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\section*{principal 2-bundle}

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

\hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles}

[[!include bundles - contents]]

\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Classification}{Classification}\dotfill \pageref*{Classification} \linebreak
\noindent\hyperlink{connections}{Connections}\dotfill \pageref*{connections} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{tfolds}{T-Folds}\dotfill \pageref*{tfolds} \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}

A \emph{principal 2-bundle} is the generalization of a $G$-[[principal bundle]] over a [[group]] $G$ to a principal structure over a [[2-group]]. It is a special case of a [[principal ∞-bundle]].

For $G = AUT(H)$ the [[automorphism 2-group]] of a group $H$, $G$-principal bundles are equivalent to $H$-[[gerbe]] (see [[gerbe (general idea)]] for more background.). An $H$-[[nonabelian bundle gerbe]] is a model for the total space of an $AUT(H)$-principal 2-bundle.

An expository introduction to the concepts is at [[infinity-Chern-Weil theory introduction]].

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

For $G$ a topological [[Lie 2-group]], a topological or smooth \textbf{$G$-principal 2-bundle} $P \to X$ is a topological or [[Lie groupoid]] that arises as the [[homotopy fiber]] of a [[cocycle]] $X \to \mathbf{B}G$ in [[ETop∞Grpd]] or [[Smooth∞Grpd]], respectively, i.e. as an [[(∞,1)-pullback]] of the form

\begin{displaymath}
\itexarray{
    P &\to& *
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow
    \\
    X &\to& \mathbf{B}G
  }
  \,.
\end{displaymath}
By the general rules of [[homotopy pullbacks]], this may be modeled by an ordinary pullback of topological or [[Lie 2-groupoid]]s of the form

\begin{displaymath}
\itexarray{
    P &\to& \mathbf{E}G
    \\
    \downarrow && \downarrow
    \\
    C(U) &\to& \mathbf{B}G
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    X
  }
  \,,
\end{displaymath}
where $C(U)$ is the [[Cech nerve]] of a [[good open cover]] $U \to X$ and where $\mathbf{E}G$ is the [[universal principal infinity-bundle|universal principal 2-bundle]] (\hyperlink{RS}{RS}). This says that principal 2-bundles are classified by [[Cech cohomology]] with coefficients in [[delooping]]s of (sheaves of) [[2-group]]s.

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

\hypertarget{Classification}{}\subsubsection*{{Classification}}\label{Classification}

\begin{theorem}
\label{}\hypertarget{}{}
Let $G$ be a [[well-pointed simplicial topological group|well pointed topological 2-group]]. Write $\mathbf{B}G := \bar W G$ for its [[delooping]] [[simplicial topological space]] and

\begin{displaymath}
B G := \vert \mathbf{B}G\vert \in Top
\end{displaymath}
for the corresponding [[geometric realization of simplicial topological spaces]]. Then $B G$ is a classifying space for topological $G$-principal 2-bundles: for $X$ a (sufficiently nice\ldots{}) [[topological space]] we have that the [[nonabelian cohomology|nonabelian]] [[Cech cohomology]] on $X$ with coefficients in $\mathbf{B}G$ is naturally in bijections with the set of [[homotopy]] classes of [[continuous function]]s $X \to B G$

\begin{displaymath}
H_{Top}(X, \mathbf{B}G) \simeq [X, B G]
  \,.
\end{displaymath}
\end{theorem}
This appears as (\hyperlink{BaezStevenson}{BaezStevenson, theorem 1}). It is also a special case of the analogous theorem for topological [[principal infinity-bundles]] in (\hyperlink{RobertsStevenson}{RobertsStevenson}).

\begin{theorem}
\label{}\hypertarget{}{}
Let $G$ be a [[Lie 2-group]] with the property that $\pi_0 G$ is a [[smooth manifold]] and the projection $G_0 \to \pi_0 G$ is a [[submersion]]. Then equivalence classes of smooth $G$-principal bundles on a [[smooth manifold]] $X$ are in natural bijection with equivalence classes of topological $G$-principal 2-bundles (regarding $G$ as a topological 2-group)

\begin{displaymath}
H_{Top}(X, \mathbf{B}G) \simeq H_{smooth}(X, \mathbf{B}G)
\end{displaymath}
induced by the natural [[forgetful functor]] [[SmoothMfd]] $\to$ [[Top]].

\end{theorem}
This appears as (\hyperlink{NikolausWaldorf11}{Nikolaus-Waldorf 11, prop. 4.1}).

\begin{cor}
\label{}\hypertarget{}{}
For [[Lie 2-group]]s with the above property, also smooth $G$-principal 2-bundles have [[classifying space]] $B G = \vert \mathbf{B}G\vert$.

