\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*{twisted bundle}

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

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - 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{Setup}{Setup}\dotfill \pageref*{Setup} \linebreak
\noindent\hyperlink{the_abstract_definition}{The abstract definition}\dotfill \pageref*{the_abstract_definition} \linebreak
\noindent\hyperlink{explicit_cocycles}{Explicit cocycles}\dotfill \pageref*{explicit_cocycles} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{twisted_ktheory}{Twisted K-theory}\dotfill \pageref*{twisted_ktheory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{GeneralReferences}{General}\dotfill \pageref*{GeneralReferences} \linebreak
\noindent\hyperlink{in_twisted_ktheory}{In twisted K-theory}\dotfill \pageref*{in_twisted_ktheory} \linebreak
\noindent\hyperlink{ReferencesAsSectionsOf2Bundles}{As 2-sections of 2-bundles}\dotfill \pageref*{ReferencesAsSectionsOf2Bundles} \linebreak


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

A \emph{twisted principal-bundle} is the object classified by a cocycle in [[twisted cohomology]] the way an ordinary [[principal bundle]] is the object classified by a cocycle in plain [[cohomology]] (generally in [[nonabelian cohomology]]).

For $\hat G$ a [[group]], a $\hat G$-[[principal bundle]] is classified in degree 1 [[nonabelian cohomology]] with coefficients in the [[delooping|delooped]] [[groupoid]] $\mathbf{B} \hat G$.

Given a realization of $\hat G$ as an abelian extension

\begin{displaymath}
A \to \hat G \to G
\end{displaymath}
of groups, i.e. given a [[fibration sequence]]

\begin{displaymath}
\mathbf{B}A \to \mathbf{B}\hat G \to \mathbf{B}G
\end{displaymath}
of [[groupoid]]s such that $\mathbf{B}A$ is once [[delooping|deloopable]] so that the [[fibration sequence]] continues to the right at least one step as

\begin{displaymath}
\mathbf{B}\hat G \to \mathbf{B}G \to \mathbf{B}^2 A
\end{displaymath}
the general mechanism of [[twisted cohomology]] induces a notion of \emph{twisted} $\hat G$-cohomology. The fibrations classified by this are the twisted $\hat G$-bundles.

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

We give a discussion of twisted bundles as a realization of [[twisted cohomology]] in any [[cohesive (∞,1)-topos]] $\mathbf{H}$ as described in the section . For the cases that $\mathbf{H} =$ [[ETop∞Grpd]] or $\mathbf{H} =$ [[Smooth∞Grpd]] this reproduces the traditional notion of [[topology|topological]] and [[smooth structure|smooth]] twisted bundles, respectively, whose twists are correspondingly topological or smooth [[bundle gerbe]]s/[[circle n-bundle]]s.

\hypertarget{Setup}{}\subsubsection*{{Setup}}\label{Setup}

Let $\mathbf{B}^{n-1}U(1) \in \mathbf{H}$ be the [[circle n-group]]. We shall concentrate here for definiteness on twists in $\mathbf{B}^2 U(1)$-[[cohomology]], since that reproduces the usual notions of twisted bundles found in the literature. But every other choice would work, too, and yield a corresponding notion of twisted bundles.

Fix once and for all an [[∞-group]] $G \in \mathbf{H}$ and a [[cocycle]]

\begin{displaymath}
\mathbf{c} : \mathbf{B}G \to \mathbf{B}^2 U(1)
\end{displaymath}
representing a [[characteristic class]]

\begin{displaymath}
[\mathbf{c}] \in H_{Smooth}^2(\mathbf{B}G,U(1))
\end{displaymath}
Notice that if $G$ is a [[compact space|compact]] [[Lie group]], as usual for the discussion of twisted bundles where $G = P U(n)$ is the [[projective unitary group]] in some dimension $n$, then by  we have that

\begin{displaymath}
H_{Smooth}^2(\mathbf{B}G, U(1)) \simeq H^3(B G, \mathbb{Z})
  \,,
\end{displaymath}
where on the right we have the ordinary [[integral cohomology]] of the [[classifying space]] $B G \in$ [[Top]] of $G$.

