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\begin{document}

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\section*{principal infinity-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{GeneralPrincBund}{$G$-principal $\infty$-bundles}\dotfill \pageref*{GeneralPrincBund} \linebreak
\noindent\hyperlink{connections_on_principal_bundles}{Connections on $G$-principal $\infty$-bundles}\dotfill \pageref*{connections_on_principal_bundles} \linebreak
\noindent\hyperlink{concrete_realizations}{Concrete realizations}\dotfill \pageref*{concrete_realizations} \linebreak
\noindent\hyperlink{in_topological_spaces}{In topological spaces}\dotfill \pageref*{in_topological_spaces} \linebreak
\noindent\hyperlink{in_simplicial_sets__kan_complexes}{In simplicial sets / Kan complexes}\dotfill \pageref*{in_simplicial_sets__kan_complexes} \linebreak
\noindent\hyperlink{in_a_petit_topos}{In a petit $(\infty,1)$-topos}\dotfill \pageref*{in_a_petit_topos} \linebreak
\noindent\hyperlink{in_a_gros_topos}{In a gros $(\infty,1)$-topos}\dotfill \pageref*{in_a_gros_topos} \linebreak
\noindent\hyperlink{smooth_principal_bundles}{Smooth principal $\infty$-bundles}\dotfill \pageref*{smooth_principal_bundles} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{ordinary_principal_bundles}{Ordinary principal bundles}\dotfill \pageref*{ordinary_principal_bundles} \linebreak
\noindent\hyperlink{circle_bundles}{Circle $n$-bundles}\dotfill \pageref*{circle_bundles} \linebreak
\noindent\hyperlink{bundle_gerbes}{Bundle gerbes}\dotfill \pageref*{bundle_gerbes} \linebreak
\noindent\hyperlink{normal_morphisms_of_groups}{Normal morphisms of $\infty$-groups}\dotfill \pageref*{normal_morphisms_of_groups} \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}

The notion of \emph{principal $\infty$-bundle} is a [[vertical categorification|categorification]] of [[principal bundle]] from [[groups]] and [[groupoids]] to [[∞-groupoids]], or rather from parameterized groupoids (generalized spaces called [[stacks]]) to parameterized $\infty$-groupoids (generalized spaces called [[∞-stacks]]).

For motivation, background and further details see

\begin{itemize}%
\item [[motivation for sheaves, cohomology and higher stacks]]
\item [[principal bundle]]
\item [[gerbe]]
\item [[principal 2-bundle]].

\end{itemize}
A model for principal $\infty$-bundles is given by

\begin{itemize}%
\item [[simplicial principal bundle]]s.

\end{itemize}
See also

\begin{itemize}%
\item [[universal principal ∞-bundle]]


\item [[groupal model for universal principal ∞-bundles]].



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

We define $G$-principal $\infty$-bundles in the general context of an [[∞-stack]] [[(∞,1)-topos]] $\mathbf{H}$, with $G$ a [[groupoid object in an (∞,1)-category|group object in the (∞,1)-topos]].

Recall that for $A \in \mathbf{H}$ an object equipped with a point $pt_A : {*} \to A$, its corresponding [[loop space object]] $\Omega A$ is the [[homotopy pullback]]

\begin{displaymath}
\itexarray{
    \Omega A &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    {*} &\to& A
  }
  \,.
\end{displaymath}
Conversely, for $G \in \mathbf{H}$ we say an object $\mathbf{B}G$ is a [[delooping]] of $G$ if it has an essentially unique point and if $G \simeq \Omega \mathbf{B}G$. We call $G$ an \textbf{[[∞-group]]}. More in detail, its structure as a [[groupoid object in an (∞,1)-category|group object in an (∞,1)-category]] is exhibited by the [[?ech nerve]]

\begin{displaymath}
\left(
 \itexarray{
   &\cdots&
   {*} \times_{\mathbf{B}G} {*} 
   \times_{\mathbf{B}G} {*}
   &\stackrel{\to}{\stackrel{\to}{\to}}&
   {*} \times_{\mathbf{B}G} {*} 
   &\stackrel{\to}{\to}& 
   {*} 
 }
 \right)
 \simeq
 \left(
  \itexarray{
   &\cdots&
   G \times G
   &\stackrel{\to}{\stackrel{\to}{\to}}&
   G 
   &\stackrel{\to}{\to}&
   {*} 
  }
 \right)
\end{displaymath}
of ${*} \to \mathbf{B}G$.

