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\begin{document}

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\section*{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{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{the_different_model_structures_and_their_interrelation}{The different model structures and their interrelation}\dotfill \pageref*{the_different_model_structures_and_their_interrelation} \linebreak
\noindent\hyperlink{injectiveprojective__localglobal__presheavessheaves}{Injective/projective - local/global - presheaves/sheaves}\dotfill \pageref*{injectiveprojective__localglobal__presheavessheaves} \linebreak
\noindent\hyperlink{reedy_and_intermediate_model_structures}{Reedy and intermediate model structures}\dotfill \pageref*{reedy_and_intermediate_model_structures} \linebreak
\noindent\hyperlink{DependencyOnSite}{Dependency on the underlying site}\dotfill \pageref*{DependencyOnSite} \linebreak
\noindent\hyperlink{PresentationOfInfiniToposes}{Presentation of $(\infty,1)$-toposes}\dotfill \pageref*{PresentationOfInfiniToposes} \linebreak
\noindent\hyperlink{FibAndCofibObjects}{Fibrant and cofibrant objects}\dotfill \pageref*{FibAndCofibObjects} \linebreak
\noindent\hyperlink{fibrant_objects}{Fibrant objects}\dotfill \pageref*{fibrant_objects} \linebreak
\noindent\hyperlink{CofibrantObjects}{Cofibrant objects}\dotfill \pageref*{CofibrantObjects} \linebreak
\noindent\hyperlink{CofibrantReplacement}{Cofibrant replacement}\dotfill \pageref*{CofibrantReplacement} \linebreak
\noindent\hyperlink{local_fibrations}{Local fibrations}\dotfill \pageref*{local_fibrations} \linebreak
\noindent\hyperlink{Descent}{Localization and descent}\dotfill \pageref*{Descent} \linebreak
\noindent\hyperlink{ČechLocalization}{Čech localization at Grothendieck (pre)topologies}\dotfill \pageref*{ČechLocalization} \linebreak
\noindent\hyperlink{LocalizationAtCoverage}{Čech localization at a coverage}\dotfill \pageref*{LocalizationAtCoverage} \linebreak
\noindent\hyperlink{for_values_in_strict_and_abelian_groupoids}{For values in strict and abelian $\infty$-groupoids}\dotfill \pageref*{for_values_in_strict_and_abelian_groupoids} \linebreak
\noindent\hyperlink{Properness}{Properness}\dotfill \pageref*{Properness} \linebreak
\noindent\hyperlink{MonoidalStructure}{Closed monoidal structure}\dotfill \pageref*{MonoidalStructure} \linebreak
\noindent\hyperlink{HomotopyLimits}{Homotopy (co)limits}\dotfill \pageref*{HomotopyLimits} \linebreak
\noindent\hyperlink{InclusionOfChainComplexes}{Inclusion of chain complexes of sheaves}\dotfill \pageref*{InclusionOfChainComplexes} \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}

[[model category|Model structures]] on [[simplicial presheaf|simplicial presheaves]] [[presentable (∞,1)-category|present]] [[(∞,1)-category of (∞,1)-presheaves|(∞,1)-categories of (∞,1)-presheaves]] and [[localization of an (∞,1)-category|localizations of these]], such as notably the left exact localizations that are [[(∞,1)-category of (∞,1)-sheaves|(∞,1)-categories of (∞,1)-sheaves]]: these model structures are [[models for ∞-stack (∞,1)-toposes]].

Recall that

\begin{itemize}%
\item a [[model category]] is a way to [[presentable (infinity,1)-category|present]] an [[(∞,1)-category]];


\item in the context of [[(∞,1)-categories]] [[presheaf|presheaves]] on an $(\infty,1)$-category $C$ are given by [[(∞,1)-functors]] $C^{op} \to$ [[SSet]].



\end{itemize}
This suggests that the [[(∞,1)-category of (∞,1)-sheaves]] on some [[site]] $C$ can be [[presentable (infinity,1)-category|presented]] by a [[model category]] structure on the ordinary [[functor category]]

\begin{displaymath}
[C^{op},SSet] \simeq [\Delta^{op}, PSh(C)]
\end{displaymath}
-- the category of [[simplicial presheaf|simplicial presheaves]] .

Various interrelated flavors of model structures on the category of simplicial presheaves on $C$ have been introduced and studied since the 1970s, originally by K. Brown and [[Andre Joyal]] and then developed in detail by [[Rick Jardine]].

Notice that when regarded as a presentation of an [[(∞,1)-sheaf]], i.e. of an [[∞-stack]], a simplicial presheaf -- being an ordinary functor instead of a [[pseudofunctor]] -- corresponds to a [[rectified ∞-stack]]. It might therefore seem that a model given by simplicial presheaves is too restrictive to capture the full expected flexibility of a notion of [[∞-stack]]. But this is not so.

In

\begin{itemize}%
\item [[Jacob Lurie]], [[Higher Topos Theory]]

\end{itemize}
a fully general definition of a [[(infinity,1)-category of (infinity,1)-sheaves|(∞,1)-category of ∞-stacks]] is given it is shown -- proposition 6.5.2.1 -- that, indeed, the Brown--Joyal--Jardine model is a [[presentable (infinity,1)-category|presentation]] of that.

More precisely

\begin{itemize}%
\item the [[global model structure on simplicial presheaves]] on a category is a [[presentable (infinity,1)-category|presentation]] of the [[(∞,1)-category of (∞,1)-presheaves]];


\item the [[Čech model structure on simplicial presheaves]] on a [[site]] is a [[presentable (infinity,1)-category|presentation]] of the [[(∞,1)-category of (∞,1)-sheaves]];


\item the [[local model structure on simplicial presheaves]] on a [[site]] is a [[presentable (infinity,1)-category|presentation]] of the \emph{hypercompletion} of the [[(∞,1)-category of (∞,1)-sheaves]] (see the discussion at [[hypercover]]).