\hypertarget{the_abstract_definition}{}\subsubsection*{{The abstract definition}}\label{the_abstract_definition}

Let $G$ and $\mathbf{c}$ be as \hyperlink{Setup}{above}.

\begin{defn}
\label{TheGroupExtension}\hypertarget{TheGroupExtension}{}
Write

\begin{displaymath}
\mathbf{B}\hat G \to \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^2 U(1)
\end{displaymath}
for the [[homotopy fiber]] of $\mathbf{c}$.

\end{defn}
This identifies $\hat G$ as the [[group extension]] of $G$ by the 2-[[cocycle]] $\mathbf{c}$.

\begin{note}
\label{TheGroupExtensionAsACircleBundle}\hypertarget{TheGroupExtensionAsACircleBundle}{}
Equivalently this means that

\begin{displaymath}
\mathbf{B}U(1) \to \mathbf{B}\hat G \to \mathbf{B}G
\end{displaymath}
is the smooth [[circle n-bundle|circle 2-bundle]]/[[bundle gerbe]] classified by $\mathbf{c}$; and its [[loop space object]]

\begin{displaymath}
U(1) \to \hat G \to G
\end{displaymath}
the corresponding [[circle group]] [[principal bundle]] on $G$.

\end{note}
Let $X \in \mathbf{H}$ be any object. From \emph{[[twisted cohomology]]} we have the following notion.

\begin{defn}
\label{AbstractDefinition}\hypertarget{AbstractDefinition}{}
The degree-1 \textbf{total twisted cohomology} $H_{tw}^1(X, \hat G)$ of $X$ with coefficients in $\hat G$, def. \ref{TheGroupExtension}, relative to the characteristic class $[\mathbf{c}]$ is the set

\begin{displaymath}
H^1_{tw}(X, \hat G) := \pi_0 \mathbf{H}_{tw}(X, \mathbf{G}\hat H)
\end{displaymath}
of connected components of the [[(∞,1)-pullback]]

\begin{displaymath}
\itexarray{  
    \mathbf{H}_{tw}(X, \mathbf{B}\hat G) &\stackrel{tw}{\to}& 
      H_{Smooth}^2(X,U(1))
    \\
     \downarrow && \downarrow
    \\
    \mathbf{H}(X, \mathbf{B}G)
     &\stackrel{\mathbf{c}_*}{\to}&
    \mathbf{H}(X, \mathbf{B}^2 U(1))
  }
  \,,
\end{displaymath}
where the right verticsl morphism is any [[section]] of the truncation projection from cocycles to cohomology classes.

Given a twisting class $[\alpha] \in H^2_{Smooth}(U(1))$ we say that

\begin{displaymath}
H_{[\alpha]}^1(X,\hat G)
  :=
  H^1_{tw}(X, \hat G) \times_{[\alpha]} *
\end{displaymath}
is the $[\alpha]$-\textbf{twisted cohomology} of $X$ with coefficients in $\hat G$ relative to $\mathbf{c}$.

\end{defn}
\begin{note}
\label{VanishingTwistGivesOrdinaryBundles}\hypertarget{VanishingTwistGivesOrdinaryBundles}{}
For $[\alpha] = 0$ the trivial twist, $[\alpha]$-twisted cohomology coincides with ordinary cohomology:

\begin{displaymath}
H^1_{[\alpha] = 0}(X, \hat G)
  \simeq
  H^1_{Smooth}(X, \hat G)
  \,.
\end{displaymath}
\end{note}
By the discussion at \emph{[[principal ∞-bundle]]} we may identify the elements of $H^1_{Smooth}(X, \hat G)$ with $\hat G$-[[principal ∞-bundle]]s $P \to X$. In particular if $\hat G$ is an ordinary [[Lie group]] and $X$ is an ordinary [[smooth manifold]], then these are ordinary $\hat G$-[[principal bundle]]s over $X$. This justifies equivalently calling the elements of $H^1_{tw}(X,\hat G)$ \textbf{twisted principal $\infty$-bundles}; and we shall write

\begin{displaymath}
\hat G TwBund(X) := H^1_{tw}(X, \hat G)
  \,,
\end{displaymath}
where throughout we leave the characteristic class $[\mathbf{c}]$ with respect to which the twisting is defined implcitly understood.