\hypertarget{GeneralPrincBund}{}\subsubsection*{{$G$-principal $\infty$-bundles}}\label{GeneralPrincBund}

To every cocycle $g : X \to \mathbf{B}G$ is canonically associated its [[homotopy fiber]] $P \to X$, the [[(∞,1)-pullback]]

\begin{displaymath}
\itexarray{
    P &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{g}{\to}& \mathbf{B}G
    \,.
  }
  \,.
\end{displaymath}
We discuss now that $P$ canonically has the structure of a $G$-[[principal ∞-bundle]] and that $\mathbf{B}G$ is the [[fine moduli space]] for $G$-principal $\infty$-bundles.

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(principal $G$-action)}

Let $G$ be a [[groupoid object in an (∞,1)-category|group object in the (∞,1)-topos]] $\mathbf{H}$. A \textbf{principal [[action]]} of $G$ on a morphism $(P \to X) \in \mathbf{H}$ is a [[groupoid object in an (∞,1)-category|groupoid object]] $P//G$ that sits over $*//G$ in that we have a morphism of [[simplicial object|simplicial diagram]]s $\Delta^{op} \to \mathbf{H}$

\begin{displaymath}
\itexarray{
    \vdots && \vdots
    \\
    P \times G \times G
    &\stackrel{(p_2, p_3)}{\to}& G \times G
    \\
   \downarrow\downarrow\downarrow 
    && \downarrow\downarrow\downarrow
    \\
    P \times G
    &\stackrel{p_2}{\to}& G
    \\
    \downarrow\downarrow && \downarrow\downarrow
    \\
    P &\stackrel{}{\to}& {*}
  }
\end{displaymath}
in $\mathbf{H}$;

and such that $P \to X$ exhibits the [[(∞,1)-colimit]]

\begin{displaymath}
X \simeq \lim_\to (P//G : \Delta^{op} \to \mathbf{H})
\end{displaymath}
called the \textbf{base space} over which the action takes place.

\end{defn}
We may think of $P//G$ as the \textbf{[[action groupoid]]} of the $G$-action on $P$. For us it \emph{defines} this $G$-action.

\begin{prop}
\label{}\hypertarget{}{}
The $G$-principal action as defined above satisfies the \textbf{principality condition} in that we have an equivalence of groupoid objects

\begin{displaymath}
\itexarray{
     \vdots && \vdots
     \\
     P \times_X P \times_X P
     &\stackrel{\simeq}{\to}& 
     P \times G \times G
     \\
    \downarrow\downarrow\downarrow 
     && \downarrow\downarrow\downarrow
     \\
     P \times_X P
     &\stackrel{\simeq}{\to}& 
     P \times G
     \\
     \downarrow\downarrow && \downarrow\downarrow
     \\
     P &\stackrel{\simeq}{\to}& P
   }
   \,.
\end{displaymath}
\end{prop}
\begin{proof}
This principality condition asserts that the groupoid object $P//G$ is [[groupoid object in an (infinity,1)-category|effective]]. By [[(∞,1)-topos|Giraud's axioms characterizing (∞,1)-toposes]], every groupoid object in $\mathbf{H}$ is effective.

\end{proof}
\begin{uprop}
For $X \to \mathbf{B}G$ any morphism, its [[homotopy fiber]] $P \to X$ is canonically equipped with a principal $G$-action with base space $X$.

