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\section*{groupal model for universal principal infinity-bundles}

\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{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos theory}}\label{topos_theory}

[[!include (infinity,1)-topos - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{OrdinaryGroups}{For ordinary groups}\dotfill \pageref*{OrdinaryGroups} \linebreak
\noindent\hyperlink{for_strict_2groups}{For strict 2-groups}\dotfill \pageref*{for_strict_2groups} \linebreak
\noindent\hyperlink{for_groups}{For $\infty$-Groups}\dotfill \pageref*{for_groups} \linebreak
\noindent\hyperlink{for_groups_in_an_arbitrary_topos}{For $\infty$-groups in an arbitrary $(\infty,1)$-topos}\dotfill \pageref*{for_groups_in_an_arbitrary_topos} \linebreak
\noindent\hyperlink{for_lie_groups}{For $\infty$-Lie groups}\dotfill \pageref*{for_lie_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}

For $G$ a model for an [[∞-group]], there is often a model for the [[universal principal ∞-bundle]] $\mathbf{E}G$ that itself carries a group structure such that the canonical inclusion $G \to \mathbf{E}G$ is a homomorphism of group objects. This extra groupal structure is important for various constructions.

\hypertarget{OrdinaryGroups}{}\subsection*{{For ordinary groups}}\label{OrdinaryGroups}

For $G$ an ordinary bare [[group]], the [[action groupoid]] $\mathbf{E}G = G//G$ of the right multiplcation action of $G$ on itself

\begin{displaymath}
G//G
  =
  \left(
  G \times G \stackrel{\overset{\cdot}{\to}}{\underset{p_1}{\to}}
  G
  \right)
\end{displaymath}
is [[contractible]]. We may think of this as the groupoid $INN(G) \subset AUT(G)$ of [[inner automorphism]]s inside the [[automorphism 2-group]] of $G$. But it is also simply isomorphic to the [[codiscrete groupoid]] on the set underlying $G$. In this latter form the 2-group structure on $\mathbf{E}G$ is manifest, which corresponds to the [[crossed module]] $(G \stackrel{Id}{\to} G)$ .

It is also manifest then that this fits with the one-object [[delooping]] groupoid $\mathbf{B}G$ of $G$ into a sequence

\begin{displaymath}
G \to \mathbf{E}G \to \mathbf{B}G
  \,,
\end{displaymath}
where the first morphism is a homomorphism of [[strict 2-group]]s.

Regarded as a sequence of morphisms in the model [[Kan complex|KanCplx]] of the [[(∞,1)-topos]] [[∞-Grpd]] this \emph{is} already a model for the universal $G$-bundle.

If $G$ here is refined to a [[Lie group]] or [[topological group]] then this is a sequence of [[∞-Lie groupoid]]s or [[topological ∞-groupoid]]s, respectively, and also then, this \emph{is} already a model for the universal $G$-principal bundle, as discussed at .

By applying the [[geometric realization]] functor

\begin{displaymath}
|-| : \infty Grpd \stackrel{\simeq}{\to} Top
\end{displaymath}
we obtain a sequence of topological spaces

\begin{displaymath}
G \to \mathcal{E}G \to \mathcal{B}G
  \,,
\end{displaymath}
where $\mathcal{E}G$ carries the structure of a [[topological group]] and the morphism $G \to \mathcal{E}G$ is a topological group homomorphism. For bare groups $G$ and under mild assumptions also for general topological groups $G$, this groupal topological model for the universal $G$-bundle obtained from the realization of the groupoid $G//G$ was consider in \hyperlink{Segal}{Segal68}.

\hypertarget{for_strict_2groups}{}\subsection*{{For strict 2-groups}}\label{for_strict_2groups}

For $G \in$ [[∞Grpd]] a [[strict 2-group]] a groupal model for $\mathbf{E}G$ was given in \hyperlink{RobertsSchreiber}{RobertsSchr07} generalizing the $INN(G) \subset AUT(G)$ construction mentioned above. This yields a weak 3-group structure on $\mathbf{E}G$ (A [[Gray-group]]).

In \hyperlink{RobertsUniversal}{Roberts07} it is observed that there is also an analog of $codisc(G)$ and that this yields a strict group structure on $\mathbf{E}G$. In fact, this strictly groupal model of $\mathbf{E}G$ turns out to be isomorphic to the standard model for the universal [[simplicial principal bundle]] traditionally denoted $W G$. And this statement generalizes\ldots{}

\hypertarget{for_groups}{}\subsection*{{For $\infty$-Groups}}\label{for_groups}

Every [[∞-group]] may be modeled by a [[simplicial group]] $G$. There is a standard [[Kan complex]] model for the universal $G$-[[simplicial principal bundle]] $\mathbf{E}G \to \mathbf{B}G$ denoted $W G \to \bar W G$.