\item the [[Bousfield localization]] of the global [[model category]] structure to the local one presents the corresponding [[localization of an (∞,1)-category]] from presheaves to sheaves, mimicking the corresponding statement for a [[category of sheaves]].



\end{itemize}
Originally K. Brown had considered in [[BrownAHT]] not a model structure on simplicial presheaves but

\begin{itemize}%
\item a [[category of fibrant objects]] structure on locally Kan simplicial sheaves (see there for details)

\end{itemize}
and Joyal had originally considered a

\begin{itemize}%
\item [[local model structure on simplicial sheaves]].

\end{itemize}
Joyal's [[local model structure on simplicial sheaves]] is [[Quillen equivalence|Quillen equivalent]] to the injective [[local model structure on simplicial presheaves]].

By repackaging [[Kan complex]]es as [[simplicial groupoids]] one obtains a [[model structure on presheaves of simplicial groupoids]] which is also Quillen equivalent to the above.

If K. Brown's [[category of fibrant objects]] on locally Kan simplicial sheaves is restricted to globally Kan simplicial sheaves on a [[topos]] with [[point of a topos|enough point]] then it is the full subcategory on the fibrant objects in the projective [[local model structure on simplicial sheaves]].

But since in all cases the weak equivalences are the same (where they apply, for Brown's model if the topos has enough points), all these local [[homotopical category|homotopical categories]] define equivalent [[homotopy category|homotopy categories]].

By Lurie's result these are in each case in turn equivalent to the [[homotopy category of an (infinity,1)-category|homotopy category of]] the [[(∞,1)-topos]] of [[∞-stacks]]. So in particular they serve as a home for general [[cohomology]].

Various old results appear in a new light this way. For instance using the old result of [[BrownAHT]] on the way ordinary [[abelian sheaf cohomology]] is embedded in the [[homotopy theory]] of simplicial sheaves, one sees that the old right derived functor definition of [[abelian sheaf cohomology]] really computes the [[∞-stackification]] of a sheaf of [[chain complex]]es regarded under the [[Dold?Kan correspondence]] as a simplicial sheaf.

\hypertarget{the_different_model_structures_and_their_interrelation}{}\subsection*{{The different model structures and their interrelation}}\label{the_different_model_structures_and_their_interrelation}

It is the very point of [[model category]] structures on a given [[homotopical category]] that there may be several of them: each [[presentable (infinity,1)-category|presenting]] the same [[(∞,1)-category]] but also each suited for different computational purposes.

So it is good that there are many model structures on simplicial (pre)sheaves, as there are.

\hypertarget{injectiveprojective__localglobal__presheavessheaves}{}\subsubsection*{{Injective/projective - local/global - presheaves/sheaves}}\label{injectiveprojective__localglobal__presheavessheaves}

The following diagram is a map for part of the territory:

Here

\begin{itemize}%
\item ``inj'' denotes the injective model structure: cofibrations are objectwise cofibrations


\item ``proj'' denotes the projective model structure: fibrations are objectwise fibrations


\item no ``loc'' subscript means global model structure: weak equivalences are the objectwise weak equivalences:


\item ``l loc'' denotes \textbf{left} [[Bousfield localization]] at [[hypercovers]] (at [[stalk]]wise acyclic fibrations if the [[point of a topos|topos has enough points]])



\end{itemize}
The identity functor on the category $SPSh(C)$ of [[simplicial presheaves]] is a [[Quillen adjunction]] for the projective and injective [[global model structure on simplicial presheaves|global model structure]] and this is a [[Quillen equivalence]].

The [[local model structure on simplicial sheaves|local model structures on simplicial sheaves]] are just the restrictions of the [[local model structure on simplicial presheaves|those on simplicial presheaves]]. (For the injective structure this is in \hyperlink{JardineLecture}{Jardine}, for the projective one in \hyperlink{Blander}{Blander, theorem 2.1, 2.2}).

These are related by a [[Quillen adjunction]] given by the usual [[geometric embedding]] of the [[category of sheaves]] as a full [[subcategory]] of that of presheaves -- with [[sheafification]] the [[left adjoint]] -- and this is also [[Quillen equivalence]].

The characteristic of the \emph{left} Bousfield localizations is that for them the fibrant objects are those that satisfy [[descent]]: see [[descent for simplicial presheaves]].

In either case

\begin{itemize}%
\item the global model structures [[presentable (infinity,1)-category|presents]] the [[(∞,1)-category of (∞,1)-presheaves]]

\end{itemize}
while

\begin{itemize}%
\item the local model structures [[presentable (infinity,1)-category|presents]] the [[(∞,1)-category of (∞,1)-sheaves]] (i.e. [[∞-stacks]]).

\end{itemize}
The following diagram collection [[model category|model categories]] that are [[presentable (infinity,1)-category|presentations]] for the [[(∞,1)-category of (∞,1)-sheaves]]. All indicated morphism pairs are [[Quillen equivalences]].

On the right this lists the model structures on simplicial (pre)sheaves, here displayed as (pre)sheaves with values in [[simplicial sets]], using $SPSh(C) \simeq PSh(C,SSet)$.

On the left we have the Joyal--Tierney and Luo--Bubenik--Tim [[model structures on presheaves of simplicial groupoids]].

(\ldots{}have to check here the relation $Sh(X,SGrpd)\leftrightarrow PSh(X, SGrpd)$)

\hypertarget{reedy_and_intermediate_model_structures}{}\subsubsection*{{Reedy and intermediate model structures}}\label{reedy_and_intermediate_model_structures}

To some extent the injective and projective model structures on simplicial presheaves are the two extremes of a larger family of model structures on simplicial presheaves that all have the same weak equivalences but different classes of cofibrations.

Notably if the domain $C$ has the special property that it is a [[Reedy category]] there is the [[Reedy model structure]] on $[C, sSet]$. Its class of cofibrations is intermediate that of the projective and the injective [[model structure on functors]] and we have [[Quillen equivalence]]s

\begin{displaymath}
[C,sSet]_{proj} \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}}
  [C,sSet]_{Reedy}
  \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}}
  [C,sSet]_{inj}
  \,.
\end{displaymath}
For general $C$, there is still a whole family of model structures on $[C^{op}, sSet]$ that interpolates between the injective and the projective model structure. See [[intermediate model structure]].

