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\begin{document}

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\section*{model structure on simplicial groupoids}

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

\hypertarget{model_category_theory}{}\paragraph*{{Model category theory}}\label{model_category_theory}

[[!include model category theory - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


The term \emph{simplicial groupoid} is often used for a [[simplicial object]] in the [[category]] [[Grpd]] of [[groupoids]] whose [[simplicial set]] of objects is simplicially constant. We will write $s Grpd$ for the category of such simplicial groupoids.

(BEWARE: perhaps a more accurate term for this concept is \textbf{simplicially enriched groupoid}, and conceptually it is often the enriched category structure that is useful. Because of this it is advisable to check the use being made of the term when consulting the literature. This is more fully discussed at [[simplicial category]].)

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

There is a [[model category]] structure on $sGrpd$ whose

\begin{itemize}%
\item fibrations are the morphisms $f : H \to K$ such that

\begin{enumerate}%
\item for every [[object]] $x$ of $H$ and every morphism $\omega : f(x) \to y$ in $K_0$ there is a morphism $\hat \omega : x \to z$ of $H_0$ such that $f(\hat \omega) = \omega$;


\item for every object $x$ in $H$ the induced morphism $f : H(x,x) \to K(f(x), f(x))$ is a [[Kan fibration]].



\end{enumerate}

\item weak equivalences are morphisms $f : H \to K$ such that

\begin{enumerate}%
\item $f$ induces in [[isomorphism]] on connected components $\pi_0 f : \pi_0 H \to \pi_0 K$;


\item for each object $x$ of $H$ the induced morphism $H(x,x) \to K(f(x), f(x))$ is a weak equivalence in the [[model structure on simplicial groups]] or equivalently in the [[model structure on simplicial sets]].



\end{enumerate}


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

\begin{prop}
\label{}\hypertarget{}{}
We have a [[Quillen adjunction]]

\begin{displaymath}
(G \dashv \bar W) : 
  Grpd^\Delta
   \stackrel{\overset{G}{\leftarrow}}{\underset{\bar W}{\to}}
  sSet_{Quillen}
\end{displaymath}
where both $G$ and $\bar W$ preserve all [[weak equivalence]]s.

\end{prop}
This appears for instance as (\hyperlink{GoerssJardine}{GoerssJardine, theorem 7.8})

\begin{remark}
\label{}\hypertarget{}{}
When restricted to simplicial groupoids of the form $(B G)_\bullet$ for $G_\bullet$ a [[simplicial group]] and $B G_n$ its [[delooping]] [[groupoid]] this produces a standard presentation of [[looping and delooping]] for [[infinity-group]]s. See there for more details.

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

\begin{itemize}%
\item [[model structure on simplicial sets]]


\item [[model structure on presheaves of simplicial groupoids]]



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

The model structure is discussed after corollary 7.3 in chapter V of

\begin{itemize}%
\item [[Paul Goerss]] and [[Frederick 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{document}