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\begin{document}

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\section*{simplicial principal 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{twisted_cartesian_products}{Twisted cartesian products}\dotfill \pageref*{twisted_cartesian_products} \linebreak
\noindent\hyperlink{UniversalSimplicialBundle}{The universal simplicial $G$-principal bundle}\dotfill \pageref*{UniversalSimplicialBundle} \linebreak
\noindent\hyperlink{definition_4}{Definition}\dotfill \pageref*{definition_4} \linebreak
\noindent\hyperlink{properties_2}{Properties}\dotfill \pageref*{properties_2} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

[[simplicial group|Simplicial groups]] model all [[∞-group]]s in [[∞Grpd]]. Accordingly [[principal ∞-bundle]]s in [[∞Grpd]] (a [[discrete ∞-groupoid|discrete]] $\infty$-bundle) should be modeled by [[Kan complex]]es $E \to X$ equipped with a principal [[action]] by a [[simplicial group]]. It is suficient to assume the action to be strict. This yields the notion of \emph{simplicial principal bundles} .

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

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

Let $G$ be a simplicial group. For $P$ a [[Kan complex]], an [[action]] of $G$ on $E$

\begin{displaymath}
\rho : E \times G \to E
\end{displaymath}
is called \textbf{principal} if it is degreewise principal, i.e. if for all $n \in \mathbb{N}$ the only elements $g \in G_n$ that have any fixed point $e \in E_n$ in that $\rho(e,g) = e$ are the neutral elements.

\end{defn}
\begin{example}
\label{}\hypertarget{}{}
The canonical action

\begin{displaymath}
G \times G \to G
\end{displaymath}
of any simplicial group on itself is principal.

\end{example}
\begin{defn}
\label{}\hypertarget{}{}
\textbf{(simplicial principal bundle)}

For $G$ a simplicial group, a morphism $P \to X$ of [[Kan complex]]es equipped with a $G$-[[action]] on $P$ is called a $G$-\textbf{simplicial principal bundle} if

\begin{itemize}%
\item the action is principal;


\item the base is isomorphic to the quotient $E/G := \lim_{\to}(E \times G \stackrel{\overset{\rho}{\to}}{\underset{p_1}{\to} E})$ by the action:

\begin{displaymath}
E/G \simeq X
  \,.
\end{displaymath}


\end{itemize}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

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

\begin{prop}
\label{}\hypertarget{}{}
A simplicial $G$-principal bundle $P \to X$ is necessarly a [[Kan fibration]].

\end{prop}
\begin{proof}
This appears as (\hyperlink{May}{May, Lemma 18.2}).

\end{proof}
\hypertarget{twisted_cartesian_products}{}\subsubsection*{{Twisted cartesian products}}\label{twisted_cartesian_products}

\begin{prop}
\label{}\hypertarget{}{}
Let $E \to B$ be a [[twisted cartesian product]] of the [[simplicial set]] $B$ with a [[simplicial group]] $G$. Then with respect to the canonical $G$-[[action]] this is a simplicial principal bundle.

\end{prop}
This is (\hyperlink{May}{May, prop. 18.4}).

\begin{remark}
\label{}\hypertarget{}{}
It is simplicial principal bundles of this form that one is mainly interested in. These are the objects that are classified by the evident [[classifying space]] $\bar W G$. This is discussed \href{UniversalSimplicialBundle}{below}.

\end{remark}
\hypertarget{UniversalSimplicialBundle}{}\subsection*{{The universal simplicial $G$-principal bundle}}\label{UniversalSimplicialBundle}

Recall from [[generalized universal bundle]] that a universal $G$-principal simplicial bundle should be a principal bundle $\mathbf{E}G \to \mathbf{B}G$ such that every other $G$-principal simplicial bundle $P \to X$ arises up to equivalence as the [[pullback]] of $\mathbf{E}G$ along a morphism $X \to \mathbf{B}G$.

A standard model for the [[delooping]] [[Kan complex]] $\mathbf{B}G$ for $G$ a simplicial group goes by the name

\begin{displaymath}
\bar W G
  \,.
\end{displaymath}
This is described at . The following establishes a model for the universal simplicial bundle over this model of $\mathbf{B}G$.