\hypertarget{DependencyOnSite}{}\subsubsection*{{Dependency on the underlying site}}\label{DependencyOnSite}

\begin{prop}
\label{SiteDependence}\hypertarget{SiteDependence}{}
Let $C,D$ be [[site]]s and let $f : C \to D$ be a [[functor]] that induces a morphism of [[site]]s in that $f_* : PSh(D) \to PSh(C)$ preserves [[sheaf|sheaves]] and its [[left adjoint]] $f^* : PSh(C) \to PSh(D)$ (given by left [[Kan extension]]) is left [[exact functor]] in that it preserves [[finite limit]]s.

Then the induced [[adjunction]]

\begin{displaymath}
f_* : SPSh(D)_{inj}^{loc} \stackrel{\leftarrow}{\to} SPSh(C)_{inj}^{loc} : f^*
\end{displaymath}
is a [[Quillen adjunction]] for the local injective model structure on presheaves on both sides.

\end{prop}
\begin{proof}
This is ``little fact 5)'' on page 10, 11 of (\hyperlink{JardineLecture}{JardineLectures}).

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
Let $C$ be a [[site]] and $f : D \hookrightarrow$ a [[full subcategory|full]] [[dense sub-site]]. Then right [[Kan extension]] $f_* : [D^{op}, sSet] \to [C^{op}, sSet]$ along $f$ yields a [[simplicial Quillen adjunction]]

\begin{displaymath}
(f^* \dashv f_*) 
   : 
  [D^{op}, sSet]_{inj,loc} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} 
  [C^{op}, sSet]_{inj,loc}
\end{displaymath}
between the [[Bousfield localization of model categories|left Bousfield localizations]] of the projective model structures at the [[sieve]] inclusions $S(\{U_i\}) \to U$ for each [[covering]] family $\{U_i \to U\}$.

\end{prop}
\begin{proof}
It is immediate that we have a simplicial Quillen adjunction on the global injective model structure: by definition of right [[Kan extension]] we have an [[sSet]]-[[adjunction]] and the [[left adjoint]] restriction functor $f^*$ trivially preserves injective cofibrations and acyclic cofibrations.

Since we have [[left proper model categories]] it is sufficient (by the discussion at ) for deducing that the Quillen adjunction descends to the local strucuture to check that $f_*$ preserves locally fibrant objects, which in turn by properties of [[Bousfield localization of model categories|left Bousfield localization]] is equivalent to checking that $f^*$ sends covering sieve inclusions to weak equivalences in $[D^{op}, sSet]_{proj,loc}$.

By the \hyperlink{GeneralizedCover}{result on generalized covers}, for this it is sufficient to check that for every covering sieve $S(\{U_i\}) \to X$ and every representable $K \in D$ and morphism $K \to f^* X$, there is a [[covering]] $\{K_j \to K\}$ in $D$ and local lifts

\begin{displaymath}
\itexarray{
    K_j &\to& f^*(S(\{U_i\}))
    \\
    \downarrow && \downarrow
    \\
    K &\to& f^* X
  }
  \,.
\end{displaymath}
This follows directly from the single defining condition on a [[coverage]] on $C$.

\end{proof}
\hypertarget{PresentationOfInfiniToposes}{}\subsection*{{Presentation of $(\infty,1)$-toposes}}\label{PresentationOfInfiniToposes}

\begin{def}
\label{}\hypertarget{}{}
Let $C$ be a [[site]]. Let $[C^{op}, sSet]_{proj}$ be the projective [[model structure on simplicial presheaves]] over $C$.

Let $W = \{C(\{U_i\}) \to U\}$ be the set of [[Čech nerve]] projections in $[C, sSet]$ for each [[covering]] $\{U_i \to U\}$ in $C$.

Write

\begin{displaymath}
(Id \dashv Id)
  : 
  [C^{op}, sSet]_{proj,loc}
  \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}}
  [C^{op}, sSet]_{proj}
\end{displaymath}
for the [[Bousfield localization of model categories|left Bousfield localization]] at $W$.

Write $([C^{op}, sSet]_{proj})^\circ$ for the full sub-[[simplicially enriched category]] on the fibrant-cofibrant objects, similarly for $([CartSp^{op}, sSet]_{proj,loc})^\circ$.

\end{def}
\begin{prop}
\label{PresentationOfTheInfinTopos}\hypertarget{PresentationOfTheInfinTopos}{}
We have an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
\itexarray{
    Sh_{(\infty,1)}(C)
    &\stackrel{\overset{L}{\leftarrow}}{\hookrightarrow}& 
    PSh_{(\infty,1)}(C)
    \\
    \uparrow^{\mathrlap{\simeq}} &&
    \uparrow^{\mathrlap{\simeq}}
    \\
    ([C^{op}, sSet]_{proj,loc})^\circ
    &
    \stackrel
       { \overset{\mathbb{L} Id}{\leftarrow} }
       { \underset{\mathbb{R}Id}{\to} }
    &
    ([C^{op}, sSet]_{proj})^\circ
  }
  \,,
\end{displaymath}
where at the bottom we have the left and right [[derived functor]]s of the identity functors, as discussed at [[simplicial Quillen adjunction]].

\end{prop}
\begin{proof}
This follows using the arguments in the proof of [[Higher Topos Theory|HTT, 6.5.2.14]] and [[Higher Topos Theory|HTT, prop. A.3.7.6]].

\end{proof}
\hypertarget{FibAndCofibObjects}{}\subsection*{{Fibrant and cofibrant objects}}\label{FibAndCofibObjects}

\hypertarget{fibrant_objects}{}\subsubsection*{{Fibrant objects}}\label{fibrant_objects}

The fibrant objects in the [[local model structure on simplicial presheaves]] are those that

\begin{itemize}%
\item are fibrant with respect to the respective [[global model structure on simplicial presheaves|global model structure]]


\item and satisfy [[descent for simplicial presheaves]]. See there for more details.



