\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*{geometry of physics -- de Rham coefficients}

\begin{quote}%
This entry is gouing to contain one chapter of \emph{[[geometry of physics]]}.

previous chapters: \emph{\href{geometry+of+physics#FlatConnections}{flat connections}}

next chapter: \emph{\href{geometry+of+physics#PrincipalConnections}{principal connections}}


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{de_rham_coefficients}{}\subsection*{{De Rham coefficients}}\label{de_rham_coefficients}

\hypertarget{model_layer}{}\subsubsection*{{Model Layer}}\label{model_layer}

\hypertarget{liealgebra_valued_differential_1forms}{}\paragraph*{{Lie-algebra valued differential 1-forms}}\label{liealgebra_valued_differential_1forms}

\begin{defn}
\label{SheafOfLieAlgebraValuedForms}\hypertarget{SheafOfLieAlgebraValuedForms}{}
Let $G$ be a [[Lie group]], and write $\mathfrak{g}$ for its [[Lie algebra]]. The set of [[Lie algebra valued differential 1-forms]] is the [[tensor product]]

\begin{displaymath}
\Omega^1(U,\mathfrak{g}) = \Omega^1(U) \otimes_{\mathbb{R}} \mathfrak{g}
  \,.
\end{displaymath}
flat forms:

\begin{displaymath}
\Omega^1_{flat}(U, \mathfrak{g}) = 
  \left\{
    \omega \in \Omega^1(U,\mathfrak{g}) |
    F_\omega = \mathbf{d} \omega + [\omega, \omega] = 0
  \right\}

  \,.
\end{displaymath}
\end{defn}
(\ldots{})

This is a [[smooth space]]

\begin{displaymath}
\Omega^1_{flat}(-,\mathfrak{g}) \in Smooth 0 Type
\end{displaymath}
For $\mathfrak{g} = Lie(\mathbb{R})$ we have

\begin{displaymath}
\Omega^1(-,Lie(\mathbb{R})) = \Omega^1
\end{displaymath}
and we write

\begin{displaymath}
\Omega^1_{flat}(-,Lie(\mathbb{R})) = \Omega^1_{cl}
\end{displaymath}
Below we see

\begin{displaymath}
\flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g})
\end{displaymath}
\hypertarget{the_de_rham_complex}{}\paragraph*{{The de Rham complex}}\label{the_de_rham_complex}

\begin{itemize}%
\item [[de Rham complex]]


\item [[de Rham cohomology]]


\item [[de Rham theorem]]



\end{itemize}
Below we see that

\begin{displaymath}
\flat_{dR}\mathbf{B}^n \mathbb{R} 
    \simeq
  \flat_{dR}\mathbf{B}^n U(1)
   \simeq
  DK[ \Omega^1(-) \stackrel{\mathbf{d}}{\to} \Omega^2(-) \stackrel{\mathbf{d}}{\to}\cdots \stackrel{\mathbf{d}}{\to} \Omega^n_{cl}(-)]
 \,.
\end{displaymath}
\hypertarget{semantic_layer}{}\subsubsection*{{Semantic Layer}}\label{semantic_layer}

\hypertarget{de_rham_coefficient_objects}{}\paragraph*{{De Rham coefficient objects}}\label{de_rham_coefficient_objects}

\begin{defn}
\label{deRhamCoefficientObject}\hypertarget{deRhamCoefficientObject}{}
For $G \in Gpr(\mathbf{H})$, its \textbf{de Rham coefficient object} is the [[homotopy pullback]]

\begin{displaymath}
\flat_{dR} \mathbf{B}G \coloneqq \flat \mathbf{B}G \times_{\mathbf{B}G} *
\end{displaymath}
in

\begin{displaymath}
\itexarray{
     \flat_{dR} \mathbf{B}G &\stackrel{UnderlyingConnection}{\to}& \flat \mathbf{B}G
     \\
     \downarrow &pb& \downarrow^{\mathrlap{UnderlyingBundle}}
     \\
     * &\to& \mathbf{B}G
  }
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This pullback diagram expresses that [[generalized element|elements]] of $\flat_{dR}\mathbf{B}G$ are flat $G$-connections $\nabla \colon X \to \flat \mathbf{B}G$, def. \ref{FlatCohesiveConnection} equipped with a trivialization of their underlying $G$-principal bundle, def. \ref{UnderlyingBundleOfFlatConnection}.