\hypertarget{explicit_cocycles}{}\subsubsection*{{Explicit cocycles}}\label{explicit_cocycles}

We unwind the abstract definition, def. \ref{AbstractDefinition}, to obtain the explicit definition of twisted bundles by [[Cech cohomology|Cech cocycles]] the way they appear in the traditional literature (see the \hyperlink{GeneralReferences}{General References} below).

\begin{prop}
\label{CechCocyclesForTwisted1Bundles}\hypertarget{CechCocyclesForTwisted1Bundles}{}
Let $U(1) \to \hat G \to G$ be a [[group extension]] of [[topological group]]s.

Let $X \in$ [[Mfd]] $\hookrightarrow$ [[ETop∞Grpd]] $=: \mathbf{H}$ be a [[paracompact topological space|paracompact]] [[topological manifold]] with [[good open cover]] $\{U_i \to X\}$.

\begin{enumerate}%
\item Relative to this every twisting cocycle $[\alpha] \in H^2_{ETop}(X, U(1))$ is a [[Cech cohomology]] representative given by a collection of functions

\begin{displaymath}
\{ \alpha_{i j k} : U_i \cap U_j \cap U_k \to U(1) \}
\end{displaymath}
satisfying on every quadruple intersection the equation

\begin{displaymath}
\alpha_{i j k} \alpha_{i k l} = \alpha_{j k l} \alpha_{i j l}
  \,.
\end{displaymath}

\item I terms of this cocycle data the twisted cohomology $H^1_{[\alpha]}(X, \hat G)$ is given by [[equivalence class]]es of [[cocycle]]s consisting of

\begin{enumerate}%
\item collections of functions

\begin{displaymath}
\{g_{i j} : U_i \cap U_j \to \hat G \}
\end{displaymath}
subject to the condition that on each triple overlap the equation

\begin{displaymath}
g_{i j} \dot g_{j k} = g_{i k} \cdot \alpha_{i j k}
\end{displaymath}
holds, where on the right we are injecting $\alpha_{i j k}$ via $U(1) \to \hat G$ into $\hat G$\newline and then form the product there;


\item subject to the [[equivalence relation]] that identifies two such collections of cocycle data $\{g_{i j}\}$ and $\{g'_{i j}\}$ if there exists functions

\begin{displaymath}
\{h_i : U_i \to \hat G\}
\end{displaymath}
and

\begin{displaymath}
\{\beta_{i j} : U_i \cap U_j \to \hat U(1)\}
\end{displaymath}
such that

\begin{displaymath}
\beta_{i j} \beta_{j k} = \beta_{i k}
\end{displaymath}
and

\begin{displaymath}
g'_{i j} = h_i^{-1} \cdot g_{i j} \cdot h_j \cdot \beta_{i j}   
  \,.
\end{displaymath}


\end{enumerate}


\end{enumerate}
\end{prop}
\begin{proof}
We pass to the standard [[presentable (∞,1)-category|presentation]] of [[ETop∞Grpd]] by the projective local [[model structure on simplicial presheaves]] over the [[site]] [[CartSp]]. We then compute the defining [[(∞,1)-pullback]] by a [[homotopy pullback]] there.

Write $\mathbf{B}\hat G_{c}, \mathbf{B}^2 U(1)_c \in [CartSp^{op}, sSet]$ etc. for the standard models of the abstract objects of these names by simplicial presheaves. Write accordingly $\mathbf{B}(U(1) \to \hat G)_c$ for the delooping of the [[crossed module]] associated to the central extension $\hat G \to G$.

In terms of this the [[characteristic class]] $\mathbf{c}$ is represented by the [[∞-anafunctor]]

\begin{displaymath}
\itexarray{
    \mathbf{B}(U(1) \to \hat G)_c &\stackrel{\mathbf{c}}{\to}&
     \mathbf{B}(U(1) \to 1)_c
     =
     \mathbf{B}^2 U(1)_c
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B}G_c
  }
  \,,
\end{displaymath}
where the top horizontal morphism is the evident projection onto the $U(1)$-labels. Moreover, the [[Cech nerve]] of the good open cover $\{U_i \to X\}$ forms a cofibrant [[resolution]]

\begin{displaymath}
\emptyset \hookrightarrow C(\{U_i\}) \stackrel{\simeq}{\to} X
\end{displaymath}
and so $\alpha$ is presented by an [[∞-anafunctor]]

