\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{simplicial de Rham complex}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{differential_forms_on_}{Differential forms on $\mathbf{B}G$}\dotfill \pageref*{differential_forms_on_} \linebreak
\noindent\hyperlink{reformulations_in_synthetic_differential_geometry}{Reformulations in synthetic differential geometry}\dotfill \pageref*{reformulations_in_synthetic_differential_geometry} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{simplicial de Rham complex} of a [[simplicial manifold]] $X_\bullet$ is the analog of the [[de Rham complex]] of [[differential form]]s of an ordinary [[manifold]]: it is the complex whose elements in degree $k$ may be thought of as $k$-[[differential form|form]]s on $X_\bullet$.

One useful conceptual way to think of this is to notice that a a [[simplicial object|simplicial]] [[manifold]] may be thought of as a degreewise [[representable functor|representable]] [[simplicial presheaf]] on [[Diff]] and then to realize that by way of the standard [[model structure on simplicial presheaves]] such a simplicial presheaf presents an [[∞-stack]] on [[Diff]] which we may think of as a [[Lie ∞-groupoid]]. From that perspective we expect that

\textbf{Slogan}. The simplicial de Rham complex is the complex of differential forms on a geometric [[∞-Lie groupoid]].

We shall discuss this in more detail below.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

There are several definitions that are [[quasi-isomorphism|quasi-isomorphic]]. The first one we give is the conceptually most straightforward one. The second one we give is sometimes more useful in computations.

Let $X_\bullet : \Delta^{op} \to Diff$ be a [[simplicial object|simplicial]] [[manifold]].

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(simplicial de Rahm complex, first version)}

Write

\begin{displaymath}
\mathcal{A}(X_\bullet)
  :=
  Tot \Omega^\bullet(X_\bullet)
\end{displaymath}
for the [[cochain complex]] that is the [[total complex]] of the [[double complex]] on the degreewise [[de Rham complex]] with one differential the simplicial-degree-wise de Rham differential and the other one the de Rham-degree-wise alternating sum of pullback along the face maps

\begin{displaymath}
\itexarray{
   \Omega^p(X_q) &\stackrel{\sum_i (-1)^i \delta_i^*}{\to}
    & \Omega^{p}(X_{q+1})
    \\
    \downarrow^{d_{dR}} && \downarrow^{d_{dR}}
    \\
    \Omega^{p+1}(X_q) 
     &\stackrel{\sum_i (-1)^i \delta_i^*}{\to}
    & \Omega^{p+1}(X_{q+1})
  }
  \,.
\end{displaymath}
\end{defn}
So an element $\omega \in \mathcal{A}(X_\bullet)$ in degree $n$ is a collection $(\omega^p_q \in \Omega^p(X_q))_{p+q = n}$ of ordinary differential forms.

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(simplicial de Rahm complex, second version)}

Write $\Delta^n_{Diff}$ for the standard $n$-[[simplex]] in its standard incarnations as a smooth [[manifold]] (with boundary). These arrange in the obvious way into the [[cosimplicial object]] $\Delta_{Diff} : \Delta \to Diff$.

Say that a differential form $\omega^p_q \in \Omega^\bullet(\Delta^p_{Diff}\times X_q)$ is \emph{compatible} if for each face map $\delta_i$ we have

\begin{quote}%
(some condition, need to look something up\ldots{})


\end{quote}
There is a decomposition

\begin{displaymath}
\Omega^n(\Delta^p_{Diff} \times X_q)
  \simeq
  \oplus_{k+l=n} \Omega^k(\Delta^p_{Diff}) 
   \otimes_{C^\infty(\Delta^p_{Diff} \times X_q)}
   \Omega^l(X_q)
  \,.
\end{displaymath}
This defines a bidegree and $A(X_\bullet)$ is the obvious total complex of the obvious double complex here

\begin{quote}%
will polish this up later\ldots{}


\end{quote}
\end{defn}
The following proposition says that and how these two complexes are related.

\begin{prop}
\label{}\hypertarget{}{}
\textbf{(Dupont)}

Consider the map between the two [[double complex]]es involved above which integrates each element in degree $(p,q)$ over $\Delta^p_{Diff}$. This induces a map on the corresponding total complexes

\begin{displaymath}
\int_\Delta : A(X_\bullet) \to \mathcal{A}(X_\bullet)
  \,.
\end{displaymath}
This is a [[morphism]] of [[cochain complex]]es which is a [[quasi-isomorphism]].

\end{prop}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{differential_forms_on_}{}\subsubsection*{{Differential forms on $\mathbf{B}G$}}\label{differential_forms_on_}

A main application of this technology is to the simplicial manifold $\mathbf{B}G = (\cdots G \times G \stackrel{\stackrel{\to}{\to}}{\to} G \stackrel{\to}{\to} {*})$ that represents the smooth groupoid which is the [[delooping]] of a [[Lie group]] $G$.