\hypertarget{definition_4}{}\subsubsection*{{Definition}}\label{definition_4}

\begin{defn}
\label{}\hypertarget{}{}
For $G$ a simplicial group, define the [[simplicial set]] $W G$ to be the [[decalage]] of $\overline{W}G$

\begin{displaymath}
W G := Dec \overline{W}G
  \,.
\end{displaymath}
\end{defn}
By the discussion at [[homotopy pullback]] this means that for $X_\bullet$ any [[Kan complex]], an ordinary [[pullback]] diagram

\begin{displaymath}
\itexarray{
    P_\bullet &\to& W G
    \\
    \downarrow && \downarrow
    \\
    X_\bullet &\stackrel{g}{\to}& \overline{W}G
  }
\end{displaymath}
in [[sSet]] exhibits $P_\bullet$ as the [[homotopy pullback]] in $sSet_{Quillen}$ / [[(∞,1)-pullback]] in [[∞Grpd]]

\begin{displaymath}
\itexarray{
    P_\bullet &\to& *
    \\
    \downarrow &\swArrow& \downarrow
    \\
    X_\bullet &\stackrel{g}{\to}& \overline{W}G
  }
  \,,
\end{displaymath}
i.e. as the [[homotopy fiber]] of the cocycle $g$.

\begin{defn}
\label{}\hypertarget{}{}
We call $P_\bullet := X_\bullet \times^g W G$ the simplicial $G$-[[principal bundle]] corresponding to $g$.

\end{defn}
\hypertarget{properties_2}{}\subsubsection*{{Properties}}\label{properties_2}

\begin{prop}
\label{}\hypertarget{}{}
Let $\{\phi : X_n \to G_{(n-1)}\}$ be the [[twisting function]] corresponding to $g : X_\bullet \to \overline{W}G$ by the above discussion.

Then the simplicial set $P_\bullet := X_\bullet \times_{g} W G$ is explicitly given by the formula called the [[twisted Cartesian product]] $X_\bullet \times^\phi G_\bullet$:

its cells are

\begin{displaymath}
P_n  = X_n \times G_n
\end{displaymath}
with face and degeneracy maps

\begin{itemize}%
\item $d_i (x,g) = (d_i x , d_i g)$ if $i \gt 0$


\item $d_0 (x,g) = (d_0 x, \phi(x) d_0 g)$


\item $s_i (x,g) = (s_i x, s_i g)$.



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

Here are some pointers on where precisely in the literature the above statements can be found.

One useful reference is

\begin{itemize}%
\item [[Peter May]], \emph{Simplicial Objects in Algebraic Topology} (\href{http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu}{djvu}).

\end{itemize}
There the abbreviation PCTP ( \emph{principal twisted cartesian product} ) is used for what above we called just [[twisted Cartesian product]]s.

The fact that every PTCP $X \times_\phi G \to X$ defined by a [[twisting function]] $\phi$ arises as the pullback of $W G \to \overline{W}G$ along a morphism of simplicial sets $X \to \overline{W}G$ can be found there by combining

\begin{enumerate}%
\item the last sentence on p. 81 which asserts that pullbacks of PTCPs $X \times_\phi G \to X$ along morphisms of simplicial sets $f : Y \to X$ yield PTCPs corresponding to the composite of $f$ with $\phi$;


\item section 21 which establishes that $W G \to \bar W G$ is the PTCP for some universal twisting function $r(G)$.


\item lemma 21.9 states in the language of composites of twisting functions that every PTCP comes from composing a cocycle $Y \to \bar W G$ with the universal twisting function $r(G)$. In view of the relation to pullbacks in item 1, this yields the statement in the form we stated it above.



\end{enumerate}
An explicit version of the statement that [[twisted Cartesian product]]s are nothing but pullbacks of a [[generalized universal bundle]] is on \href{http://ncatlab.org/timporter/files/menagerie10.pdf#page=148}{page 148} of

\begin{itemize}%
\item [[Tim Porter]], \emph{\href{http://ncatlab.org/timporter/show/crossed+menagerie}{Crossed Menagerie}}

\end{itemize}
On \href{http://ncatlab.org/timporter/files/menagerie10.pdf#page=239}{page 239} there it is mentioned that

\begin{displaymath}
G \to W G \to \overline{W}G
\end{displaymath}
is a model for the [[loop space object]] [[fiber sequence]]

\begin{displaymath}
G \to * \to \mathbf{B}G
  \,.
\end{displaymath}
One place in the literature where the observation that $W G$ is the [[decalage]] of $\overline{W}G$ is mentioned fairly explicitly is page 85 of

\begin{itemize}%
\item [[John Duskin]], \emph{Simplicial methods and the interpretation of ``triple'' cohomology}, number 163 in Mem. Amer. Math. Soc., 3, Amer. Math. Soc. (1975)

\end{itemize}
Discussion of \emph{topological} simplicial principal bundles is 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}
[[!redirects simplicial principal bundles]]

[[!redirects principal simplicial bundle]] [[!redirects principal simplicial bundles]]

[[!redirects universal simplicial principal bundle]] [[!redirects universal simplicial principal bundles]]



\end{document}