The connection $\Theta$ determines, and is equivalent to, an equivariant splitting of the tangent bundle
%\[ \xymatrix@!@1{TP \ar[r]^-{\pi_* \oplus \Theta}_-{\iso} &\pi^* TM \oplus \fg.}\]
%The \textit{curvature} $\Omega \= d\Theta + \frac{1}{2}[\Theta \wedge \Theta] \in \Omega^2(P;\fg)$, which satisfies $R_g^* \Omega = \Ad_{g^{-1}} \Omega$, measures the extent to which this splitting is not integrable.
%
%The splitting induces a bi-grading on the differential forms $\Omega^*(P)$, with
%\[ \Omega^{i,j}(P) = C^\infty(P, \pi^* \Lambda^i TM^* \otimes \Lambda^j \fg^*), \]
%and the exterior derivative $d$ decomposes as
%\[ d = (-1)^i \dCE + d_\nabla + (-1)^i d_\Omega \]
%
%Homomorphism


%\reminder{Explain integration of $\O_G(M)$.}

%The \textit{Maurer--Cartan form} 
%\[ \theta \in \Omega^1(G; \fg) \iso \Hom(TG,\fg) \]
%is defined by extending a vector to a left-invariant vector field, and it trivializes the tangent bundle $TG$.  It is commonly denoted $g^{-1}dg$ because, under the common identification $\fg \iso T_1 G$, it corresponds to the left-translation $L_{g^{-1}*}\colon T_g G \to T_1 G$.  
%

\section{Introduction}

Let $E$ be a generalized cohomology theory (or a spectrum), viewed as a functor from some category of ``nice-enough" topological spaces to abelian groups that satisfies all the Eilenberg--Steenrod axioms except for the dimension axiom.  There is now a well-established notion of a \textit{differential extension} of $E$.  It is a functor $\check E$ equipped with natural transformations to $E$ and differential forms $\Omega^*(-;\pi_*E \otimes \R)$ satisfying certain short exact sequences \cite{HS05, SS08a, BS10, 2012arXiv1208.3961B}.  There are many functors from some geometric category to an algebraic category that are called cohomology theories, but are not generalized cohomology theories in the sense of Eilenberg--Steenrod.  An interesting problem, with many potential applications, is to consider differential extensions for more general cohomology theories.  A well-known example is Deligne cohomology in the holomorphic category \cite{Bry93}.

This paper contains baby steps in the direction of constructing differential extensions of equivariant cohomology theories.  We only consider Borel $G$-equivariant cohomology, where $G$ is a compact Lie group.  For a $G$-manifold $M$, we define the differential equivariant cohomology $\D H_G(M)$ to be the differential cohomology of the Borel quotient $EG \times_G M$, where the differential forms $\Omega(EG)$ are defined as the Weil algebra $W\fg = S \fg^* \otimes \Lambda \fg^*$.  The functor $\D H_G$ fits into the usual short exact sequences for $\D H$ but with the subscript $G$ appearing everywhere, including differential forms $\Omega_G(M) = (W\fg \otimes \Omega(M))_{\basic}$.  Similar statements hold for a generalized cohomology theory $E$.

While we use the notation $\D E_G(M)$, it may be more appropriate to write $\D E_G(EG\times_G M)$ in other contexts.  For example, if one were to construct a differential extension of equivariant $K$-theory (in the sense of Atiyah--Segal), one should call that $\D K_G$.  Presumably, there would be a natural transformation $\D K_G(-) \to \D K (EG\times_G -)$ refining the usual $K_G(-) \to K (EG\times_G -)$.  Differential extensions $\D K_G$ have been constructed for finite groups \cite{Ortiz09, SzaboValentino10, 2009arXiv0905.4181B} but not yet for more general Lie groups.  The difficulty comes from the subtleties of delocalized Chern characters and de Rham models for equivariant K-theory.

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\section{Equivariant forms}\label{sec:EquivForms}

The material in this sections is standard, and useful references include \cite{GuilleminSternberg99, MathaiQuillen, Meinrenken-EquivCoh}.  Assume $G$ is a compact Lie group with Lie algebra $\fg$.  While most of the following makes for non-compact $G$, the crucial de Rham theorem fails to hold.

