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\begin{document}

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\section*{model structure on cosimplicial simplicial sets}

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

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

[[!include model category theory - contents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

For $\Delta$ the [[simplex category]] the [[functor category]] $sSet^{\Delta}$ is that of [[cosimplicial object]]s in [[simplicial set]]s: [[cosimplicial simplicial set]]s.

There are various standard [[model category]] structures on this category. The [[Reedy model structure]] is discussed in (\hyperlink{BousfieldKan}{BousfieldKan}), the injective structure is discussed in (\hyperlink{Jardine}{Jardine}).

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

\begin{remark}
\label{}\hypertarget{}{}
The [[totalization]] of a cosimplicial simplicial set $X^\bullet$ coincides with the [[sSet]]-enriched [[hom-object]]

\begin{displaymath}
Tot(X_\bullet) = sSet^{\Delta}(\Delta, X_\bullet)
  \,,
\end{displaymath}
where $\Delta : [k] \mapsto \Delta[k]$ is the canonical cosimplicial simplicial set given by the [[simplex]]-assignment.

Since $\Delta$ is cofibrant in the [[Reedy model structure]] it follows that totalization of Reedy-fibrant cosimplicial simplicial sets preserves weak equivalences. The following lists situations in which totalization respects weak equivalences even without this assumption.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Totalization is closely related to [[descent objects]]. If $A$ is a [[simplicial presheaf]] and $Y \to X$ is a [[hypercover]], then the descent object is the [[sSet]]-[[hom object|enriched hom]]

\begin{displaymath}
Desc(Y,X) = sPsh(Y,A)
  \,.
\end{displaymath}
If we decompose

\begin{displaymath}
Y = \int^{[n] \in \Delta} \Delta[n] \cdot Y_n
\end{displaymath}
into its cells by a [[coend]], where now each $Y_n$ is a Set-valued [[presheaf]] (see [[co-Yoneda lemma]]), then this is

\begin{displaymath}
\begin{aligned}
    Desc(Y,A) 
    &= 
    sPSh(\int^{n \in \Delta} \Delta[n] \cdot Y_n, A)
    \\
    & = 
    \int_{n \in \Delta} sPSh(\Delta[n] \cdot Y_n , A)
    \\
    & = \int_{n \in \Delta} sSet(\Delta[n], sPSh(Y_n,A))
   \\
    &= Tot( sPSh(Y_\bullet, A))
  \end{aligned}
  \,,
\end{displaymath}
where the equality signs are [[isomorphisms]] of [[simplicial sets]], the outside integral sign denotes the [[end]], and in the integrand we are using that [[simplicial presheaves]] are [[simplicially enriched category|simplicially enriched]] and [[tensoring|tensored]] over simplicial sets.

So a standard class of examples of cosimplicial simplicial sets to keep in mind are those obtained by evaluating a [[simplicial presheaf]] degreewise on the components of a [[hypercover]]. Its totalization then is the corresponding [[descent object]].

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
For $G^\bullet \to H^\bullet$ a morphism of [[cosimplicial object|cosimplicial]] [[groupoid]]s which is degreewise an [[equivalence of categories|equivalence]], also the induced morphism of [[totalization]]s

\begin{displaymath}
Tot(G^\bullet) \to Tot(H^\bullet)
\end{displaymath}
is a weak equivalence (of [[simplicial set]]s).

\end{prop}
This is (\hyperlink{Jardine}{Jardine, corollary 12}).

\begin{defn}
\label{RestrictedTotalization}\hypertarget{RestrictedTotalization}{}
Let $\Delta_+ \hookrightarrow \Delta$ be the [[subcategory]] of the [[simplex category]] on the co-face maps. Write $rTot$ for the corresponding [[totalization]], called the \textbf{restricted totalization}.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
For $G^\bullet \to H^\bullet$ a degreewise weak equivalence of [[strict 2-category|strict]] [[2-groupoid]]s, the resulting morphism of connected components of \hyperlink{RestrictedTotalization}{restricted totalizations}

\begin{displaymath}
rTot(G^\bullet) \to rTot(H^\bullet)
\end{displaymath}
is a [[weak equivalence]].

\end{prop}
This is (\hyperlink{Prezma}{Prezma, theorem 6.1}).

Retsricted to $\pi_0$ this statement appeared as (\hyperlink{Yekutieli}{Yekutieli, theorem 2.4}). Notice that it is indeed necessary to use the restricted totalization instead of the ordinary totalization here.

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

The [[Reedy model structure]] on $sSet^{\Delta}$ is discussed in Chapter X of

\begin{itemize}%
\item [[Aldridge Bousfield]] and [[Dan Kan]], \emph{Homotopy limits, completions and localizations} Springer-Verlag, Berlin, 1972. Lecture Notes in Mathematics, Vol. 304.

\end{itemize}
The [[model structure on functors|injective model structure]] is discussed in

\begin{itemize}%
\item [[Rick Jardine]], \emph{Cosimplicial spaces and cocycles} (\href{http://www.math.uwo.ca/~jardine/papers/preprints/cosimp5.pdf}{pdf})

\end{itemize}
Totalization of cosimplicial strict [[2-groupoids]] is considered in

\begin{itemize}%
\item [[Amnon Yekutieli]], \emph{Combinatorial Descent Data for Gerbes} (\href{http://arxiv.org/abs/1109.1919}{arXiv:1109.1919})

\end{itemize}
and

\begin{itemize}%
\item [[Matan Prezma]], \emph{Descent data of cosimplicial 2-groupoids} (\href{http://arxiv.org/abs/1112.3072}{arXiv:1112.3072})

\end{itemize}
[[!redirects model structure on cosimplicial spaces]]



\end{document}