\end{cor}
\hypertarget{connections}{}\subsection*{{Connections}}\label{connections}

An ordinary [[principal bundle]] may be equipped with a [[connection on a bundle|connection]] by refining the cocycle

\begin{displaymath}
X \to \mathbf{B} G
\end{displaymath}
to a cocycle

\begin{displaymath}
P_1(X) \to \mathbf{B} G
\end{displaymath}
where $P_1(X)$ is the [[path groupoid]] of $X$.

Similarly, 2-bundles may be equipped with connections by refining their cocycles $X \to \mathbf{B}H$ to cocycles out of a higher path groupoid. Details on this are at [[schreiber:differential cohomology in a cohesive topos]].

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

\hypertarget{tfolds}{}\subsubsection*{{T-Folds}}\label{tfolds}

Principal 2-bundles for the [[T-duality 2-group]] serve to model [[T-folds]] in [[string theory]] (\hyperlink{NikolausWaldorf18}{Nikolaus-Waldorf 18})

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

\begin{itemize}%
\item [[principal bundle]] / [[torsor]] / [[associated bundle]]


\item \textbf{principal 2-bundle} / [[gerbe]] / [[bundle gerbe]]

\begin{itemize}%
\item [[principal 2-connection]]


\item [[central extension of groupoids]]


\item [[line 2-bundle]]



\end{itemize}

\item [[principal 3-bundle]] / [[2-gerbe]] / [[bundle 2-gerbe]]


\item [[principal ∞-bundle]] / [[associated ∞-bundle]] / [[∞-gerbe]]

\begin{itemize}%
\item [[circle n-bundle with connection]]

\end{itemize}


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

The general description of higher bundles internal to generalized spaces modeled as [[∞-stacks]] is discussed in

\begin{itemize}%
\item Jardine, \emph{Cocycle categories} (\href{http://arxiv.org/abs/math.AT/0605198}{pdf}) .

\end{itemize}
The above situation of ordinary $G$-[[principal bundles]] is section 2.1 \emph{[[torsor|Torsors]] for [[sheaves]] of groups} in that article. The generalization to principal 2-bundles and [[principal ∞-bundles]] is then briefly indicated in section 2.2, \emph{Diagrams and torsors} .

The point is that in the [[(∞,1)-topos]] of topological or smooth or whatever [[∞-groupoids]] (i.e. in the [[(∞,1)-category]] of [[∞-stacks]] on our category of [[space and quantity|test spaces]]) the above situation generalizes straightforwardly:

For $G$ a [[2-group]], a $G$-principal $2$-bundle is a fibration of groupoids $P \to X$ that arises as the [[fibration sequence|homotopy fiber]] of a classifying morphism $X \to \mathbf{B}G$ (a $2$-[[2-anafunctor|anafunctor]])

\begin{displaymath}
\itexarray{
  P &\to& {*}
  \\
  \downarrow && \downarrow
  \\
  X &\to& \mathbf{B}G
 }
  \,.
\end{displaymath}
This may be modeled by the pullback of the [[universal principal infinity-bundle|universal principal 2-bundle]] as described in

\begin{itemize}%
\item [[Urs Schreiber]], [[David Roberts]], \emph{The inner automorphism 3-group of a strict 2-group} (\href{http://arxiv.org/abs/0708.1741}{arXiv})

\end{itemize}
As ordinary [[principal bundles]], the gadgets obtained this way may be described from various points of view, using [[anafunctor]] cocycles $g : X \stackrel{\simeq}{\to}\leftarrow Y \to \mathbf{B}H$ in [[nonabelian cohomology]], or the corresponding total spaces being 2-[[torsors]] equipped with 2-group [[action]], or certain variants of this.

Maybe the earliest explicit description of a principal $\infty$-bundle using a [[geometric definition of higher category]] is

\begin{itemize}%
\item P. Glenn, \emph{Realization of cohomology classes in arbitrary exact categories}, J. Pure Appl. Algebra 25, 1982, no. 1, 33- 105.

\end{itemize}
This describes [[torsors]] over [[∞-groupoids]] in terms of the corresponding $\infty$-[[action groupoids]].