\end{uprop}
\begin{proof}
First we show that we have a morphism of simplicial diagrams

\begin{displaymath}
\itexarray{
    \vdots && \vdots && \vdots
    \\
    P \times_X P \times_X P 
    &\stackrel{\simeq}{\to}& P \times G \times G
    &\to& G \times G
    \\
    \downarrow\downarrow\downarrow
    && 
   \downarrow\downarrow\downarrow 
    && \downarrow\downarrow\downarrow
    \\
    P \times_X P &\stackrel{\simeq}{\to}& P \times G
    &\stackrel{p_2}{\to}& G
    \\
    \downarrow\downarrow
    && \downarrow\downarrow && \downarrow\downarrow
    \\
    P &\stackrel{=}{\to}& P &\stackrel{}{\to}& {*}
    \\
    \downarrow && \downarrow && \downarrow
    \\
    X &\stackrel{=}{\to}& X &\stackrel{g}{\to}& \mathbf{B}G
  }
  \,,
\end{displaymath}
with the right square swhere the left horizontal morphisms are equivalences, as indicated. We proceed by induction through the height of this diagram.

The defining [[(∞,1)-pullback]] square for $P \times_X P$ is

\begin{displaymath}
\itexarray{
    P \times_X P &\to& P
    \\
    \downarrow && \downarrow
    \\
    P &\to& X
  }
\end{displaymath}
To compute this, we may attach the defining $(\infty,1)$-pullback square of $P$ to obtain the [[pasting]] diagram

\begin{displaymath}
\itexarray{
    P \times_X P &\to& P &\to& {*}
    \\
    \downarrow && \downarrow && \downarrow
    \\
    P &\to& X &\to& \mathbf{B}G
    \,.
  }
\end{displaymath}
and use the \href{http://ncatlab.org/nlab/show/pullback#Pasting}{pasting law for pullbacks}, to conclude that $P \times_X P$ is the pullback

\begin{displaymath}
\itexarray{
    P \times_X P &\to& &\to& {*}
    \\
    \downarrow && && \downarrow
    \\
    P &\to& X &\to& \mathbf{B}G
    \,.
  }
\end{displaymath}
By definition of $P$ as the homotopy fiber of $X \to \mathbf{B}G$, the lower horizontal morphism is equivbalent to $P \to {*} \to \mathbf{B}G$, so that $P \times_X P$ is also equivalent to the pullback

\begin{displaymath}
\itexarray{
    P \times_X P &\to& &\to& {*}
    \\
    \downarrow && && \downarrow
    \\
    P &\to& {*} &\to& \mathbf{B}G
    \,.
  }
\end{displaymath}
This finally may be computed as the pasting of two pullbacks

\begin{displaymath}
\itexarray{
    P \times_X P &\simeq& P \times G &\to& G &\to& {*}
    \\
    &&\downarrow && \downarrow && \downarrow
    \\
    &&P &\to& {*} &\to& \mathbf{B}G
    \,.
  }
\end{displaymath}
of which the one on the right is the defining one for $G$ and the remaining one on the left is just an [[(∞,1)-product]].

Proceeding by induction from this case we find analogously that $P^{\times_X^{n+1}} \simeq P \times G^{\times_n}$: suppose this has been shown for $(n-1)$, then the defining pullback square for $P^{\times_X^{n+1}}$ is

\begin{displaymath}
\itexarray{
    P \times_X P^{\times_X^n}
    &\to&
    P
    \\
    \downarrow && \downarrow
    \\
    P^{\times_X^n}&\to& X
  }
  \,.
\end{displaymath}
We may again paste this to obtain

\begin{displaymath}
\itexarray{
    P \times_X P^{\times_X^n}
    &\to&
    P
    &\to&
    *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    P^{\times_X^n}&\to& X &\to& \mathbf{B}G
  }
\end{displaymath}
and use from the previous induction step that

\begin{displaymath}
(P^{\times_X^n} \to X \to \mathbf{B}G) 
  \simeq
  (P^{\times_X^n} \to * \to \mathbf{B}G)
\end{displaymath}
to conclude the induction step with the same arguments as before.