\end{remark}
\hypertarget{OrdinaryDifferentialFormsFromSmoothCohesion}{}\paragraph*{{Recovering smooth differential forms from cohesive de Rham coefficients}}\label{OrdinaryDifferentialFormsFromSmoothCohesion}

Let $\mathbf{H} =$ [[Smooth∞Grpd]]. All [[smooth manifolds]] and sheaves on smooth manifolds etc. in the following are canonically regarded as objects in this $\mathbf{H} = Sh_\infty(CartSp)$.

\begin{prop}
\label{DeRhamCoefficientsOfLieGroup}\hypertarget{DeRhamCoefficientsOfLieGroup}{}
For $G$ a [[Lie group]], the de Rham coefficient object $\flat_{dR}\mathbf{B}G$, def. \ref{deRhamCoefficientObject} of its [[delooping]] is given by the [[sheaf]] of flat [[Lie algebra valued differential 1-forms]] $\Omega^1_{flat}(-,\mathfrak{g})$, def. \ref{SheafOfLieAlgebraValuedForms}, for $\mathfrak{g}$ the [[Lie algebra]] of $G$:

\begin{displaymath}
\flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g})
  \,.
\end{displaymath}
\end{prop}
This is discussed at \emph{\href{smooth+infinity-groupoid+--+structures#deRhamWithCoefficientsInBOfLieGroup}{smooth ∞-groupoid - structures - de Rham coefficients for BG with G a Lie group}}.

Write $U(1)$ for the [[circle group]] regared as a [[Lie group]] in the standard way.

\begin{prop}
\label{}\hypertarget{}{}
For $n \in \mathbb{N}$, the de Rham coefficient object $\flat_{dR}\mathbf{B}^n U(1)$, def. \ref{deRhamCoefficientObject}, of the $n$-fold [[delooping]] of $U(1)$ is given by the image under the [[Dold-Kan correspondence]]

\begin{displaymath}
DK \colon : Sh(CartSp, Ch_\bullet) \to Sh(CartSp, sSet)
  \to L_{lwhe} Sh(CartSp, sSet) \simeq \mathbf{H}
\end{displaymath}
of the truncated [[de Rham complex]] of sheaves of differential forms,

\begin{displaymath}
\begin{aligned}
   \flat_{dR}\mathbf{B}^n U(1)
   &\simeq
   \flat_{dR} \mathbf{B}^n \mathbb{R}
   \\
   & \simeq
   DK[\Omega^1(-) \stackrel{\mathbf{d}}{\to} \cdots \stackrel{\mathbf{d}}{\to} \Omega^n_{cl}(-)]
    \\ 
   & \simeq DK[\Omega^1_{cl}(-) \to 0 \to  \cdots \to 0]
  \end{aligned}
  \,.
\end{displaymath}
\end{prop}
This is discussed at \emph{\href{smooth+infinity-groupoid+--+structures#deRhamCoefficientsInBnU1}{smooth ∞-groupoid - structures - de Rham coefficients for the circle n-groups}}.

\hypertarget{syntactic_layer}{}\subsubsection*{{Syntactic Layer}}\label{syntactic_layer}

\begin{displaymath}
\begin{aligned}
    \flat_{dR}(\mathbf{B}G \colon Type)\; \colon & Type
    \\
    \coloneqq & 
     \;\;
    \sum_{\nabla \colon \flat \mathbf{B}G}
    (  UnderlyingBundle(\nabla) = * )
  \end{aligned}
\end{displaymath}
\hypertarget{MaurerCartanForms}{}\subsection*{{\textbf{Maurer-Cartan forms}}}\label{MaurerCartanForms}

\hypertarget{MaurerCartanLayerMod}{}\subsubsection*{{Model Layer}}\label{MaurerCartanLayerMod}

\hypertarget{MaurerCartanFormOnLieGroup}{}\paragraph*{{Maurer-Cartan form on a Lie group}}\label{MaurerCartanFormOnLieGroup}

\begin{itemize}%
\item [[Maurer-Cartan form]]

\end{itemize}
\begin{displaymath}
\theta_G \colon G \to \Omega^1_{flat}(-,\mathfrak{g})
\end{displaymath}
Consider

\begin{displaymath}
\flat_{dR} \mathbf{B}\mathbb{R}  = \Omega^1_{cl}
\end{displaymath}
the Maurer-Cartan form on $\mathbb{R}$ is the [[de Rham differential]]

\begin{displaymath}
\theta_{\mathbb{R}} = \mathbf{d} \colon \mathbb{R} \to \Omega^1_{cl} \hookrightarrow \Omega^1
  \,.
\end{displaymath}
\hypertarget{semantic_layer_2}{}\subsubsection*{{Semantic Layer}}\label{semantic_layer_2}

