\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*{differential fivebrane structure}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

\hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology}

[[!include differential cohomology - contents]]

\hypertarget{chernweil_theory}{}\paragraph*{{$\infty$-Chern-Weil theory}}\label{chernweil_theory}

[[!include infinity-Chern-Weil theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{ChernWeilTheory}{Construction in terms of $L_\infty$-Cech cocycles}\dotfill \pageref*{ChernWeilTheory} \linebreak
\noindent\hyperlink{presentation_of_the_differential_class_by_a_fibration}{Presentation of the differential class by a fibration}\dotfill \pageref*{presentation_of_the_differential_class_by_a_fibration} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Where a [[fivebrane structure]] is a trivialization of a class in [[integral cohomology]], a \emph{differential fivebrane structure} is the trivialization of this class refined to [[ordinary differential cohomology]]:

the second fractional [[Pontryagin class]]

\begin{displaymath}
\frac{1}{6} p_2 : B String \to B^8 \mathbb{Z}
\end{displaymath}
in the [[(∞,1)-topos]] [[∞Grpd]] $\simeq$ [[Top]] has a refinement to $\mathbf{H} =$ [[Smooth∞Grpd]] of the form

\begin{displaymath}
\frac{1}{6}\mathbf{p}_2 : \mathbf{B}String \to \mathbf{B}^7 U(1)
\end{displaymath}
-- the .

The induced morphism on [[cocycle]] [[∞-groupoid]]s

\begin{displaymath}
\frac{1}{6}\mathbf{p}_2
  :
  \mathbf{H}(X,\mathbf{B} String)
  \stackrel{}{\to}
  \mathbf{H}(X,\mathbf{B}^7 U(1))
\end{displaymath}
sends a [[string 2-group]]-[[principal 2-bundle]] $P$ to its corresponding [[Chern-Simons circle 7-bundle]] $\frac{1}{6}\mathbf{p}_2(P)$.

A choice of trivialization of $\frac{1}{6}p_2(P)$ is a [[fivebrane structure]]. The [[n-groupoid|6-groupoid]] of smooth fivebrane structures is the [[homotopy fiber]] of $\frac{1}{6}\mathbf{p}_2$ over the trivial [[circle n-bundle with connection|circle 7-bundle]].

By [[Chern-Weil theory in Smooth∞Grpd]] this morphism may be further refined to a [[differential characteristic class]] $\frac{1}{6}\hat \mathbf{p}_2$ that lands in the [[ordinary differential cohomology]] $\mathbf{H}_{diff}(X, \mathbf{B}^7 U(1))$, classifying [[circle n-bundle with connection|circle 7-bundles with connection]]

\begin{displaymath}
\frac{1}{6}\hat \mathbf{p}_2
  :
  \mathbf{H}_{conn}(X,\mathbf{B} String)
  \stackrel{}{\to}
  \mathbf{H}_{diff}(X,\mathbf{B}^7 U(1))
\end{displaymath}
The [[n-groupoid|7-groupoid]] of \textbf{differential fivebrane structures} is the [[homotopy fiber]] of this refinement $\frac{1}{6}\hat \mathbf{p}_2$ over the [[circle n-bundle with connection|trivial circle 7-bundle with trivial connection]] or more generally over the trivial circle 7-bundles with possibly non-trivial connection.

Such a differential fivebrane structure over a [[smooth manifold]] $X$ is characterized by a tuple consisting of

\begin{enumerate}%
\item a [[connection on a 2-bundle|2-connection]] $\nabla$ on a [[string 2-group|String]]-[[principal 2-bundle]] on $X$;


\item a choice of trivial [[circle n-bundle with connection|circle 7-bundle]] with connection $(0, H_7)$, hence a differential 7-form $H_7 \in \Omega^7(X)$;


\item a choice of [[equivalence in an (∞,1)-category|equivalence]] $\lambda$ of the [[Chern-Simons circle 7-bundle]] with connection $\frac{1}{6}\hat\mathbf{p}_2(\nabla)$ of $\nabla$ with this chosen 7-bundle



\end{enumerate}
\begin{displaymath}
\lambda : \frac{1}{6}\hat \mathbf{p}_2(\nabla) \stackrel{\simeq}{\to}
    (0,H_7)
    \,.
\end{displaymath}
More generally, one can consider the [[homotopy fiber]]s of $\frac{1}{6}\hat \mathbf{p}_2$ over arbitrary circle 7-bundles with connection $\hat \mathcal{G}_8 \in \mathbf{H}_{diff}^8(X, \mathbf{B}^3 U(1))$ and hence replace $(0,H_7)$ in the above with $\hat \mathcal{G}_8$. According to the general notion of [[twisted cohomology]], these may be thought of as \textbf{twisted differential string structures}, where the class $[\mathcal{G}_8] \in H^8_{diff}(X)$ is the twist.