This shows that $P//G$ is the [[Cech nerve]] of $P \to X$. It remains to show that indeed $X = {\lim_\to}_n P \times G^{\times^n}$. For this notice that $* \to \mathbf{B}G$ is an [[effective epimorphism in an (infinity,1)-category]]. Hence so is $P \to X$. This proves the claim, by definition of effective epimorphism.

using this we have

\begin{displaymath}
\begin{aligned}
    X & \simeq \mathbf{B}G \prod_{\mathbf{B}G} X
    \\
      & \simeq \left({\lim_{\to}}_n G^{\times^n}\right) \prod_{\mathbf{B}G} X
      \\
      & \simeq {\lim_{\to}}_n ( G^{\times^n} \prod_{\mathbf{B}G} X )
      \\
      & \simeq {\lim_\to}_n ( P\times G^{\times^n} )
      \\
      & \simeq {\lim_\to} P//G
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
We have established that every [[cocycle]] $X \to \mathbf{B}G$ canonically induced a $G$-principal action on the homotopy fiber $P \to X$. The following definition declares the \emph{$G$-principal $\infty$-bundles} to be those $G$-principal actions that do arise this way.

\begin{defn}
\label{}\hypertarget{}{}
We say a $G$-principal action of $G$ on $P$ over $X$ is a $G$-[[principal ∞-bundle]] if the colimit over $P//G \to *//G$ produces a pullback square: the bottom square in

\begin{displaymath}
\itexarray{
    \vdots && \vdots
    \\
    P \times G \times G
    &\to& G \times G
    \\
   \downarrow\downarrow\downarrow 
    && \downarrow\downarrow\downarrow
    \\
    P \times G
    &\stackrel{p_2}{\to}& G
    \\
    \downarrow\downarrow && \downarrow\downarrow
    \\
    P &\stackrel{}{\to}& {*}
    \\
    \downarrow && \downarrow
    \\
    X = \lim_\to (P \times G^\bullet) &\stackrel{g}{\to}& 
   \mathbf{B}G = \lim_\to( G^\bullet)
  }
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
For $G$ an [[infinity-group]] in $\mathbf{H}$ and $X \in \mathbf{H}$ any object, write

\begin{displaymath}
G Bund(X) \subset Grpd(\mathbf{H})/{*//G}
\end{displaymath}
for the [[sub-(infinity,1)-category]] on the [[over-(infinity,1)-category]] of the groupoid objects over $*//G$ on the $G$-principal $\infty$-bundles as above.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
We have an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
G Bund(X) \simeq \mathbf{H}(X, \mathbf{B}G)
\end{displaymath}
of $G$-orincipal $\infty$-bundles over $X$ with [[cocycles]] $X \to \mathbf{B}G$.

\end{prop}
\begin{proof}
The [[arrow category]] $\mathbf{H}^I$ is still an [[(infinity,1)-topos]] and hence the Griraud-Lurie axioms still hold. This means that by the discussion at [[groupoid object in an (infinity,1)-category]] (using the statement below [[Higher Topos Theory|HTT, cor. 6.2.3.5]]) we have an equivalence

\begin{displaymath}
Grpd(\mathbf{H}^I) \simeq 
  (\mathbf{H}^{I})^{I}_{eff}
\end{displaymath}
between groupoid objects in $\mathbf{H}^I$ and [[effective epimorphism in an (infinity,1)-category|effective epimorphisms]] in the arrow category.

Notice that groupoid objects and effective epis in $\mathbf{H}^I$ are given objectwise over the two objects of the inerval $I = \Delta[1]$.

Restricting this equivalence along the inclusion

\begin{displaymath}
\mathbf{H}(X, \mathbf{B}G)
    \hookrightarrow
  (\mathbf{H}^I)^I
\end{displaymath}
given by sending a cocycle to its homotopy fiber diagram

\begin{displaymath}
(X \to \mathbf{B}G)
  \mapsto 
  \left(
     \itexarray{
        P &\to& * 
        \\
        \downarrow && \downarrow
        \\
        X &\to& \mathbf{B}G
     }
  \right)
\end{displaymath}
therefore yields precisely the equivalence in question

\begin{displaymath}
\itexarray{
     G Bund(X) &\hookrightarrow& Grpd(\mathbf{H}^I)
     \\
     \downarrow^\simeq && \downarrow^\simeq
     \\
     \mathbf{H}(X, \mathbf{B}G)
     &\stackrel{hofib}{\hookrightarrow}&
     (\mathbf{H}^I)^I
  }
  \,.
\end{displaymath}
\end{proof}
In words this says that the [[cohomology]] on $X$ with coefficients in $G$ classified $G$-principal $\infty$-bundles, and in fact does so on the level of [[cocycle]]s.