\begin{displaymath}
\itexarray{
     C(\{U_i\}) &\stackrel{\alpha}{\to}& \mathbf{B}^2 U(1)_c
     \\
     \downarrow^{\mathrlap{\simeq}}
     \\
     X
  }
  \,.
\end{displaymath}
Using that $[CartSp^{op}, sSet]_{proj}$ is a [[simplicial model category]] this means in conclusion that the [[homotopy pullback]] in question is given by the ordinary [[pullback]] of [[simplicial set]]s

\begin{displaymath}
\itexarray{
     \mathbf{H}^1_{[\alpha]}(X,\hat G) &\to& *
     \\
     \downarrow && \downarrow^{\mathrlap{\alpha}}
     \\
     [CartSp^{op}, sSet](C(\{U_i\}), \mathbf{B}(U(1) \to \hat G)_c)
     &\stackrel{\mathbf{c}_*}{\to}&
     [CartSp^{op}, sSet](C(\{U_i\}), \mathbf{B}^2 U(1)_c)
  }
  \,.
\end{displaymath}
An object of the resulting simplicial set is then seen to be a simplicial map $g : C(\{U_i\}) \to \mathbf{B}(U(1) \to \hat G)_c$ that assigns

\begin{displaymath}
g
  \;\;
  :
  \;\;
  \itexarray{
    && (x,j)
    \\
    & \nearrow &\Downarrow& \searrow
    \\
    (x,i)
    &&\to&&
    (x,k)
  }
  \;\;\;\;
  \mapsto
  \;\;\;\;
  \itexarray{
    && \bullet
    \\
    & {}^{\mathllap{g_{i j}(x)}}\nearrow 
     &\Downarrow^{\alpha_{i j k}(x)}& 
     \searrow^{\mathrlap{g_{j k}(x)}}
    \\
    \bullet
    &&\underset{g_{i k}(x)}{\to}&&
    \bullet
  }
\end{displaymath}
such that projection out along $\mathbf{B}(U(1) \to \hat G)_c \to \mathbf{B}(U(1) \to 1)_c = \mathbf{B}^2 U(1)_c$ produces $\alpha$.

Similarily for the morphisms. Writing out what these diagrams in $\mathbf{B}(U(1) \to \hat G)_c$ mean in equations, one finds the formulas claimed above.

\end{proof}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

(\ldots{})

Consider the extension $U(1) \to U(n) \to P U(n)$ of the [[projective unitary group]] to the [[unitary group]] for all $n$. Then [[direct sum]] of matrices gives a sum operation

\begin{displaymath}
H^1_{[\alpha]}(X, P U(n_1))
  \times
  H^1_{[\alpha]}(X, P U(n_2))
  \to 
  H^1_{[\alpha]}(X, P U(n_1 + n_2))
\end{displaymath}
and a tensor product operation

\begin{displaymath}
H^1_{[\alpha_1]}(X, P U(n))
  \times
  H^1_{[\alpha_2]}(X, P U(n))
  \to 
  H^1_{[\alpha_1]+ [\alpha_2]}(X, P U(n_1 \cdot n_2))
\end{displaymath}
(\ldots{})

\hypertarget{twisted_ktheory}{}\subsubsection*{{Twisted K-theory}}\label{twisted_ktheory}

[[equivalence class|Equivalence classes]] of twisted $U(n)$-bundles for fixed $\mathbf{B}U(1)$-twist $\alpha$ form a model for [[topological K-theory|topological]] $\alpha$-[[twisted K-theory]]. See there for details.

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

\begin{itemize}%
\item [[twisted K-theory]]


\item [[Chan-Paton bundle]]


\item [[twisted spin structure]]


\item [[twisted ∞-bundle]]



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

\hypertarget{GeneralReferences}{}\subsubsection*{{General}}\label{GeneralReferences}

The notion and term \emph{twisted bundle} (with finite rank) apparently first appears in

\begin{itemize}%
\item [[Marco Mackaay]], \emph{A note on the holonomy of connections in twisted bundles} (\href{http://arxiv.org/abs/math/0106019}{arXiv:math/0106019}) .

\end{itemize}
The equivalent notion of [[gerbe module]] apparently appears first in

\begin{itemize}%
\item [[Ernesto Lupercio]], [[Bernardo Uribe]], \emph{Gerbes over Orbifolds and Twisted K-theory} (\href{http://arxiv.org/abs/math/0105039}{arXiv:math/0105039}) ,

\end{itemize}
there explicitly in terms of [[Cech cohomology|Cech cocycles]] relative to an [[open cover]]. The generalization to infinite rank and arbitrary covering morphisms was amplified in (\hyperlink{CBMMS}{CBMMS}) below.