\end{itemize}
This descent condition is the analog in this model of the [[sheaf]]-condition and the [[stack]]-condition. In fact, it reduces to these for truncated simplicial presheaves.

Since the fibrancy condition in the global projective model structure is simple -- it just requires that the [[simplicial presheaf]] is in fact a presheaf of [[Kan complex]]es -- the local projective model structure has slightly more immediate characterization of fibrant objects than the local injective model structures. (In fact, for suitable choices of [[site]]s it may become very simple, as the above discussion of site dependence of the model structure shows).

On the other hand the cofibrancy condition on objects is entirely \emph{trivial} in the global and local injective model structure: since a cofibration there is just an objectwise cofibration, and since every [[simplicial set]] is cofibrant, every object is injective cofibrant.

But the cofibrant objects in the projective structure are not too nasty either: every object that is degreewise a coproduct of representables is cofibrant. In particular the [[Čech nerve]]s of any \emph{[[good cover]]} (see below for more details) is a projectively cofibrant object.

A \textbf{cofibrant replacement} functor in the local projective structure is discussed in

\begin{itemize}%
\item [[Daniel Dugger]], \emph{Universal homotopy theories} (\href{http://hopf.math.purdue.edu/Dugger/dduniv.pdf}{pdf})

\end{itemize}
Something related to a \textbf{fibrant replacement} functor (``$\infty$-stackification'') is discussed in section 6.5.3 of

\begin{itemize}%
\item [[Jacob Lurie]], [[Higher Topos Theory]]

\end{itemize}
\hypertarget{CofibrantObjects}{}\subsubsection*{{Cofibrant objects}}\label{CofibrantObjects}

In the injective [[local model structure on simplicial presheaves]] all objects are cofibrant. For the projective local structure

necessary and sufficient conditions are given here:

\begin{itemize}%
\item [[Richard Garner]], \href{https://mathoverflow.net/a/127187/381}{MO comment}

\end{itemize}
More specifically, there is the following useful statement (see also \emph{[[projectively cofibrant diagram]]}) (see also \hyperlink{Low}{Low, remark 8.2.3}).

\begin{defn}
\label{}\hypertarget{}{}
A simplicial presheaf $X \in sPSh(C)$ is said to have \textbf{free degeneracies} or the \textbf{degenerate cells split off} if in each degree there is a [[subobject|sub]]-presheaf $N_k \hookrightarrow X_k$ such that the canonical mophism

\begin{displaymath}
\coprod_{\underset{surj.}{\sigma : [k] \to [n]}} 
  N_n \stackrel{\coprod_{\underset{surj.}{\sigma : [k] \to [n]}} \sigma^*}{\to} 
  F_k
\end{displaymath}
is an [[isomorphism]].

\end{defn}
So if degenerate cells split off we have in particular that

\begin{displaymath}
X_k = X_k^{nd} \coprod X_k^{dg}
  \,,
\end{displaymath}
where $X_k^{nd}$ is the presheaf of non-degenerate $k$-cells and $X_k^{dg}$ is a separate presheaf containing all the degenerate cells (and itself a coproduct over separate presheaves for each degree and order of degeneracy).

\begin{prop}
\label{}\hypertarget{}{}
In the \emph{projective} [[local model structure on simplicial presheaves|local model structure]] all objects that are

\begin{enumerate}%
\item degreewise [[coproduct]]s of [[representable functor|representable]]s


\item and whose degenerate cells split off



\end{enumerate}
are cofibrant.

\end{prop}
This is in the proof of lemma 2.7 in section 9 of

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

\end{itemize}
\begin{example}
\label{}\hypertarget{}{}
\textbf{(split hypercovers)}

If $Y \to X$ is an acyclic fibration in the local projective model structure with $X$ a representable and $Y$ cofibration in the above way, it is called a \textbf{[[split hypercover]]} .

All [[Čech nerve]]s $C(\{U_i\})$ coming from an [[open cover]] have split degeneracies. The condition that the Čech nerve be degreewise a coproduct of representables is a condition akin to that of [[good open cover]]s (which is precisely the special case for $C =$ [[CartSp]]). This is then a split hypercover of \emph{height} 0.

\end{example}
\begin{defn}
\label{}\hypertarget{}{}
\textbf{(good cover)}

A [[Čech nerve]] $U$ with a weak equivalence $U \stackrel{\simeq}{\to} X$ in $SPSh(C)^{loc}$ is a \textbf{[[good cover]]} if it is degreewise a coproduct of [[representable functor|representable]]s.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This reduces to the ordinary notion of [[good cover]] as an open cover by contractible spaces such that all finite intersections of these are again contractibe when using a [[site]] like $C =$ [[CartSp]].

\end{remark}
\hypertarget{CofibrantReplacement}{}\subsubsection*{{Cofibrant replacement}}\label{CofibrantReplacement}

In

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

\end{itemize}
a useful cofibrant replacement functor for the projective local model structure is discussed.

\begin{defn}
\label{}\hypertarget{}{}
For $A  \in PSh(C) \hookrightarrow SPSh(C)$ an ordinary presheaf (simplicially discrete simplicial presheaf) let $\tilde Q A$ be the simplicial presheaf which in degree $k$ is

\begin{displaymath}
(\tilde Q A)_k := \coprod_{U_k \to U_{k-1} \to \cdots \to U_0 \to A} U_k
  \,,
\end{displaymath}
where the $U_k$ range over the representables, i.e. the objects in $C \hookrightarrow SPSh(C)$. The face and degeneracy maps are the obvious ones coming from composing maps and inserting identity maps in the labels over which the coproduct ranges.