This standard model $W G$ does happen to have the structure of a [[simplicial group]] itself, and this structure is compatible with that of $G$ in that the canonical inclusion $G \to W G$ is a homomorphisms of simplicial groups.

This sounds like a statement that must be well known, but it was maybe not in the literature. The simplicial group structure is spelled out explicitly in \hyperlink{RobertsUniversal}{Roberts07}

\hypertarget{for_groups_in_an_arbitrary_topos}{}\subsection*{{For $\infty$-groups in an arbitrary $(\infty,1)$-topos}}\label{for_groups_in_an_arbitrary_topos}

Since the $W$ construction discussed above is functorial, this generalizes to prresheaves of simplicial groups and hence gives models for group objects and groupal universal [[principal ∞-bundle]]s in [[(∞,1)-topos]]es modeled by the [[model structure on simplicial presheaves]].

(\ldots{})

\hypertarget{for_lie_groups}{}\subsection*{{For $\infty$-Lie groups}}\label{for_lie_groups}

In the [[(∞,1)-topos]] [[?LieGrpd]] of [[∞-Lie groupoid]]s we can obtain [[∞-group]]s by [[Lie integration]] of [[L-∞-algebra|∞-Lie algebra]]s.

Corresponding to this is a construction of Lie-integrated groupal universal principal $\infty$-bundles:

for $\mathfrak{g}$ an $L_\infty$-algebra, there is an $L_\infty$-algebra $inn(\mathfrak{g})$, defined such that its [[Chevalley-Eilenberg algebra]] is the [[Weil algebra]] $W(\mathfrak{g})$ of $\mathfrak{g}$:

\begin{displaymath}
CE(inn(\mathfrak{g})) = W(\mathfrak{g})
  \,.
\end{displaymath}
Under [[Lie integration]] this gives a groupal model for the universal principal $\infty$-bundle over the [[∞-Lie group]] that integrates $\mathfrak{g}$.

This is described at



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

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


\item [[universal principal ∞-bundle]] , \textbf{groupal model for universal principal ∞-bundle}


\item [[universal connection]]



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

The observation that for $G$ an ordinary [[group]], its [[action groupoid]] sequence $G \to G//G \to \mathbf{B}G$ -- which is the [[strict 2-group]] coming from the [[crossed module]] $(G \stackrel{Id}{\to} G)$ - maps under the [[nerve]] to the universal $G$-bundle appeared in

\begin{itemize}%
\item G. B. Segal, \emph{Classifying spaces and spectral sequences} , Publ. Math. IHES No. 34 (1968) pp. 105--112

\end{itemize}
A weak 3-group structure on $G \to \mathbf{E}G$ for $G$ a [[strict 2-group]] is descibed in

\begin{itemize}%
\item [[David Roberts]], [[Urs Schreiber]], \emph{The inner automorphism 3-group of a strict 2-group} , Journal of Homotopy and Related Structures, vol. 3(1) (\href{http://arxiv.org/abs/0708.1741}{arXiv:0708.1741})

\end{itemize}
The [[simplicial group]] structure on $G \to \mathbf{E}G$ for $G$ a general [[simplicial group]] is stated explicitly in

\begin{itemize}%
\item [[David Roberts]], \emph{The universal simplicial bundle is a simplicial group}, \href{http://nyjm.albany.edu/}{New York Journal of Mathematics}, Volume 19 (2013) 51-60, \href{http://nyjm.albany.edu/j/2013/19-5.html}{journal version}, \href{http://arxiv.org/abs/1204.4886}{arXiv:1204.4886}.

\end{itemize}
A general abstract construction of this simplicial group structure is discussed in

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

\end{itemize}
The use of [[L-∞-algebra]]s $inn(\mathfrak{g})$ as $L_\infty$-algebraic models for universal $\mathfrak{g}$-principal bundle (evident as it is) was considered as such in

\begin{itemize}%
\item [[nLab:Hisham Sati]], [[nLab:Urs Schreiber|U.S.]], [[nLab:Jim Stasheff]], \emph{$L_\infty$-algebra connections} in Fauser (eds.) Recent Developments in QFT, Birkh\"a{}user (\href{http://arxiv.org/abs/0801.3480}{arXiv:0801.3480})

\end{itemize}
[[!redirects groupal model for universal principal ∞-bundle]] [[!redirects groupal model for universal principal ∞-bundles]] [[!redirects groupal models for universal principal ∞-bundles]]

[[!redirects groupal models for universal principal infinity-bundles]]



\end{document}