The Weil algebra $W\fg$ is a differential graded algebra (dga) 
\begin{gather*}W\fg \= S \fg^* \otimes \Lambda \fg^*, \\
\deg S^1\fg^* = 2, \quad \deg \Lambda^1\fg^* = 1,
\end{gather*}
where, $S$ and $\Lambda$ denote the total symmetric and exterior powers, respectively.  (Following a remark from the end of Section 3.2 in \cite{GuilleminSternberg99}) The Weil differential $d_W$ is defined by
\[ d_W \= d_{CE} + d_K,\]
the sum of the Chevalley--Eilenberg differential $d_{CE}$ \cite{CE48} and the Koszul differential $d_K$.  The Koszul differential $d_K$ is induced by the natural isomorphism $\Lambda^1 \fg^* \iso S^1 \fg^*$.
\begin{align*}
\Lambda^1 \fg^* &\overset{d_K}\longrightarrow S^1 \fg^* \oplus \Lambda^2 \fg^* \\
\psi &\longmapsto \psi + 0 \\
\ \\
S^1 \fg^* &\overset{d_K}\longrightarrow \Lambda^2 \fg^* \\
\eta &\longmapsto 0
\end{align*}
The Chevalley--Eilenberg differential $d_{CE}$ is the natural extension to $W\fg$ of the exterior derivative on the complex of left-invariant vector fields, denoted $(\Lambda \fg^*, d_\fg)$.
\begin{align*}
\Lambda^1 \fg^* &\overset{d_{CE}}\longrightarrow S^1 \fg^* \oplus \Lambda^2 \fg^* \\
\psi &\longmapsto 0 +  d_\fg \psi \\
\ \\
S^1 \fg^* &\overset{d_{CE}}\longrightarrow S^1 \fg^* \otimes \Lambda^1 \fg^* \\
\eta &\longmapsto d_{CE} \eta
\end{align*}
where
\[ \langle d_{CE} \eta, X_1 \otimes X_2 \rangle = \langle \eta, [X_1,X_2]  \rangle  \]

\begin{lemma}These above definitions are the same as the coordinate definitions appearing in the literature.
\begin{proof}
Definitions from Matthai--Quillen \cite{MathaiQuillen}:
\begin{align*}
d_W \theta^a &= \Omega^a - \frac 12 C^a_{bc} \theta^b \theta^c \\
d_W \Omega^a &= C^a_{bc} \Omega^b \theta^c 
\end{align*} 

Checking:
\begin{align*}
\langle d_{CE} \theta^a, (X_b, X_c) \rangle &= \langle -\frac 12 C^a_{rs} \theta^r \wedge \theta^s, (X_b,X_c) \rangle = -\frac 12 C^a_{rs} \left( \delta^r_b \delta^s_c - \delta^r_c \delta^s_b  \right) = -C^a_{bc} \\
\langle d_{\fg} \theta^a, (X_b, X_c) \rangle &= - \langle \theta^a, [X_b, X_c] \rangle = - \langle \theta^a, C^r_{bc} X_r \rangle = - C^a_{bc}
\end{align*}