For $A \in SPSh(C)$ an arbitrary simplicial presheaf let $Q A$ be the diagonal of the bisimplicial presheaf obtained by applying $\tilde Q$ degreewise

\begin{displaymath}
Q A =
  \left(
    \cdots
    \coprod_{U_1 \to U_0 \to A_1}
     U_1
    \stackrel{\to}{\to}\coprod_{U_0 \to A_0} U_0
  \right)
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
For all $A \in SPSh(C)$ the object $Q A$ is cofibrant and is weakly equivalent to $A$ in $SPSh(C)_{proj}^{loc}$.

\end{prop}
This is in prop 2.8 of

\begin{itemize}%
\item [[Daniel Dugger]], \emph{[[Universal Homotopy Theories]]}

\end{itemize}
\hypertarget{local_fibrations}{}\subsection*{{Local fibrations}}\label{local_fibrations}

A \emph{[[local fibration]]} or \emph{local weak equivalence} of simplicial (pre)sheaves is defined to be one whose lifting property is satisfied after refining to some cover.

\textbf{Warning}. Notice that this is a priori unrelated to equivalences and fibrations with respect to any local model structure.

If the [[site]] $C$ has [[point of a topos|enough points]], then local fibrations of simplicial presheaves are equivalently those that are [[stalk]]wise fibrations of simplicial sets.

This is discussed in (\hyperlink{Jardine96}{Jardine 96}).

\hypertarget{Descent}{}\subsection*{{Localization and descent}}\label{Descent}

\hypertarget{ČechLocalization}{}\subsubsection*{{Čech localization at Grothendieck (pre)topologies}}\label{ČechLocalization}

We discuss some aspects of the [[Bousfield localization of model categories|left Bousfield localization]] of the projective global model structure on simplicial presheaves at [[Grothendieck topologies]] and [[covering]] families. By the discussion at [[topological localization]] these are models for [[topological localization]]s leading to [[(∞,1)-categories of (∞,1)-sheaves]].

The central reference is (\hyperlink{DuggerHollanderIsaksen}{DuggerHollanderIsaksen}) with the central theorem being this one:

\begin{theorem}
\label{}\hypertarget{}{}
Let $C$ be a [[site]] given by a [[Grothendieck topology]]. The left [[Bousfield localization of model categories|Bousfield localization]] of $sPSh(C)_{proj}$ and $sPSh(C)_{inj}$, respectively, at the following classes of morphisms exist and coincide:

\begin{enumerate}%
\item the set of all [[covering]] [[sieve]] [[subfunctor]]s $R \hookrightarrow j(X)$;


\item the set of all morphisms $hocolim_R \to U \to X$ for $R$ a covering sieve of $X$;


\item the set of all [[Čech nerve]] projections $C(\{U_i\}) \to X$ for $\{U_i \to X\}$ a covering sieve;


\item the class of all bounded [[hypercover]]s $U \to X$;


\item the class of morphisms $F \to \bar F$ from a [[simplicial presheaf]] $F$ to the simplicial [[sheaf]] obtained by degreewise [[sheafification]].


\item if the topology is generated from a [[basis for a topology|basis]], then: the set of covering sieve subfunctors $R_U \to X$ for each covering family $\{U_i \to X\}$ in the basis.



\end{enumerate}
\end{theorem}
This is theorem A5 in \href{http://front.math.ucdavis.edu/0205.5027}{DugHolIsak}.

This localization $sPSh(C)_{proj,cov}$ is the \textbf{Čech localization} of $sPSh(C)$ with respect to the given [[Grothendieck topology]]. It is a presentation of [[topological localization]] of an [[(∞,1)-category of (∞,1)-presheaves]] to an [[(∞,1)-category of (∞,1)-sheaves]].

\begin{displaymath}
\itexarray{
    Sh_{(\infty,1)}(C) &\stackrel{\overset{L}{\to}}{\hookrightarrow}&
    Psh_{(\infty,1)}(C)
    \\
    \uparrow^{\mathrlap{\simeq}}
    &&
    \uparrow^{\mathrlap{\simeq}}
    \\
    (sPSh(C)_{proj,cov})^\circ
    &\stackrel{\overset{left. Bousf.}{\leftarrow}}{\underset{}{\to}}&
    (sPSh(C)_{proj})^\circ
  }
  \,.
\end{displaymath}
The following definition and proposition provides information on what the general morphisms are which become weak equivalences after localization at

\begin{def}
\label{GeneralizedCover}\hypertarget{GeneralizedCover}{}
\textbf{([[generalized cover]])}

Let $C$ be a [[site]]. A \textbf{[[local epimorphism]]} (or \textbf{[[generalized cover]]}) in $sPSh(C)$ is a morphism $f : E \to B$ of simplicial presheaves with the property that for every [[representable functor|representable]] $U$ and every morphism $j(U) \to B$ there exists a [[covering]] [[sieve]] $\{U_i \to U\}$ such that for every $U_i \to U$ the composite $U_i \to U \to B$ has a lift $\sigma$ through $f$

\begin{displaymath}
\itexarray{
    j(U_i) &\stackrel{\exists \sigma}{\to}& E
    \\
    \downarrow && \downarrow
    \\
    j(U) &\stackrel{\forall}{\to} & B
  }
  \,.
\end{displaymath}
\end{def}
(\hyperlink{DuggerHollanderIsaksen}{Dugger-Hollander-Isaksen, corollary A.3})

\begin{prop}
\label{}\hypertarget{}{}
For $f : E \to B$ a [[local epimorphism]] in $sPSh(C)$ in the above sense, its [[Čech nerve]] projection

\begin{displaymath}
C(E) \to B
\end{displaymath}
is a weak equivalence in the projective local model structure $sPSh(C)_{proj, loc}$.

\end{prop}
This is \hyperlink{DuggerHollanderIsaksen}{Dugger-Hollander-Isaksen, corollary A.3}.

$\,$

\hypertarget{LocalizationAtCoverage}{}\subsubsection*{{Čech localization at a coverage}}\label{LocalizationAtCoverage}

In the literature localization of categories of simplicial presheaves is typically discussed with respect to a [[Grothendieck topology]] or a [[basis for a topology]]. Here we discuss aspects of localization at a [[coverage]].