Dupont exhibits a [[Chern-Weil homomorphism]]

\begin{displaymath}
I(G) \to A(\mathbf{B}G)
\end{displaymath}
and constructs a canonical [[connection on a bundle|connection]] on the [[generalized universal bundle|universal G-principal-bundle]]

\begin{displaymath}
\mathbf{E}G \to \mathbf{B}G
\end{displaymath}
(which, in traditional [[simplicial group]] notation, reads $W G \to \bar W G$ -- but recall that these are ``Lie'' simplicial groups here).

\hypertarget{reformulations_in_synthetic_differential_geometry}{}\subsection*{{Reformulations in synthetic differential geometry}}\label{reformulations_in_synthetic_differential_geometry}

[[Urs Schreiber]]: This section is supposed to provide a useful reformulation of the simplicial de Rham complex in the context described at [[schreiber:∞-Lie theory]].

In the language uses there, the statement we establish is the following:

\begin{prop}
\label{}\hypertarget{}{}
Let $C =$ [[CartSp]]${}_{th}$ and $\mathbf{H} = (sPSh(C)_{proj}^{loc})^\circ$ the [[(∞,1)-category of (∞,1)-sheaves]] on [[CartSp]], the [[(∞,1)-topos]] of [[Lie ∞-groupoid]]s. This is a [[locally contractible (∞,1)-topos]] (as discussed there). Accordinly we have its [[schreiber:path ∞-groupoid]] and [[schreiber:infinitesimal path ∞-groupoid]] $\mathbf{\Pi}_{inf}(-)$.

Then

\begin{itemize}%
\item the [[schreiber:Chevalley-Eilenberg algebra]] of $\mathbf{\Pi}_{inf}(X)$ is [[quasi-isomorphism|quasi-isomorphic]] to the simplicial de Rham complex

\begin{displaymath}
CE( \Pi^{inf}(X)) \simeq \mathcal{A}(X_\bullet)
  \,.
\end{displaymath}


\end{itemize}
\end{prop}
The following discussion breaks this down and then describes the proof.

As a preparation, recall from the discussion at [[differential forms in synthetic differential geometry]] that if we pass from [[Diff]] to a [[smooth topos]] $(\mathcal{T},R)$ that models the axioms of [[synthetic differential geometry]], then for sufficiently well-behaved objects $X \in \mathcal{T}$ there is the [[infinitesimal singular simplicial complex]] $X^{(\Delta^\bullet_{inf})} : \Delta^{op} \to \mathcal{T}$, the [[simplicial object]] that in degree $k$ is the [[space]] of [[infinitesimal object|infinitesimal]] $k$-[[simplex|simplices]] in $X$.

As discussed there, this is such that under the [[Dold-Kan correspondence]] the [[cosimplicial algebra]] $Hom( X^{\Delta^\bullet_{inf}}, R )$ maps to the [[de Rham complex]] (and under the [[monoidal Dold-Kan correspondence]] to the full [[de Rham dg-algebra]]):

\begin{displaymath}
C_{Moore} : 
  C^\infty( X^{(\Delta^\bullet_{inf}}))
  \mapsto
  \Omega^\bullet(X)
  \,.
\end{displaymath}
It would be nice to have an analog of this statement for simplicial objects and the simplicial de Rham complex. I am thinking that the answer should be the following:

Let $X_\bullet : \Delta^{op} \to \mathcal{T}$ be a [[simplicial object]] that is degreewise of the sort such that the [[infinitesimal singular simplicial complex]] $(X_n)^{\Delta^\bullet_{inf}}$ exists. Use that $\mathcal{T}$ is canonically [[copower|tensored]] over $Set$ to find that simplicial objects in $\mathcal{T}$ are canonically tensored over [[simplicial set]]s. Then consider the \emph{realization}

\begin{displaymath}
\mathbf{\Pi}_{inf}(X_\bullet) := \int^{[n] \in \Delta}
    \Delta[n] \cdot X_n^{(\Delta^{\bullet}_{inf})}
\end{displaymath}
where

\begin{itemize}%
\item $\Delta[n]$ is the standard [[simplicial set|simplicial]] $n$-[[simplex]]


\item the integrand is the tensor operatoin of simplicial objects by simplicial sets


\item the integral sign denotes the [[coend]].



\end{itemize}
By the lemma \emph{expression in terms of simplicial realization} at [[schreiber:infinitesimal path ∞-groupoid]] this is the same as $\mathbf{\Pi}_{inf}(X)$.

The above proposition now reads in pedestrian terms:

\begin{prop}
\label{}\hypertarget{}{}
The [[Moore complex|Moore cochain complex]] of the [[cosimplicial algebra]] $C^\infty(\mathbf{\Pi}_{inf}(X_\bullet)) := Hom_{\mathcal{T}}(\mathbf{\Pi}_{inf}(X_\bullet),R)$ is [[quasi-isomorphism|quasi-isomorphic]] to the simplicial de Rham complex of $X_\bullet$.