\hypertarget{connections_on_principal_bundles}{}\subsubsection*{{Connections on $G$-principal $\infty$-bundles}}\label{connections_on_principal_bundles}

For some comments on the generalization of the notion of [[connection on a bundle]] to principal $\infty$-bundles see [[schreiber:differential cohomology in an (∞,1)-topos -- survey]].

\hypertarget{concrete_realizations}{}\subsection*{{Concrete realizations}}\label{concrete_realizations}

We discuss realizations of the general definition in various [[(∞,1)-toposes]] $\mathbf{H}$.

\hypertarget{in_topological_spaces}{}\subsubsection*{{In topological spaces}}\label{in_topological_spaces}

The following general construction was originally due to Quillen and defines principal \emph{groupoid} $\infty$-bundles in the [[(∞,1)-topos]] [[Top]] in its [[presentable (infinity,1)-category|presentation]] by the [[model structure on simplicial sets]].

Let $C$ be a small [[category]] and let

\begin{displaymath}
\rho_P : C \to SSet
\end{displaymath}
be a functor with values in [[SSet]] such that it sends all morphisms in $C$ to weak equivalences in [[SSet]] ([[simplicial homotopy group|weak homotopy equivalences]] of simplicial sets).

Consider first the case that $C$ has a single object, so that it is the [[delooping]] $\mathbf{B}G$ of a [[monoid]] or [[group]] $G$. Then

Let

\begin{displaymath}
P := \rho_P(\bullet)
\end{displaymath}
be the simplicial set assigned to this single object and let

\begin{displaymath}
X := P//G := hocolim \rho_P
\end{displaymath}
be the corresponding [[action groupoid]] (see there for the description as a weak colimit).

Notice that, as every [[action group]], this comes with a canonical map $P//G \to \mathbf{B}G$.

\begin{utheorem}
Given the above, the diagram

\begin{displaymath}
\itexarray{  
     P &\to& {*}
     \\
     \downarrow && \downarrow
     \\
     X &\stackrel{g}{\to}& \mathbf{B}G
  }
\end{displaymath}
is a homotopy pullback (i.e. defines a [[fibration sequence]]).

\end{utheorem}
\begin{proof}
This is originally due to

\begin{itemize}%
\item D. Quillen, \emph{Higher algebraic K-theory I}, Springer Lecture notes in Math. 341 (1973) 85--147.

\end{itemize}
The statement is reproduced in section IV of

\begin{itemize}%
\item P. G. Goerss and J. F. Jardine, 1999, \emph{Simplicial Homotopy Theory}, number 174 in Progress in Mathematics, Birkhauser. (\href{http://www.maths.abdn.ac.uk/~bensondj/html/archive/goerss-jardine.html}{ps})

\end{itemize}
\end{proof}
\begin{uremark}
For the simple case that $G$ is [[group]], in which case $\rho_C$ necessarily takes values not just in weak equivalences but is isomorphisms of simplicial sets, this says that $P \to X$ is a $G$-principal $\infty$-bundle. In particular the \emph{principality} of the action is manifestly exhibited by the fact that the base space $X$ is the (weak) quotient of $P$ by the action of $G$.

The above reproduces manifest the description of ordinary $G$-principal topological bundles in the incarnation as groupoids as described in detail at [[generalized universal bundle]].

More generally, when $G$ is just a [[monoid]] the above descibes something a bit more general than an ordinary $G$-[[principal bundle]] (as then the action of $G$ on the total space may be by weak equivalences that are not isomorphisms).

\end{uremark}
Quillen's original construction is more general than this, concerning in fact 1-[[groupoid]]-principal $\infty$-bundles:

\begin{utheorem}
Let now $C$ be a [[category]] and for

\begin{displaymath}
\rho_P : C \to SSet
\end{displaymath}
a functor that sends all morphisms to weak equivalences of simplicial sets.

Let now for each object $c \in C$

\begin{displaymath}
P_c := \rho_C(c)
\end{displaymath}
be the ``bundle of $c$-fibers''.

Then for each $c$ the diagram

\begin{displaymath}
\itexarray{  
     P_c &\to& {*}
     \\
     \downarrow && \downarrow^{{*} \mapsto c}
     \\
     X &\stackrel{g}{\to}& C
  }
\end{displaymath}
is a homotopy pullback (i.e. defines a [[fibration sequence]]).