Let $C$ be a category equipped with a [[coverage]], i.e. a collection of families of morphisms $\{U_i \to U\}$ for each object $U$ in $C$, called \emph{covering families} such that for any morphism $f : V \to U$ there exist diagrams

\begin{displaymath}
\itexarray{
    V_j &\to& U_{i(j)}
    \\
    \downarrow && \downarrow
    \\
    V &\stackrel{f}{\to}& U
  }
\end{displaymath}
such that $\{V_i \to V\}$ is itself a covering family.

Write $S(\{U_i\}) \to j(U)$ for the [[sieve]] corresponding to a covering family, regarded as a [[subfunctor]] of the [[representable functor]] $j(U)$, which we both regard as simplicially discrete objects in $sPSh(C)$.

Write $sPSh(C)_{inj,cov}$ for the left [[Bousfield localization of model categories|Bousfield localization]] of $sPSh(C)_{inj}$ at these morphisms $S(\{U_i\}) \to j(U)$ corresponding to covering families.

\begin{prop}
\label{}\hypertarget{}{}
A subfunctor inclusion $\tilde S \hookrightarrow j(U)$ corresponding to a sieve that \emph{contains} a covering sieve $S(\{U_i\})$ is a weak equivalence in $sPSh(C)_{inj,cov}$

\end{prop}
\begin{proof}
Write $J$ for the set of morphisms in $\tilde S$ but not in $S$.

Let $j(V_j) \to j(U)$ be a morphism not in $S(\{U_i\})$. By assumption we can find a covering family $\{V_{j,k} \to V_j\}$ such that for all $j,i$ we have commuting diagrams

\begin{displaymath}
\itexarray{
    V_{j,k} &\to& U_{i}
    \\
    \downarrow && \downarrow
    \\
    V_j &\stackrel{f}{\to}& U
  }
  \,.
\end{displaymath}
Consider the commuting diagram

\begin{displaymath}
\itexarray{
    \coprod_j S(\{V_{j,k}\}) &\hookrightarrow& S(\{U_i\} \cup \{V_{j,k}\})
    \\
    {}^{\mathllap{\simeq}}\downarrow && \downarrow
    \\
    \coprod_j j(V_j) &\to& S(\{U_i\} \cup \{V_j\})
  }
  \,.
\end{displaymath}
Observe that this is a [[pushout]] in $sPSh(C)$, that the top morphism is a cofibration in $sPSh(C)_{inj}$ and hence in $sPSh(C)_{inj,cov}$, that the left morphism is a local weak equivalence, that by general properties of [[Bousfield localization of model categories|left Bousfield localization]] the localization $sPSh(C)_{inj,cov}$ is [[proper model category|left proper]]. Therefore the morphism $S(\{U_i\} \cup \{V_{j,k}\}) \to S(\{U_i\} \cup \{V\}) = \tilde S$ is a weak equivalence.

Next observe that from the horizontal morphisms of the above commuting diagrams that defined the covers $\{V_{j,k} \to V_j\}$ we have an induced morphism $S(\{U_i\} \cup \{V_{j,k}\}) \to S(\{U_i\})$, and this exhibits $S(\{U_i\})$ as a [[retract]]

\begin{displaymath}
\itexarray{
    S(\{U_i\}) &\to& S(\{U_i\} \cup \{V_{j,k}\}) &\to& S(\{U_i\})
    \\
    \downarrow && \downarrow && \downarrow
    \\
    \tilde S &=& \tilde S &=& \tilde S 
  }
  \,.
\end{displaymath}
By closure of weak equivalences under retracts, this shows that the inclusion $S(\{U_i\}) \to \tilde S$ is a weak equivalence. By 2-out-of-3 this finally means that $\tilde S \hookrightarrow j(U)$ is a weak equivalence.

\end{proof}
\begin{cor}
\label{}\hypertarget{}{}
For $S(\{U_i\}) \to j(U)$ a covering sieve, its [[pullback]] $f^*S(\{U_i\}) \to j(V)$ in $sPSh(C)$ along any morphism $j(f) : j(V) \to j(U)$

\begin{displaymath}
\itexarray{
    f^* S(\{U_i\}) &\to& S(\{U_i\})
    \\
    \downarrow && \downarrow
    \\
    j(V) &\stackrel{j(f)}{\to}& j(U)
  }
\end{displaymath}
is also a weak equivalence.

\end{cor}
\begin{lemma}
\label{}\hypertarget{}{}
If $S(\{U_i\}) \to j(U)$ is the sieve of a covering family and $\tilde S \hookrightarrow j(U)$ is any sieve such that for every $f_i : U_i \to U$ the pullback $f_i^* \tilde S$ is a weak equivalence, then $\tilde S \to j(U)$ becomes an [[isomorphism]] in the [[homotopy category]].

\end{lemma}
\begin{proof}
First notice that if $f_i^* \tilde S$ is a weak equivalence for all $i$, then the pullback of $\tilde S$ to any element of the sieve $S(\{U_i\})$ is a weak equivalence. Use the [[co-Yoneda lemma]] to write

\begin{displaymath}
S(\{U_i\}) = \lim_{\underset{V \to U_i \to U}{\to}} j(V)
  \,.
\end{displaymath}
Now consider these objects in the [[(∞,1)-category of (∞,1)-presheaves]] that is presented by $sPSh(C)_{inj}$.

Since that has [[universal colimits]] we have the pullback square

\begin{displaymath}
\itexarray{
    i^* \lim_\to j(V)
    &\simeq& \lim_{\to} f_V^* \tilde S
    &\to& \tilde S
    \\
    && \downarrow && \downarrow^{\mathrlap{i}}
    \\
    S(\{U_i\}) &=& 
    \lim_{\underset{f_V : V \to U_i \to U}{\to}} j(V)
    &\stackrel{(f_V)}{\to}&
    j(U)
  }
\end{displaymath}
and the left vertical morphism is a colimit over morphisms that are weak equivalences in $sPSh(C)_{inj,loc}$. By the general properties of [[reflective sub-(∞,1)-categories]] this means that the total left vertical morphism becomes an isomorphism in the homotopy category of $sPSh(C)_{inj,cov}$. Also the bottom morphism is an isomorphism there, and hence the right vertical one is.