\hypertarget{maurercartan_form_on_a_cohesive_group}{}\paragraph*{{Maurer-Cartan form on a cohesive $\infty$-group}}\label{maurercartan_form_on_a_cohesive_group}

Let $\mathbf{H}$ be a [[cohesive (infinity,1)-topos]] $(\mathbf{\Pi} \dashv \flat \dashv \sharp)$. We discuss a general formulation of [[Maurer-Cartan forms]] on cohesive [[infinity-groups]]

Let $G \in Grp(\mathbf{H})$ be a [[group object in an (infinity,1)-category|group object]].

Use the [[pasting law]] together with the fact that $\flat$ is [[right adjoint]] and hence preserves [[limits]] to get $\theta$ in

\begin{displaymath}
\itexarray{
    G &\to& *
    \\
    \downarrow^{\mathrlap{\theta}} & pb & \downarrow
    \\
    \flat_{dR} \mathbf{B}G &\to& \flat \mathbf{B}G
    \\
    \downarrow &pb& \downarrow
    \\
    * &\to& \mathbf{B}G
  }
\end{displaymath}
\begin{defn}
\label{GeneralAbstractMaurerCartanForm}\hypertarget{GeneralAbstractMaurerCartanForm}{}
This is the \textbf{[[Maurer-Cartan form]]} on $G$

\begin{displaymath}
\theta \;\colon\; G \to \flat_{dR} \mathbf{B}G
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{DifferentiationOfInfinityGroupValuedFunction}\hypertarget{DifferentiationOfInfinityGroupValuedFunction}{}
For $S \;\colon\; X \to G$ a morphism, write

\begin{displaymath}
S^{-1} \mathbf{d} S 
  \coloneqq S^* \theta_G
  \;\colon\;
  X \stackrel{S}{\to}
  G
  \stackrel{\theta_G}{\to}
  \flat_{dR}\mathbf{B}G
\end{displaymath}
for its composite with the map of def. \ref{GeneralAbstractMaurerCartanForm}, hence the \textbf{pullback of the Maurer-Cartan form along $S$}. We also call this the \textbf{[[de Rham differential]]} of $S$.

\end{defn}
\hypertarget{maurercartan_forms_on_smooth_groups}{}\paragraph*{{Maurer-Cartan forms on smooth $\infty$-groups}}\label{maurercartan_forms_on_smooth_groups}

\begin{prop}
\label{}\hypertarget{}{}
For $G$ a [[Lie group]] canonically regarded in $\mathbf{H} =$[[Smooth∞Grpd]] the general abstract morphism

\begin{displaymath}
\theta_G \colon G \to \flat_{dR}\mathbf{B}G
\end{displaymath}
is identified, via the identification $\flat_{dR}\mathbf{B}G \simeq \Omega^1_{flat}(-,\mathfrak{g})$ of prop. \ref{DeRhamCoefficientsOfLieGroup} and the [[Yoneda lemma]], with the traditional [[Maurer-Cartan form]]

\begin{displaymath}
\theta_G \in \Omega^1_{flat}(G, \mathfrak{g})
  \,.
\end{displaymath}
\end{prop}
\hypertarget{CohesiveDifferentiation}{}\paragraph*{{Cohesive differentiation}}\label{CohesiveDifferentiation}

The Maurer-Cartan form on the [[line object]]

\begin{displaymath}
\theta_{\mathbb{R}} \colon \mathbb{R} \to \Omega^1_{cl}(-,\mathbb{R})
\end{displaymath}
is the [[de Rham differential]],

\begin{displaymath}
\mathbf{d} = \theta_{\mathbb{R}}
  \,.
\end{displaymath}
\hypertarget{universal_curvature_characteristic_forms}{}\paragraph*{{Universal curvature characteristic forms}}\label{universal_curvature_characteristic_forms}

For $G = \mathbf{B}^n U(1)$

\begin{displaymath}
curv \colon \mathbf{B}^n U(1) \to \flat_{dR} \mathbf{B}^{n+1}U(1)
\end{displaymath}
sends a circle $n$-bundle to the curvature of a pseudo-connection on it.

\hypertarget{syntactic_layer_2}{}\subsubsection*{{Syntactic Layer}}\label{syntactic_layer_2}

(\ldots{})



\end{document}