Discussion of a [[splitting principle]] for twisted vector bundles (phrased in terms of [[gerbe modules]]) is in

\begin{itemize}%
\item [[Atsushi Tomoda]], \emph{On the splitting principle of bundle gerbe modules}, Osaka J. Math. Volume 44, Number 1 (2007), 231-246. (\href{https://projecteuclid.org/euclid.ojm/1174324334}{Euclid}, talk slides \href{http://ton.prosou.nu/official/ryousi2005.pdf}{pdf})

\end{itemize}
\hypertarget{in_twisted_ktheory}{}\subsubsection*{{In twisted K-theory}}\label{in_twisted_ktheory}

Just as [[vector bundle]]s model cocycles in [[K-theory]], twisted vector bundles model cocycles in [[twisted K-theory]].

For twists $c$ that are torsion class (i.e. have finite order as group elements in the [[cohomology group]] $H(X,\mathbf{B}^2 A)$ ) this was realized in

\begin{itemize}%
\item [[Alan Carey]], [[Peter Bouwknegt]], [[Varghese Mathai]], [[Michael Murray]] and [[Danny Stevenson]], \emph{K-theory of bundle gerbes and twisted K-theory} , Commun Math Phys, 228 (2002) 17-49 (\href{http://arxiv.org/abs/hep-th/0106194}{arXiv})

\end{itemize}
which also, apparently, is the source where gerbe modules as such were first introduced.

The generalization of this construction to non-torsion twists requires using [[vectorial bundle]]s instead of plain [[vector bundle]]s. Full twisted K-theory in terms of twisted vectorial bundles was realized in

\begin{itemize}%
\item [[Kiyonori Gomi]], \emph{Twisted K-theory and finite-dimensional approximation} (\href{http://arxiv.org/abs/0803.2327}{arXiv})

\end{itemize}
There the twisted cocycle equation discussed above appears on the bottom of page 7.

Then there is

\begin{itemize}%
\item [[Max Karoubi]], \emph{Twisted bundles and twisted K-theory}, Clay Mathematics Proceedings, Volume 19 (2011) (\href{http://www.math.jussieu.fr/~karoubi/Publications/89.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item [[Ulrich Pennig]], \emph{Twisted K-theory with coefficients in $C^\ast$-algebras}, (\href{http://arxiv.org/abs/1103.4096}{arXiv:1103.4096})

\end{itemize}
\hypertarget{ReferencesAsSectionsOf2Bundles}{}\subsubsection*{{As 2-sections of 2-bundles}}\label{ReferencesAsSectionsOf2Bundles}

The observation that twisted vector bundles may be understood as higher-order [[sections]] of [[2-vector bundle]]s associated with [[circle n-bundle with connection|circle 2-bundles]]/[[bundle gerbes]] appears in

\begin{itemize}%
\item [[Urs Schreiber]], \emph{Quantum 2-States: Sections of 2-vector bundles} Talk at \emph{Higher categories and their applications}, Fields institute (2007) ([[Schreiber2States.pdf:file]]).

\end{itemize}
A discussion of this with [[connection on a bundle|2-connections]] taken into account is in section 4.4.3 of

\begin{itemize}%
\item [[Urs Schreiber]], [[Konrad Waldorf]], \emph{Connections on non-abelian Gerbes and their Holonomy} (\href{http://arxiv.org/abs/0808.1923}{arXiv:0808.1923})

\end{itemize}
A discussion in the context of [[principal infinity-bundles]] (as opposed to higher vector bundles), is in section ``2.3.5 Twisted cohomology and sections'' and then in section ``3.3.7.2 Twisted 1-bundles -- twisted K-theory''

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

\end{itemize}
The observation then re-appears independently in

\begin{itemize}%
\item [[Chris Rogers]], \emph{Higher geometric quantization}, talk at \emph{Higher Structures} in G\"o{}ttingen (2011) (\href{http://www.crcg.de/wiki/Higher_geometric_quantization}{pdf slides})

\end{itemize}
[[!redirects twisted bundles]]

[[!redirects twisted vector bundle]] [[!redirects twisted vector bundles]]

[[!redirects twisted unitary bundle]] [[!redirects twisted unitary bundles]]



\end{document}