\end{proof}
\begin{lemma}
\label{}\hypertarget{}{}
In total this shows that the localization at the [[coverage]] produces the [[topological localization]] at the [[Grothendieck topology]] generated by that coverage.

\end{lemma}
\hypertarget{for_values_in_strict_and_abelian_groupoids}{}\subsubsection*{{For values in strict and abelian $\infty$-groupoids}}\label{for_values_in_strict_and_abelian_groupoids}

Many simplicial presheaves appearing in practice are (equivalent) to objects in [[sub-(∞,1)-categories]] of $Sh_{(\infty,1)}(C)$ of abelian or at least [[strict ∞-groupoid]]s. These subcategories typically offer convenient and desireable contexts for formulating and proving statements about special cases of general simplicial presheaves.

One well-known such notion is given by the [[Dold-Kan correspondence]]. This identifies [[chain complex]]es of [[abelian group]]s with strict and strictly [[symmetric monoidal (infinity,1)-category|symmetric monoidal]] $\infty$-groupoids.

Dropping the condition on symmetric monoidalness we obtain a more general such inclusion, a kind of non-abelian Dold-Kan correspondence:

the identification of [[crossed complex]]es of groupoids as precisely the [[strict omega-groupoid|strict ∞-groupoid]]s. This has been studied in particular in [[nonabelian algebraic topology]].

So we have a sequence of inclusions

\begin{displaymath}
\itexarray{
     ChainCplx &\hookrightarrow& CrsCpl &\hookrightarrow& KanCplx
     \\
     \downarrow^{\mathrlap{simeq}} && \downarrow^{\mathrlap{simeq}} && \downarrow^{\mathrlap{simeq}}
     \\
     StrAb Str\infty Grpd &\hookrightarrow&
     Str \infty Grpd
     &\hookrightarrow&
     \infty Grpd 
  }
\end{displaymath}
of strict $\infty$-groupoids into all $\infty$-groupoids. See also the [[cosmic cube]] of [[higher category theory]].

Among the special tools for handling $\infty$-stacks on $C$ that factor at some point through the above inclusion are the following:

\begin{itemize}%
\item \textbf{restriction to abelian sheaf cohomology} -- Using the fact that the objects of $Sh_{(\infty,1)}(C)$ are modeled by [[simplicial presheaves]] symmetric monoidal $\infty$-Lie groupoids are identified under the [[Dold-Kan correspondence]] with $\mathbb{N}$-graded [[chain complex|chain complexes]] of sheaves. To these the rich set of tools for [[abelian sheaf cohomology]] apply.


\item \textbf{descent for strict $\infty$-groupoid valued sheaves} -- There is a good theory of [[descent]] for (presheaves) with values in strict $\infty$-groupoids (more restrictive than the fully general theory but more general than [[abelian sheaf cohomology]]). This goes back to [[Ross Street]] and its relation to the full theory has been clarified by [[Dominic Verity]] in \hyperlink{Verity}{Verity09}.



\end{itemize}
We state a useful theorem for the computation of [[descent]] for presheaves with values in [[strict ∞-groupoid]]s. Recall the standard terminology for [[descent]], i.e. for the $(\infty,1)$-categorical [[sheaf]]-condition:

For $U \in C$ a representable, $Y,A \in [C^{op}, sSet]$ simplicial presheaves and $p : Y \to U$ a morphism, we say that $A$ \emph{satisfies [[descent]]} along $p$ or equivalently that $A$ is a $p$-[[local object]] if the canonical morphism

\begin{displaymath}
A(U) \stackrel{=}{\to}
  [C^{op}, sSet](U,A) 
  \to 
  [C^{op}, sSet](Y,A)
\end{displaymath}
is a weak equivalence. Here the first equality is the enriched [[Yoneda lemma]]. By the [[co-Yoneda lemma]] we may decompose $Y$ into its cells as

\begin{displaymath}
Y = \int^{[n] \in \Delta} \Delta[n] \cdot Y_n
  \,,
\end{displaymath}
where in the integrand we have the [[copower|tensoring]] of $[C^{op}, sSet]$ over [[sSet]]. Using that the enriched [[hom-functor]] sends coends to ends, the enriched [[hom-functor]] on the right we may equivalently write out as an [[end]]

\begin{displaymath}
\begin{aligned}
    [C^{op}, sSet](Y,A) 
    & =
    [C^{op}, sSet](\int^{[n] \in \Delta} \Delta[n] \cdot Y_n ,A)
    \\
    & = 
    \int_{[n] \in \Delta}[C^{op}, sSet](\Delta[n] \cdot Y_n ,A)
    \\
    & =
    \int_{[n] \in \Delta} sSet(\Delta[n], [C^{op}, sSet](Y_n, A))
    \\
    & =
    \int_{[n] \in \Delta} sSet(\Delta[n], A(Y_n))
    \\
    & =:Desc(Y,A)
  \end{aligned}
\end{displaymath}
(equality signs denote [[isomorphism]]s), where in the second but last line we again used the [[copower|tensoring]] of [[simplicial presheaves]] $[C^{op}, sSet]$ over [[sSet]].

In the last line we have the \emph{[[totalization]]} of the cosimplicial [[simplicial object]]

\begin{displaymath}
A(Y_\bullet) : \Delta \to sSet
  \,,
\end{displaymath}
sometimes called the \emph{descent object} of $A$ relative to $Y$, even though in this case it is really nothing but the hom-object of $Y$ into $A$. If $A$ is fibrant and $Y$ cofibrant, then $Desc(Y,A)$ is a Kan complex: the \emph{descent $\infty$-groupoid} .