\end{utheorem}
This classical construction is recalled in the introduction of

\begin{itemize}%
\item Jardine, \emph{Diagrams and torsors} (\href{http://www.math.uiuc.edu/K-theory/0723/diagrams3.pdf}{pdf})

\end{itemize}
\hypertarget{in_simplicial_sets__kan_complexes}{}\subsubsection*{{In simplicial sets / Kan complexes}}\label{in_simplicial_sets__kan_complexes}

See [[simplicial principal bundle]].

\hypertarget{in_a_petit_topos}{}\subsubsection*{{In a petit $(\infty,1)$-topos}}\label{in_a_petit_topos}

For $X$ a [[topological space]] $C = Op(X)$ the [[category of open subsets]] of $X$, let $\mathbf{H} = Sh_{(\infty,1)}(X)$ be the [[(∞,1)-topos]] of [[∞-stacks]] on $C$. This is the [[petit topos]] incarnation of $X$.

In its [[presentable (infinity,1)-category|presentation]] by the [[model structure on simplicial presheaves]] this is the context in which princpal $\infty$-bundles are discussed in

\begin{itemize}%
\item Jardine, \emph{Diagrams and torsors} (\href{http://www.math.uiuc.edu/K-theory/0723/diagrams3.pdf}{pdf})

\end{itemize}
\hypertarget{in_a_gros_topos}{}\subsubsection*{{In a gros $(\infty,1)$-topos}}\label{in_a_gros_topos}

For $C$ a [[site]] of test [[space]], -- for instance duals of [[algebras over a Lawvere theory]] as described at [[function algebras on infinity-stacks]] -- let $\mathbf{H} = Sh_{(\infty,1)}(C)$ be the [[(∞,1)-topos]] of [[∞-stacks]] on $C$. This is a [[gros topos]].

\hypertarget{smooth_principal_bundles}{}\paragraph*{{Smooth principal $\infty$-bundles}}\label{smooth_principal_bundles}

Smooth principal $\infty$-bundles are realized in the $\infty$-[[Cahiers topos]] as described in some detail at [[∞-Lie groupoid]].

In this context there is a notion of [[connection on a principal ∞-bundle]].

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

\hypertarget{ordinary_principal_bundles}{}\subsubsection*{{Ordinary principal bundles}}\label{ordinary_principal_bundles}

For $G$ an ordinary [[Lie group]], a $G$-principal bundle in the $(\infty,1)$-topos $\mathbf{H} =$ [[?LieGrpd]] is an ordinary $G$-[[principal bundle]].

\hypertarget{circle_bundles}{}\subsubsection*{{Circle $n$-bundles}}\label{circle_bundles}

For $G = \mathbf{B}^{n-1} U(1) \in$ [[?LieGrpd]], the , a $G$-principal $\infty$-bundle is a \textbf{circle $n$-bundle}.

See [[circle n-bundle with connection]].

Classes of examples include

\begin{itemize}%
\item the [[Chern-Simons circle 3-bundle]].

\end{itemize}
\hypertarget{bundle_gerbes}{}\subsubsection*{{Bundle gerbes}}\label{bundle_gerbes}

\begin{itemize}%
\item A [[bundle gerbe]] is a concrete model for the total space [[groupoid]] of the total space of a $\mathbf{B}U(1)$-[[principal 2-bundle]].

More generally, a [[nonabelian bundle gerbe]] is a concrete model for the [[groupoid]] of the total space of a general [[principal 2-bundle]].


\item A [[bundle 2-gerbe]] is a concrete model for the total space [[2-groupoid]] of the total space of a $\mathbf{B}^2 U(1)$-principal 3-bundle.

More generally, a [[nonabelian bundle 2-gerbe]] is a concrete model for the [[2-groupoid]] of the total space of a general principal 3-bundle.

Classes of examples include

\begin{itemize}%
\item the [[Chern-Simons bundle 2-gerbe]].