This theory of higher bundles and [[gerbes]] was made to look manifestly like a systematic [[vertical categorification|categorification]] of the familiar description of ordinary [[principal bundles]] in terms of cocycles and local trivializations in

\begin{itemize}%
\item Luca Mauri, PhD thesis, 1998 (\href{http://home.aubg.bg/faculty/aganchev/Category%20Theory/Mauri%20%28papers%29/mauri-2.pdf}{pdf});

\end{itemize}
An abridged version is

\begin{itemize}%
\item L. Mauri, M. Tierney, \emph{Two-descent, two-torsors and local equivalence} , J. Pure Appl. Algebra 143 (1999), 313--327.

\end{itemize}
The first article in the differential-geometric context was

\begin{itemize}%
\item [[Toby Bartels]], \emph{2-Bundles} (\href{http://arxiv.org/abs/math.CT/0410328}{arXiv}, \href{http://toby.bartels.name/2bundles/}{web})

\end{itemize}
One should notice that if one uses categories internal to [[diffeological spaces]], then these are (under their [[nerve]]) in particular [[simplicial presheaf|simplicial presheaves]], and that the [[anafunctors]] used as morphisms between these simplicial presheaves represent precisely the morphisms the corresponding [[(∞,1)-category of (∞,1)-sheaves]] using the [[model structure on simplicial presheaves]] or, more lightweight, the structure of a Brown [[category of fibrant objects]] on $\infty$-groupoid valued sheaves.

A description of higher principal bundles (see also [[principal ∞-bundle]]) in terms of the [[model structure on simplicial presheaves]] appears in

\begin{itemize}%
\item Jardine, Luo, \emph{Higher order principal bundles} (\href{http://www.math.uiuc.edu/K-theory/0681/cocycles6.pdf}{pdf})

\end{itemize}
The relation of such 2-categorical constructions of 2-bundles to the one of simplicially modeled $\infty$-bundles by Glenn was established in

\begin{itemize}%
\item [[Igor Bakovic]], \emph{Bigroupoid 2-torsors} (\href{http://edoc.ub.uni-muenchen.de/9209/1/Bakovic_Igor.pdf}{pdf}).

\end{itemize}
Still more explicit descriptions of these constructions are given in

\begin{itemize}%
\item Christoph Wockel, \emph{A global perspective to gerbes and their gauge stacks} (\href{http://arxiv.org/abs/0803.3692}{arXiv}) .

\end{itemize}
These constructions either work internal to [[Diff]] or internal to some [[topos]].

More generally, a principal 2-bundle is a ([[n-truncated object of an (infinity,1)-category|2-truncated]] [[principal ∞-bundle]]) in a [[(∞,1)-topos]] of [[∞-stack]]s over some [[site]].

This is for instance in

\begin{itemize}%
\item [[Behrang Noohi]], E. Aldrovandi, \emph{Butterflies II: Torsors for 2-group stacks},\href{http://arxiv.org/abs/0910.1818}{arXiv}

\end{itemize}
Notice that [[torsor]] is just another word for (internal) [[principal bundle]].

Classification results of principal 2-bundles are in

\begin{itemize}%
\item [[John Baez]], [[Danny Stevenson]], \emph{The classifying space of a topological 2-group} Algebraic Topology Abel Symposia, 2009, Volume 4, 1-31 (\href{http://arxiv.org/abs/0801.3843}{arXiv:0801.3843})

\end{itemize}
\begin{itemize}%
\item [[David Roberts]], [[Danny Stevenson]], \emph{Simplicial principal bundles in parametrized spaces} ()

\end{itemize}
An extensive discussion of various models of principal 2-bundles is in

\begin{itemize}%
\item [[Thomas Nikolaus]], [[Konrad Waldorf]], \emph{Four Equivalent Versions of Non-Abelian Gerbes} (\href{http://arxiv.org/abs/1103.4815}{arXiv:1103.4815})

\end{itemize}
For a comprehensive account in the general context of [[principal infinity-bundles]] see

\begin{itemize}%
\item [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], \emph{[[schreiber:Principal ∞-bundles -- theory, presentations and applications]]} (\href{http://arxiv.org/abs/1207.0248}{arXiv:1207.0248}, \href{http://arxiv.org/abs/1207.0249}{arXiv:1207.0249})

\end{itemize}
For more references see at \emph{[[principal 2-connection]]}.

The example for structure group the [[T-duality 2-group]] is discussed, as a formalization of [[T-folds]], in

\begin{itemize}%
\item [[Thomas Nikolaus]], [[Konrad Waldorf]], \emph{Higher geometry for non-geometric T-duals} (\href{https://arxiv.org/abs/1804.00677}{arXiv:1804.00677})

\end{itemize}
[[!redirects principal 2-bundles]]



\end{document}