\begin{align*}
\langle d_{CE} \Omega^a, X_b \otimes X_c \rangle &= \langle C^a_{rs} \Omega^r \theta^s, X_b X_c \rangle = C^a_{bc} \\
\langle d_{CE} \Omega^a, X_b X_c \rangle &= \langle \Omega^a, [X_b,X_c]\rangle = \langle \Omega^a, C^r_{bc} X_r \rangle = C^a_{bc}
\end{align*}
\end{proof} \end{lemma}
A good way to think about this is the following.  The complex $W\fg$ has a natural bi-grading.  The differential $d_W$ then decomposes under the bi-grading as
\[ d_W = d^{(0,1)} + d^{(2,-1)}, \quad d^{0,1} = d_{CE}, \quad d^{(2,-1)} = d_K. \]
On the subcomplex $\Lambda \fg^*$, the differential $d_{CE}$ equals the standard Lie algebra derivative $d_{\fg}$, and we require that $d_K: \Lambda^1\fg^* = \fg^* \to \fg^* = S^1 \fg^*$ is the identity map.  Drawing this gives a useful picture.
{\scriptsize
\[ \hskip -.45in \xymatrix{ &\Lambda^3 \fg^* \ar[drr]^{d_K}&0 &S^1 \fg^* \otimes \Lambda^3 \fg^*  \ar[drr]^{d_K}&0 &S^2 \otimes \Lambda^3 \fg^* \\
&\Lambda^2\fg \ar[u]_{d_\fg} \ar[drr]^{d_K} & 0
 & S^1 \fg^* \otimes \Lambda^2\fg^* \ar[u]_{d_{CE}} \ar[drr]^{d_K}& 0 & S^2 \fg^* \otimes \Lambda^2 \fg^* \ar[u]_{d_{CE}}\\
& \fg^*  \ar[u]_{d_\fg} \ar[drr]^{\iso}& 0 & S^1 \fg^* \otimes \Lambda^1 \fg^* \ar[u]_{d_{CE}} \ar[drr]^{d_K}& 0 & S^2 \fg^* \otimes \Lambda^1 \fg^* \ar[u]_{d_{CE}}\\
&\R \ar[u]_{0} & 0 & S^1 \fg^* \ar[u]_{d_{CE}} & 0 & S^2\fg^* \ar[u]_{d_{CE}}
} \] }
In order to require that $d_W$ is a differential, it must satisfy $d_W^2 = 0$.  Therefore, we are forced into our definition of $d_{CE}$ on $S^1 \fg^*$.  To see this, let $\theta^a \in \Lambda^1 \fg^*$, where $\{X_i\}$ is a basis for $\fg$, which induces bases $\{\theta^i\}$ and $\{\Omega^i\}$ on $\Lambda^1 \fg^*$ and $S^1 \fg^*$ respectively.  Let $C^i_{jk}$ be the structure constants for $\fg$.  Then,
\begin{align*}
(d_K d_{\fg} + d_{CE} d_K)\theta^a &= 0\\
d_{CE} (d_K \theta^a) &= - d_K (d_\fg \theta^a) \\
d_{CE} \Omega^a &= - d_K( - \frac 12 C^a_{ij} \theta^i \theta^j) \\
&= \frac 12 C^a_{ij} \left( \Omega^i \theta^j - \theta^i \Omega^j \right) \\
&= C^a_{ij} \Omega^i \theta^j
\end{align*}
This is the usual coordinate-based definition of $d_W \Omega^a$.  What we are really doing is creating a bi-graded cochain complex that will map to the cochain complex of forms on a principal $G$-bundle with connection.  The differential $d_W$ is defined in such a way that there are universal connection, curvature forms on $W\fg$ satisfying the usual properties.  

Define the connection $\Theta_W$ and curvature $\Omega_W$ on $W\fg$ by the following:\reminder{insert precise notion of what I mean by connection?}
\begin{align*}
\Theta_W &\in \Lambda^1\fg^* \otimes \fg \iso \Hom(\fg, \fg), \quad \Theta_W \iso \Id \\
\Omega_W &\in S^1\fg^* \otimes \fg \iso \Hom(\fg, \fg), \quad \Omega_W \iso \Id \\
\Omega_W &= d_K(\Theta_W) = d_W \Theta_W + \frac{1}{2} [\Theta_W, \Theta_W] 
\end{align*}
(In the coordinate definition, we write $\Theta = \theta^i X_i$, $\Omega = \Omega^i X_i$.)

If $(P,\Theta)\overset{\pi}\to M$ is a (right) principal $G$-bundle with connection $\Theta_P$, the differential forms on $P$ decompose as
\[ \Omega^*(P) = \Gamma(P, \pi^*\Lambda TM^* \otimes \Lambda \fg^*) \]
via the equivariant decomposition $TP = \pi^* TM \oplus \fg$.  

The complex of right-invariant forms decomposes as
\[ \Omega^*(P)^G = \Omega^*(M; \Lambda^* \fg^*_P), \]
where $\fg_P = P \times_{G} \fg$ is the adjoint bundle.  Under the natural bi-grading, the exterior derivative $d$ decomposes in terms of the Lie algebra derivative $d_{\fg}$, covariant differentiation by the connection $d_{\nabla}$, and contraction by the curvature $\iota_{\Omega}$.

\reminder{Guess: If we consider general forms $\Omega(P)$, we will see role of $d_{CE}$ in more generality.  The picture below is for $\Omega(P)^G$, and we will have $(W\fg)^G \to \Omega(P)^G$.  On $(S\fg)^G$, the invariant symmetric polynomials, the derivation $d_{CE}$ will be 0.}