\end{itemize}


\end{itemize}
\hypertarget{normal_morphisms_of_groups}{}\subsubsection*{{Normal morphisms of $\infty$-groups}}\label{normal_morphisms_of_groups}

A principal $\infty$-bundle over a [[0-connected]] object / [[delooping]] object $\mathf{B}K$ is a \emph{[[normal morphism of ∞-groups]]}. See there for more details.

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

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


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


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


\item \textbf{principal $\infty$-bundle} / [[associated ∞-bundle]] / [[∞-gerbe]], [[twisted ∞-bundle]], [[groupoid-principal ∞-bundle]]


\item [[vector bundle]]


\item [[(∞,1)-vector bundle]] / [[(∞,n)-vector bundle]]



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

The notion of principal $\infty$-bundle (often addressed in the relevant literature in the language of [[torsors]]) appears in the context of the [[simplicial presheaf]] [[model structure on simplicial presheaves|model]] for generalized spaces in

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


\item [[Rick Jardine]], \emph{Cocycle categories} (\href{http://arxiv.org/abs/math.AT/0605198}{pdf}).



\end{itemize}
An earlier description in terms of simplicial objects 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}
In that article not the total space of the bundle $P \to X$ is axiomatized, but the $\infty$-[[action groupoid]] of the action of $G$ on it.

See the remarks at [[principal 2-bundle]].

See also

\begin{itemize}%
\item [[Matthias Wendt]], \emph{Classifying spaces and fibrations of simplicial sheaves}, J. Homotopy Relat. Struct. 6 (2011), no. 1, 1-38 (\href{http://arxiv.org/abs/1009.2930}{arXiv:1009.2930})

\end{itemize}
on [[associated ∞-bundle]]s.

The fully general abstract formalization in [[(∞,1)-topos theory]] as discussed here was first indicated in

\begin{itemize}%
\item [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{[[schreiber:Twisted Differential String and Fivebrane Structures]]}

\end{itemize}
A more comprehensive conceptual account is in

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


\item [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], \emph{[[schreiber:Principal ∞-bundles -- models and general theory]]}



\end{itemize}
The [[classifying spaces]] for a large class of principal $\infty$-bundles are discussed in

\begin{itemize}%
\item [[David Roberts]], [[Danny Stevenson]], \emph{Simplicial principal bundle in parameterized spaces} (\href{http://arxiv.org/abs/1203.2460}{arXiv:1203.2460})

\end{itemize}
\begin{itemize}%
\item [[Danny Stevenson]], \emph{Classifying theory for simplicial parametrized groups} (\href{http://arxiv.org/abs/1203.2461}{arXiv:1203.2461})

\end{itemize}
A fairly comprehensive account of the literature is also in the introduction of \hyperlink{NSS12}{NSS 12, ``Presentations''}.

For $\mathbf{H}= \infty Grpd$ the statement that homotopy types over $B G$ are equivalently $G$-[[infinity-actions]] is maybe due to

\begin{itemize}%
\item E. Dror, [[William Dwyer]], and [[Daniel Kan]], \emph{Equivariant maps which are self homotopy equivalences}, Proc. Amer. Math. Soc. 80 (1980), no. 4, 670--672 (\href{http://www.jstor.org/stable/2043448}{JSTOR})

\end{itemize}
This is mentioned for instance as exercise 4.2in

\begin{itemize}%
\item [[William Dwyer]], \emph{Homotopy theory of classifying spaces}, Lecture notes Copenhagen (June, 2008) \href{http://www.math.ku.dk/~jg/homotopical2008/Dwyer.CopenhagenNotes.pdf}{pdf}

\end{itemize}
Closely related discussion of homotopy fiber sequences and homotopy action but in terms of [[Segal spaces]] is in section 5 of

\begin{itemize}%
\item [[Matan Prezma]], \emph{Homotopy normal maps} (\href{http://arxiv.org/pdf/1011.4708v7.pdf}{arXiv})

\end{itemize}
There, conditions are given for a morphism $A_\bullet \to B_\bullet$ to a [[reduced Segal space]] to have a fixed homotopy fiber, and hence encode an action of the loop group of $B$ on that fiber.

[[!redirects principal infinity-bundles]] [[!redirects principal ∞-bundle]] [[!redirects principal ∞-bundles]]



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