\end{prop}
\begin{proof}
We use the cosimplicial and the simplicial version of the [[Eilenberg-Zilber theorem]] together with the fact that for a [[bisimplicial set]] the diagonal is given by the realization (as discussed there) $Diag F_{\bullet,\bullet} \simeq \int^{[n] \in \Delta} \Delta[n] \cdot F_{n,\bullet}$ to compute

\begin{displaymath}
\begin{aligned}
    C ( C^\infty(\mathbf{\Pi}_{inf}(X_\bullet)) )
    & :=
    C ( Hom( \int^{[n]} \Delta[n] \cdot X_n^{(\Delta^\bullet_{inf})}  
    , R))
    \\
    & \simeq
    C ( Hom( Diag X_\bullet^{(\Delta^\bullet_{inf})} , R))    
    \\
    & \simeq
    C Diag Hom( X_\bullet^{(\Delta^\bullet_{inf})} ,R)
    \\
    & \simeq_{q i}
    Tot C Hom( X_\bullet^{(\Delta^\bullet_{inf})} ,R)
    \\
    & \simeq 
    Tot \Omega^\bullet(X_\bullet)
    \,.
  \end{aligned}
\end{displaymath}
Here in the last step we used the following reasoning on the [[bisimplicial object]] $([p],[q]) \mapsto (X_p)^{(\Delta^q_{inf})}$. We know that

\begin{itemize}%
\item for fixed $p$, the normalized [[Moore complex]] of the [[cosimplicial algebra]] $Hom((X_p)^{(\Delta^\bullet_{inf})},R)$ is the [[de Rham complex]] of $X_p$ -- this is the statement about combinatorial [[differential forms in synthetic differential geometry]].


\item for fixed $q$ the [[Moore complex]] of the [[cosimplicial algebra]] $Hom((X_\bullet)^{((\Delta^q_{inf}))}, R)$ is the complex of functions on simplices whose differential is the alternating sum of pullbacks along face maps -- by the very definition of the [[Moore complex]].



\end{itemize}
This means that the simplicial de Rham complex is (quasi-isomorphic to) the total complex of the bi-cosimplicial algebra

\begin{displaymath}
\mathcal{A}(X_\bullet) \simeq
  Tot C(Hom((X_\bullet)^{(\Delta^\bullet_{inf})}, R))
  \,.
\end{displaymath}
So it remains to show that this total complex is also (quasi-isomorphic to) the [[Moore complex]] of $Hom(\mathbf{\Pi}_{inf}(X_\bullet),R)$. For this use \href{http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-4.dvi}{exercise 1.6 here} which says (transported from [[Set]] to $\mathcal{T}$) that this is the diagonal [[simplicial object]] of our [[bisimplicial object]]

\begin{displaymath}
d((X_\bullet)^{\Delta^\bullet_{inf}})
  \simeq
  \int^{[n]} \Delta^n \cdot (X_n)^{(\Delta^\bullet_{inf})}
  \,.
\end{displaymath}
This implies that the [[Moore complex]] of $Hom(\mathbf{\Pi}_{inf}(X_\bullet),R)$ is the Moore complex of the diagonal of the bisimplicial algebra $Hom(X_\bullet^{(\Delta^\bullet_{inf})}),R)$.

This way the desired statement recudes to the quasi-isomorphism

\begin{displaymath}
diag (Hom(\mathbf{\Pi}_{inf}(X_\bullet),R)
  \simeq
  Tot (Hom(\mathbf{\Pi}_{inf}(X_\bullet),R)
  \,.
\end{displaymath}
But this (even their chain-homotopy equivalence) is the content of the generalized [[Eilenberg-Zilber theorem]].

\end{proof}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Canonical references on simplicial de Rham cohomology are by [[Johan Louis Dupont]]. For instance

\begin{itemize}%
\item [[Johan Louis Dupont]], \emph{Simplicial de Rham Cohomology and characteristic classes of flat bundles} Topology 15 (1976)


\item [[Johan Louis Dupont]], \emph{A dual simplicial de Rham complex}, Lecture notes in Mathematics 1318 (1988) (\href{http://www.springerlink.com/content/65784420x146888h/}{journal})


\item chaper 3 in \emph{Curvature and Characteristic Classes}



\end{itemize}
\begin{quote}%
(I am still looking for the best survey reference\ldots{})


\end{quote}
When restricted to low degree this is closely related to or synonymous to considerations of de Rham cohomology in the context of [[differentiable stack]]s.

\begin{quote}%
(need to dig out references)


\end{quote}
A [[de Rham theorem]] for simplicial manifolds was proven in the classical

\begin{itemize}%
\item [[Raoul Bott]], [[Herbert Shulman]], [[Jim Stasheff]], \emph{On the de Rham theory of certain classifying spaces}, Advances in Mathematics, Volume 20, Issue 1, April 1976, Pages 43-56 (, \href{https://core.ac.uk/download/pdf/82496263.pdf}{pdf})

\end{itemize}
In principle closely related is the discussion of a de Rham theorem for [[∞-stack]]s as discussed in

\begin{itemize}%
\item [[nLab:Carlos Simpson|Carlos Simpson]], [[nLab:Constantin Teleman|Constantin Teleman]], \emph{de Rham theorem for $\infty$-stacks} (\href{http://math.berkeley.edu/~teleman/math/simpson.pdf}{pdf})

\end{itemize}
though a simplicial de Rham complex is only somewhat implicit in that article.

[[!redirects simplicial deRham complex]]



\end{document}