{\scriptsize
\[ \hskip -.45in \xymatrix{ &\vdots &\vdots &\vdots &\vdots &\\
&\Omega^0(M;\Lambda^2\fg^*_P) \ar[u]_{d_\fg} \ar[r]^{\nabla} \ar@{.>}[drr]^>>>>{\iota_\Omega} & \Omega^1(M;\Lambda^2\fg^*_P) \ar[u]_{-d_\fg} \ar[r]^{d_\nabla} \ar@{.>}[drr]^>>>>{-\iota_\Omega}& \Omega^2(M;\Lambda^2\fg^*_P) \ar[u]_{d_\fg} \ar[r]^{d_\nabla} \ar@{.>}[drr]^>>>>{\iota_\Omega}& \Omega^3(M;\Lambda^2\fg^*_P) \ar[u]_{-d_\fg} \ar[r]^{d_\nabla}&\cdots\\
&\Omega^0(M;\fg^*_P)  \ar[u]_{d_\fg} \ar[r]^{\nabla} \ar@{.>}[drr]^>>>>{\iota_\Omega}& \Omega^1(M;\fg^*_P) \ar[u]_{-d_\fg} \ar[r]^{d_\nabla} \ar@{.>}[drr]^>>>>{-\iota_\Omega}& \Omega^2(M;\fg^*_P) \ar[u]_{d_\fg} \ar[r]^{d_\nabla} \ar@{.>}[drr]^>>>>{\iota_\Omega}& \Omega^3(M;\fg^*_P) \ar[u]_{-d_\fg} \ar[r]^{d_\nabla}&\cdots\\
&\Omega^0(M;\R) \ar[u]_{0} \ar[r]^{d_M} & \Omega^1(M;\R) \ar[u]_{0} \ar[r]^{d_M}& \Omega^2(M;\R) \ar[u]_{0} \ar[r]^{d_M}& \Omega^3(M;\R) \ar[u]_{0} \ar[r]^{d_M}&\cdots
} \] }


\begin{lemma}Linear algebra lemma: If $\theta \in \Omega^1(X;V)$, and $\Omega \in \Omega^2(X;V)$, then $\theta^{\wedge k}, \Omega^{\wedge k} \in \Omega^*(X; TV)$ (where $TV$ is the tensor algebra) live in the subspaces
\[ \theta^{\wedge k} \in \Omega^k (X; \Lambda^k V), \qquad \Omega^{\wedge k} \in \Omega^{2k}(X; S^k V).\]
\begin{proof}
This essentially follows from the graded commutative properties of the exterior algebra.  It probably has a very clean, categorical statement, but one could also prove it quite explicitly.  For 1-forms, and $\sigma \in S_k$,
\[ \sigma \left( \theta^k (X_1, \ldots, X_k) \right) = \theta^k \left(  \sigma(X_1, \ldots, X_k) \right) = |\sigma| \>\theta^k(X_1, \ldots, X_k).\]
The second statement requires a bit more care.
\end{proof} \end{lemma}

\begin{prop}Given a principal $G$-bundle $P\to M$ with connection $\theta_P$ and curvature $\Omega_P$, there is a natural $G$-equivariant homomorphism of bi-graded dgas (with connection)
\[ W\fg \overset{w}\longrightarrow \Omega(P),\]
which we refer to as the Weil homomorphism $w$.  As a consequence, $w$ descends to homomorphisms
\begin{align*}
(W\fg)^G &\overset{w}\longrightarrow \Omega(P)^G \\
S\fg^* = (W \fg)_{\basic} &\overset{w}\longrightarrow \Omega(P)_{\basic} = \pi^* \Omega(M)
\end{align*}
\begin{proof}
The Weil homomorphism $w$ is given by
\begin{align*}
S^i \fg^* \otimes \Lambda^j \fg^* &\overset{w}\longrightarrow \Omega^{2i+j}(P)\\
\omega \otimes \eta &\longmapsto \omega(\Omega_P^{\wedge i}) \wedge \eta( \Theta_P^{\wedge j})
\end{align*}
This uses the natural maps
\begin{align*}
\Omega^{2i}(P; S^i \fg) \otimes S^i \fg^* &\to \Omega^{2i}(P;\R)\\
\Omega^{\wedge i} \otimes \omega &\mapsto \omega(\Omega^{i}) \\
\ \\
\Omega^j(P;\Lambda^j \fg) \otimes \Lambda^j \fg^* &\to \Omega^j (P;\R) \\
\Theta^j \otimes \eta &\mapsto \eta(\Theta^j) 
\end{align*}

The compatibility with differentials follows from the structure equations for the connections, curvature.

Equivariance.