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

We will assume that the reader is familiar with basics of the discussion at [[Smooth∞Grpd]]. We often write $\mathbf{H} := Smooth \infty Grpd$ for short.

Let $String(n) \in$ [[Smooth∞Grpd]] be the smooth [[String 2-group]], for some $n \gt 6$,\footnote{for $n=3,4,5,6$ $String(n)$ is not 6-connected, so one must take extra care with the intervening stages in the [[Whitehead tower]].}  regarded as a [[Lie 2-group]] and thus canonically as an [[∞-group]] object in [[Smooth∞Grpd]]. We shall notationally suppress the $n$ in the following. Write $\mathbf{B}String$ for its [[delooping]] of $Spin$ in [[Smooth∞Grpd]]. (See the discussion ). Let moreover $\mathbf{B}^6 U(1) \in Smooth \infty Grpd$ be the  and $\mathbf{B}^7 U(1)$ its [[delooping]].

At [[Chern-Weil theory in Smooth∞Grpd]] the following statement is proven (\hyperlink{FSS}{FSS}).

\begin{uprop}
The image under [[Lie integration]] of the canonical [[Lie algebra cohomology|Lie algebra 7-cocycle]]

\begin{displaymath}
\mu = \langle -,[-,-], [-,-], [-,-]\rangle : \mathfrak{so} \to b^6 \mathbb{R}
\end{displaymath}
on the [[semisimple Lie algebra]] $\mathfrak{so}$ of the [[Spin group]] -- the [[special orthogonal Lie algebra]] -- is a morphism in [[Smooth∞Grpd]] of the form

\begin{displaymath}
\frac{1}{6} \mathbf{p}_2 : 
  \mathbf{B}String
    \to
  \mathbf{B}^7 U(1)
\end{displaymath}
whose image under the  $\Pi : Smooth \infty Grpd \to$ [[∞Grpd]] is the ordinary second fractional [[Pontryagin class]]

\begin{displaymath}
\frac{1}{6}p_2 : B String \to B^8 \mathbb{Z}
\end{displaymath}
in [[Top]]. Moreover, the corresponding 

\begin{displaymath}
\frac{1}{6}\hat \mathbf{p}_2 : 
  \mathbf{H}_{conn}(-,\mathbf{B}String)
  \to
  \mathbf{H}_{diff}(-, \mathbf{B}^7 U(1))
\end{displaymath}
is in [[cohomology]] the corresponding refined [[Chern-Weil homomorphism]]

\begin{displaymath}
[\frac{1}{6}\hat \mathbf{p}_String] : 
  H^1_{Smooth}(X,String) \to H_{diff}^8(X)
\end{displaymath}
with values in [[ordinary differential cohomology]] that corresponds to the second [[Killing form]] [[invariant polynomial]] $\langle - , - ,-,-\rangle$ on $\mathfrak{so}$.

\end{uprop}
\begin{udef}
For any $X \in$ [[Smooth∞Grpd]], the [[n-groupoid|6-groupoid]] of \textbf{differential fivebrane-structures} on $X$ -- $Fivebrane_{diff}(X)$ -- is the [[homotopy fiber]] of $\frac{1}{6}\hat \mathbf{p}_2(X)$ over the trivial differential cocycle.

More generally (see [[twisted cohomology]]) the 6-groupoid of \textbf{twisted differential fivebrane structures} is the [[(∞,1)-pullback]] $Fivebrane_{diff,tw}(X)$ in

\begin{displaymath}
\itexarray{
    Fivebrane_{diff,tw}(X) &\to& H_{diff}^8(X)
    \\
    \downarrow && \downarrow
    \\
    \mathbf{H}_{conn}(X,\mathbf{B}String)
      &\stackrel{\frac{1}{6}\hat \mathbf{p}_2}{\to}&
    \mathbf{H}_{diff}(X,\mathbf{B}^7 U(1))
  }
  \,,
\end{displaymath}
where the right vertical morphism is a choice of (any) one point in each [[connected component]] (differential cohomology class) of the [[cocycle]] [[∞-groupoid]] $\mathbf{H}_{diff}(X,\mathbf{B}^7 U(1))$ (the [[homotopy type]] of the [[(∞,1)-pullback]] is independent of this choice).

