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\section*{Čech model structure on simplicial presheaves}

\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{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \textbf{ech model structure on simplicial presheaves} on a [[site]] $C$ is a model for the [[topological localization]] of an [[(∞,1)-category of (∞,1)-presheaves]] on $C$ to the [[(∞,1)-category of (∞,1)-sheaves]].

It is obtained from the the [[global model structure on simplicial presheaves]] on $C$ by [[Bousfield localization of model categories|left Bousfield localization]]s at [[Cech cover]]s: its fibrant objects are [[∞-stack]]s that satisfy [[descent]] over [[Cech cover]]s but not necessarily over [[hypercover]]s.

Further [[Bousfield localization of model categories|left Bousfield localization]] at [[hypercover]]s leads from the ech model structure to the Joyal-Jardine [[local model structure on simplicial presheaves]] that presents the [[hypercomplete (∞,1)-topos]] which is the [[hypercompletion]] of that presented by the ech model structure.

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

Let $C$ be a small [[site]] and write $[C^{op}, sSet]_{proj}$ and $[C^{op}, sSet]_{inj}$ for the projective and injective [[global model structure on simplicial presheaves]], respectively.

For $\{U_i \to V\}_i$ a covering family in the [[site]] $C$, let

\begin{displaymath}
C(\{U_i\})
  :=
  \left(
    \cdots\stackrel{\to}{\stackrel{\to}{\to}}\coprod_{i j} U_{i j}\stackrel{\to}{\to}\coprod_i U_i
  \right)
\end{displaymath}
be the corresponding [[Cech nerve]], regarded as a [[simplicial presheaf]] on $C$. This comes canonically with a morphism

\begin{displaymath}
C(\{U_i\}) \to V
\end{displaymath}
of simplicial presheaves, the corresponding \emph{ech cover morphism} .

Notice that by the discussion at \href{http://ncatlab.org/nlab/show/model+structure+on+simplicial+presheaves#FibAndCofibObjects}{model structure on simplicial presheaves - fibrant and cofibrant objects} this is a morphism between cofibrant objects.

\begin{udefn}
The injective (projective) \textbf{ech model structure on simplicial presheaves} $[C^{op},sSet]_{Cech}$ on $C$ is the [[Bousfield localization of model categories|left Bousfield localization]] of $[C^{op}, sSet]_{inj}$ ($[C^{op}, sSet]_{proj}$) at the set of ech cover morphisms.

\end{udefn}
By the general poperties of [[Bousfield localization of model categories|Bousfield localization]] this means that the fibrant-cofibrant objects $A$ of $[C^{op},sSet]_{Cech}$ are precisely those that are fibrant-cofibrant in the global model structure and in addition satisfy the [[descent]] condition that for all covers $\{U_i \to V\}$ the morphism of simplicial sets

\begin{displaymath}
A(U) \simeq [C^op,sSet](V,U) \to [C^{op},sSet](C(\{U_i\}), A)
\end{displaymath}
is a weak equivalence in the standard [[model structure on simplicial sets]].

This is the model for the $\infty$-analog of the [[sheaf]] condition, modelling the [[topological localization]] of an $(\infty,1)$-presheaf $(\infty,1)$-topos.

[[Mike Shulman]]: Two questions, one (hopefully) easy and one (perhaps) hard:

\begin{enumerate}%
\item Is there a Quillen equivalent ech model structure on simplicial \emph{sheaves}? Can we just lift the model structure for simplicial presheaves along the sheafification adjunction?


\item Is there a characterization of the weak equivalences in either ech model structure?



\end{enumerate}
I am particularly interested in this for the following reason. According to Beke in \emph{Sheafifiable homotopy model categories}, the weak equivalences in the [[local model structure on simplicial sheaves]] are precisely those maps $f\colon X\to Y$ of simplicial objects in the corresponding 1-topos of sheaves of sets such that the statement ``$f$ is a weak equivalence of simplicial sets'' is true in the [[internal logic]] of the topos (at least, interperiting ``$f$ is a weak equivalence of simplicial sets'' by one particular set of geometric sentences whose interpretation in $Set$ is equivalent to saying that a simplicial map is a weak equivalence). But if, as [[HTT]] teaches us, ech descent is often to be preferred to hyperdescent, then we should be interested in ech weak equivalences instead. So I would really like to know what it means for a map of simplicial sheaves to be a ech weak equivalence, in the \emph{internal logic} of the 1-topos of sheaves of sets. If nothing else, I think such a characterization would help me understand the real meaning of [[hypercompletion]]. But any sort of characterization of them would be better than none.

[[Urs Schreiber]]: below is a reply to the first question.

[[Mike Shulman]]: Thanks for attacking this. I thought I should also mention, for anyone listening in, that this question is evidently also relevant to what the correct notion of [[internal ∞-groupoid]] may be.

\begin{quote}%
check


\end{quote}
We may form the [[transferred model structure]] on simplicial \emph{[[sheaf|sheaves]]} by transferring along the degreewise [[sheafification]] [[adjunction]]

\begin{displaymath}
Sh(C) \stackrel{\overset{sh}{\leftarrow}}{\underset{}{\hookrightarrow}}
  PSh(C)
  \,.
\end{displaymath}
This defines fibrations and weak equivalences in $sSh(C)$ to be those morphisms that are fibrations or weak equivalences, respectively, as morphism in $sPSh(C)_{Cech} = [C^{op},sSet]_{Cech}$.

As discussed there, sufficient conditions for this to be a model structure is that

\begin{itemize}%
\item the inclusion $Sh(C) \hookrightarrow PSh(C)$ preserves [[filtered colimit]]s;


\item $sSh(C)$ has functorial fibrant replacement and functorial [[path object]]s for fibrant objects.



\end{itemize}
Since [[sheafification]] does preserve [[filtered colimit]]s the first condition is satisfied degreewise and hence is satisfied.

[[Mike Shulman]]: I believe that sheafification preserves $\kappa$-filtered colimits for some sufficiently large $\kappa$, but if the site has covers of infinite cardinality, I don't see why sheafification would preserve $\omega$-filtered colimits. But I think this is enough for the proof to work.

Since the [[small object argument]] holds in $sSh(C)$ for generating acyclic cofibrations we have functorial fibrant replacement. And a path object is obtained just by forming objectwise the standard path object in [[sSet]], as in $[C^{op}, sSet]$.

[[Mike Shulman]]: The small object argument doesn't automatically produce functorial fibrant replacements in this context\ldots{} isn't the whole question whether the map to the ``fibrant replacement'' is still a weak equivalence (in the underlying category)? I.e. whether $F(J)$-cell complexes are still weak equivalences.

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

A detailed though unfinished account of the ech model structure is given in

\begin{itemize}%
\item Daniel Dugger, \emph{Sheaves and homotopy theory} (\href{http://www.uoregon.edu/~ddugger/cech.html}{web}, \href{http://ncatlab.org/nlab/files/cech.pdf}{pdf})

\end{itemize}
But beware of this document is unfinished. Some aspects of this appeared in

\begin{itemize}%
\item [[Daniel Dugger]], \emph{[[DuggerUniv.pdf:file]]}

\end{itemize}
[[!redirects ?ech model structure on simplicial presheaves]] [[!redirects Cech model structure on simplicial presheaves]] [[!redirects ?ech model structure on simplicial presheaves]]



\end{document}