The connection $\Theta_W \mapsto \Theta_P$.
\end{proof}\end{prop}

\begin{prop}The dga $(W\fg, d_W)$ is acyclic; i.e. $H^i(W\fg) = 0$ except for $H^0( W\fg) = \R$.  For $G$ a compact group, the cohomology of 
\[ H^*((W\fg)_{\basic}) = (W\fg)_{\basic} = S \fg^* \iso H^*(BG;\R).\]
\begin{proof}
\end{proof}\end{prop}

Because of the Propositions above, we consider $W\fg$ to be a good replacement for the differential forms $\Omega(EG)$, and $(W\fg)_{\basic} = S\fg^*$ a good model for $\Omega(BG)$.  The complex $W\fg$ is acyclic with a free $G$-action, and there is a natural weak equivalence $W\fg \to \Omega(EG)$ induced by the Weil homomorphism after choosing a connection $\Theta_{EG}$ on $EG$.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Hopkins--Singer cocycle model}

Denote
\begin{align*} 
\Omega^\bullet(EG) &\= W^\bullet(\fg) = S^\bullet(\fg^*) \otimes \Lambda^\bullet(\fg^*), \\
\Omega^\bullet (EG \times M) &= W^\bullet(\fg) \otimes \Omega^\bullet(M), \\
\Omega_G^\bullet(M)  &\= \Omega^\bullet(EG \times_G M) = \big( W(\fg) \otimes \Omega(M) \big)_\bas \iso \big(S(\fg^*) \otimes \Omega(M) \big)^G.
\end{align*}
See Appendix \ref{sec:EquivForms} for more details.

Let $(EG, \Theta_{EG})\to BG$ be a classifying space with universal connection.  The specific model should not really matter, but we require that $G$ acts freely on $EG$, $EG$ is ``contractible," and there is a natural injective map $\w: W(\fg) \into C^*(EG;\R)$ given by the Weil homomorphism.  For $G$ compact, we can simply take $EG$ to be embedded in the Stiefel manifolds classifying $U(N)$ for some $N$.

EXPLICITLY: we assume $EG$ is an object in some (model) category that classifies $G$-bundles, there is a notion of singular cochains, $EG$ is acyclic (all the homotopy groups vanish, and $C^*(EG;A)$ is an acyclic complex), and $\Omega(EG) \into C^*(EG;\R)$ is an inclusion (as sets) and a weak equivalence.

Let $\L \subset V$ be a completely disconnected subgroup of a vector space (over $\R$ or $\C$).  Then, we use the Hopkins--Singer construction to define $\HG^\bullet(M;\L) \= \HG[]^\bullet(EG\times_G M;\L)$, where the right-hand side is interpreted using the above definition of $\Omega(EG\times_G M)$.  More concretely,

\begin{defn}For a left $G$-manifold $M$, define the differential equivariant cochain complex $\CG^\bullet(M;\L) \= \CG[]^\bullet(EG\times_G M;\L)$ by 
\begin{align*}
\CG^k(M;\L) &\= C^k(EG \times_G M;\L) \times C^{k-1}(EG\times_G M;V) \times \Omega_G^k(M;V) \\
d(c,h,\omega) &\= (\delta c, \w(\omega) - c - \delta h, d_G \omega).
\end{align*}
\end{defn}

Natural cochain maps
\[ \xymatrix{ & C_G^\bullet(\L) \\
\CG^\bullet(\L) \ar[ur]^{\cc} \ar[dr]_{\curv} \\
& \Omega_G^\bullet (V)
} \]

There is a natural descending filtration of cochain complexes $\CG(M;\L)$
\[ \CG^\bullet = F^0 \CG \supset F^1 \CG \supset \cdots \]
given by requiring
\[ \curv F^s \CG^k = 0 \text{ when } k < s. \]

\begin{equation}
\HG^k (M;\L) \= H^k \big(F^k\CG^\bullet(M;\L) \big) = \frac{\ZG^k(M;\L)}{d \CG^{k-1}(M;\L)_{\flat} }
\end{equation}