\end{udef}
\begin{uremark}
In terms of local [[∞-Lie algebra valued differential forms]] data this has been considered in (\hyperlink{SSSIII}{SSSIII}), as we shall discuss \hyperlink{ChernWeilTheory}{below}.

\end{uremark}
\hypertarget{ChernWeilTheory}{}\subsection*{{Construction in terms of $L_\infty$-Cech cocycles}}\label{ChernWeilTheory}

We use the [[presentable (∞,1)-category|presentation]] of the [[(∞,1)-topos]] [[Smooth∞Grpd]] (as described there) by the local [[model structure on simplicial presheaves]] $[CartSp_{smooth}^{op}, sSet]_{proj,loc}$ to give an explicit construction of twisted differential fivebrane structures in terms of [[Cech cohomology|Cech]]-cocycles with coefficients in [[∞-Lie algebra valued differential forms]].

Recall the following fact from [[Chern-Weil theory in Smooth∞Grpd]] (\hyperlink{FSS}{FSS}).

\begin{uprop}
The differential second fractional Pontryagin class $\frac{1}{6} \hat \mathbf{p}_2$ is presented in $[CartSp_{smooth}^{op}, sSet]_{proj}$ by the top morphism of simplicial presheaves in

\begin{displaymath}
\itexarray{
    \mathbf{cosk}_7\exp(\mathfrak{so})_{ChW,smp}
     &\stackrel{\exp(\mu, cs)}{\to}&
    \mathbf{B}^7 \mathbb{R}/\mathbb{Z}_{ChW,smp}     
    \\
    \downarrow && \downarrow
    \\
    \mathbf{cosk}_7\exp(\mathfrak{so})_{diff,smp}
     &\stackrel{\exp(\mu, cs)}{\to}&
    \mathbf{B}^7 \mathbb{R}/\mathbb{Z}_{smp} 
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B}String_{c}
  }
  \,.
\end{displaymath}
\end{uprop}
Here the middle morphism is the direct [[Lie integration]] of the [[infinity-Lie algebra cohomology|L-∞ algebra cocycle]] while the top morphisms is its restriction to coefficients for [[connection on an ∞-bundle|∞-connections]].

In order to compute the [[homotopy fiber]]s of $\frac{1}{6}\hat \mathbf{p}_2$ we now find a [[resolution]] of this morphism $\exp(\mu,cs)$ by a fibration in $[CartSp_{smooth}^{op}, sSet]_{proj}$. By the fact that this is a [[simplicial model category]] then also the hom of any cofibrant object into this morphism, computing the cocycle $\infty$-groupoids, is a fibration, and therefore, by the general discussion at [[homotopy pullback]], we obtain the [[homotopy fiber]]s as the ordinary [[fiber]]s of this fibration.

\hypertarget{presentation_of_the_differential_class_by_a_fibration}{}\subsubsection*{{Presentation of the differential class by a fibration}}\label{presentation_of_the_differential_class_by_a_fibration}

(\ldots{})

The discussion is analogous to that at [[differential string structure]]. To be filled in

(\ldots{})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

[[Whitehead tower]] of the [[orthogonal group]]

\begin{itemize}%
\item [[orientation]]


\item [[spin structure]]


\item [[string structure]]


\item [[fivebrane structure]]


\item [[twisted differential c-structure]]

\begin{itemize}%
\item [[differential string structure]]


\item [[supergravity C-field]]


\item \textbf{differential fivebrane structure}


\item [[differential T-duality]]



\end{itemize}


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

The local data for the [[∞-Lie algebra valued differential forms]] for the description of twisted differential fivebrane structures as above was given in

\begin{itemize}%
\item [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{Twisted differential string and fivebrane structures} ()

\end{itemize}
The full Cech-Deligne cocycles induced by this and their homotopy fibers were discussed in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{[[schreiber:Cech cocycles for differential characteristic classes]]}

\end{itemize}
A general discussion is at

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]}

\end{itemize}
in section 4.2.

[[!redirects differential fivebrane structures]]

[[!redirects twisted differential fivebrane structure]] [[!redirects twisted differential fivebrane structures]]

[[!redirects twisted differential Fivebrane structure]] [[!redirects twisted differential Fivebrane structures]]

[[!redirects smooth second fractional Pontryagin class]]



\end{document}