\begin{prop}Commutative square
\begin{equation}\label{eq:FundSquare} \xymatrix{ \HG^\bullet(M;\L) \ar[r]^{\cc} \ar[d]_{\curv}& H^\bullet_G(M;\L) \ar[d]\\
\Omega^k_G(M;V)_\L \ar[r]^{\dR}  & H^\bullet_G(M;V)_\L
}\end{equation}
and short exact sequences
\begin{align*}
0 \longrightarrow H^{\bullet - 1}_G(M; V/\L) \longrightarrow &\HG^\bullet(M;\L) \longrightarrow \Omega^\bullet_G(M;V)_\L \longrightarrow 0 \\
0 \longrightarrow \frac{\Omega^{\bullet -1}_G(M;V)}{\Omega^{\bullet -1}_G(M;V)_\L} \longrightarrow &\HG^\bullet(M;\L) \longrightarrow H^\bullet_G(M;\L) \longrightarrow 0 \\
0 \longrightarrow \frac{H^{\bullet - 1}_G(M;V)}{H^{\bullet - 1}_G(M;V)_\L} \longrightarrow &\HG^\bullet(M;\L) \longrightarrow A^\bullet_G(M;\L) \longrightarrow 0
\end{align*}
where $A^\bullet_G(M;\L)$ is the pullback (in sets) of the commutative square \eqref{eq:FundSquare}.
\begin{proof}Follows same proof as \cite{HS05}, and more formally, the outline in Appendix \ref{}.
\end{proof}\end{prop}

Character diagram:
\[ \xymatrix{ 0 \ar[dr]&&&&0 \\
&H_G^{\bullet - 1}(M;V/\L) \ar[dr] \ar@/^1pc/^{-B}[rr]&& H_G^{\bullet}(M;\L) \ar[ur]  \ar@/^1pc/[dr]\\
H_G^{\bullet -1}(M;V) \ar@/^1pc/[ur] \ar@/_1pc/[dr]&& \HG^{\bullet}(M;\L) \ar_{\cc}[ur] \ar^{\curv}[dr] \ar@{}^>>>>>{\dR}[rr]&& H_G^{\bullet}(M;V) \\
& {\displaystyle \frac{\Omega^{\bullet-1}_G(M;V)}{\Omega^{\bullet-1}_G(M;V)_\L}} \ar[ur] \ar@/_1pc/_{d_G}[rr]&& \Omega^{\bullet}_G(M;V)_\L \ar@/_1pc/[ur] \ar[dr]\\
0 \ar[ur]&&&&0
} \]


\begin{example}When $G=1$ is trivial, we can take $EG= \pt$, and we recover 
\[ \CG[1]^\bullet(M;\L) \iso \CG[]^\bullet(M;\L), \quad \HG[1]^\bullet(M;\L) \iso \HG[]^\bullet(M;\L).\]
\end{example}

\begin{example}
When $M=\pt$ and $G$ is compact, then $H^{2k}(BG;\R) \iso S^k(\fg^*)^G$ and $H^{2k+1}(BG;\R) = 0$.
\begin{align*}
0 \longrightarrow H^{2k-1}(BG;\R/\Z) \longrightarrow &\HG^{2k}(\pt;\Z) \longrightarrow H^{2k}(BG;\Z) \longrightarrow 0 \\
0 \longrightarrow \frac{S^k(\fg^*)^G}{S^k(\fg^*)^G_\Z} \longrightarrow &\HG^{2k+1}(\pt;\Z) \longrightarrow H^{2k+1}(BG;\Z) \longrightarrow 0
\end{align*}
%\[ \xymatrix{ 0 \ar[dr]&&&&0 \\
%&H^{2k - 1}(BG;\R/\Z) \ar[dr] \ar@/^1pc/[rr]&& H^{2k}(BG;\Z) \ar[ur]  \ar@/^1pc/[dr]\\
%0 \ar@/^1pc/[ur] \ar@/_1pc/[dr]&& \HG^{2k}(\pt;\Z) \ar[ur] \ar[dr]&& H^{2k}(BG;\R)_\Z \\
%& 0 \ar[ur] \ar@/_1pc/[rr]&& S^{k}(\fg^*)^G_\Z \ar@/_1pc/[ur]^{\iso} \ar[dr]\\
%0 \ar[ur]&&&&0
%} \]
%\[ \xymatrix{ &&H^{2k}(BG;\R/\Z) && H^{2k+1}(BG;\Z) \\
%&\frac{H^{2k}(BG;\R)}{H^{2k}(BG;\R)_\Z} && \HG^{2k+1}(\pt;\Z) && 0 \\
%&& \frac{S^{k}(\fg^*)}{S^{k}(\fg^*)_\Z} && 0
%} \]
\end{example}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Equivariant differential characters}

Consider the stack $[M/{G^\nabla}]$, and 
\[ [M/G^\nabla](\Delta^k) = \Hom_{stacks}(\Delta^k, [M/G^\nabla]) \]
and let 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Chern--Weil theory}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Hopkins--Singer: Generalized cohomology}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Relation to other models}


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


%\section{Homotopy pullbacks}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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