Now suppose that $\mathcal{A} : C^{op} \to Str \infty Grpd$ is a presheaf with values in [[strict ∞-groupoid]]s. In the context of strict $\infty$-groupoids the standard $n$-[[simplex]] is given by the $n$th [[oriental]] $O(n)$. This allows to perform a construction that looks like a descent object in $Str\infty Grpd$:

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(Ross Street)}

The descent object for $\mathcal{A} \in [C^{op}, Str \infty Grpd]$ relative to $Y \in [C^{op}, sSet]$ is

\begin{displaymath}
Desc(Y,\mathcal{A}) := \int_{[n] \in \Delta} Str\infty Cat(O(n), \mathcal{A}(Y_n))
  \;\in Str \infty Grpd
  \,,
\end{displaymath}
where the [[end]] is taken in $Str \infty Grpd$.

\end{defn}
This objects had been suggested by [[Ross Street]] to be the right descent object for strict $\infty$-category-valued presheaves in \hyperlink{Street03}{Street03}

Under the [[∞-nerve]] functor $N_O : Str\infty Grpd \to sSet$ this yields a [[Kan complex]] $N_0 Desc(Y,\mathcal{A})$. On the other hand, applying the $\omega$-nerve directly to $\mathcal{A}$ yields a simplicial presheaf $N_O\mathcal{A}$ to which the above simplicial descent applies.

The following theorem asserts that under certain conditions both notions coincide.

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Dominic Verity)}

If $\mathcal{A} : C^{op}, Str \infty Grpd$ and $Y : C^{op} \to sSet$ are such that $N_O \mathcal{A}(Y_\bullet) : \Delta \to sSet$ is fibrant in the [[Reedy model structure]] $[\Delta, sSet_{Quillen}]_{Reedy}$, then

\begin{displaymath}
N_O Desc(Y,\mathcal{A}) \stackrel{\simeq}{\to} Desc(Y, N_O \mathcal{A})
\end{displaymath}
is a [[weak homotopy equivalence]] of [[Kan complex]]es.

\end{theorem}
This is proven in \hyperlink{Verity}{Verity09}.

\begin{corollary}
\label{}\hypertarget{}{}
If $Y \in [C^{op}, sSet]$ is such that $Y_\bullet : \Delta \to [C^{op}, Set] \hookrightarrow [C^{op}, sSet]$ is cofibrant in $[\Delta, [C^{op}, sSet]_{proj}]_{Reedy}$ then for $\mathcal{A} : C^{op} \to Str \infty Grpd$ we have

\begin{displaymath}
N_O Desc(Y,\mathcal{A}) \stackrel{\simeq}{\to} Desc(Y, N_O \mathcal{A})
  \,.
\end{displaymath}
\end{corollary}
\begin{proof}
If $Y_\bullet$ is Reedy cofibrant, then by definition the canonical morphisms

\begin{displaymath}
\lim_{\to}( ([n] \stackrel{+}{\to} [k]) \mapsto Y_k ) \to Y_n
\end{displaymath}
are cofibrations in $[C^{op}, sSet]_{proj}$. Since the latter is an $sSet_{Quillen}$ [[enriched model category]] and $N_O \mathcal{A}$ is fibrant, it follows that the [[hom-functor]] $[C^{op}, sSet](-, N_O \mathcal{A})$ sends cofibrations to fibrations, so that

\begin{displaymath}
N_O\mathcal{A}(Y_n)
  \to
  \lim_{\leftarrow}( [n]\stackrel{+}{\to} [k] \mapsto N_O\mathcal{A}(Y_k))
\end{displaymath}
is a [[Kan fibration]]. But this says that $N_O \mathcal{A}(Y_\bullet)$ is Reedy fibrant, so that the assumption of Verity's theorem is met.

\end{proof}
\begin{corollary}
\label{}\hypertarget{}{}
For $Y$ the [[Čech nerve]] of a [[good open cover]] $\{U_i \to X\}$ of a [[manifold]] $X$ and any $\mathcal{A} : CartSp^{op} \to Str \infty Grpd$ we have that

\begin{displaymath}
[C^{op}, sSet](Y,N_O \mathcal{A})
  \simeq
  N_O Desc(Y_\bullet, \mathcal{A})
  \,.
\end{displaymath}
\end{corollary}
\begin{proof}
By the above is sufices to note that $Y_\bullet$ is cofibrant in $[\Delta^{op}, [C^{op}, sSet]_{proj}]_{Reedy}$ if $Y$ is the [[Čech nerve]] of a good open cover. By the assumption of good open cover we have that $Y$ is degreewise a coproduct of representables and that the inclusion of all degenerate $n$-cells into all $n$-cells is a full inclusion into a coproduct, i.e. an inlusion of the form

\begin{displaymath}
\coprod_{i \in I} U_i \to \coprod_j U_{j \in J}
\end{displaymath}
induced from an inclusion of subsets $I \hookrightarrow J$. Since all representables are cofibrant in $[C^{op}, sSet]_{proj}$ such an inclusion is a cofibration.

\end{proof}
In conclusion we find that for determining the $\infty$-stack condition for \emph{strict} $\infty$-Lie groupoids we may equivalently use Street's formula for strict $\infty$-groupid valued presheaves. This is sometimes useful for computations in low categorical degree.

\hypertarget{Properness}{}\subsection*{{Properness}}\label{Properness}

The global model structures on simplicial presheaves are all [[left proper model categories|left]] and [[right proper model categories]]. Since left [[Bousfield localization of model categories]] preserves left properness (as discussed there), the local model structures are also left proper.

But the local model structures are not in general right proper anymore.

\begin{prop}
\label{OverSiteWithEnoughPointsWeakEquivalencesDetectedOnStalks}\hypertarget{OverSiteWithEnoughPointsWeakEquivalencesDetectedOnStalks}{}
Let $C$ be a [[site]] with [[point of a topos|enough points]]. Then the weak equivalences in the local model structures on $sPSh(C)$ are the [[stalk]]-wise [[weak homotopy equivalences]] of simplicial sets.

\end{prop}
p. 12 \href{http://www.math.uiuc.edu/K-theory/0175/}{here})

\begin{prop}
\label{}\hypertarget{}{}
A sufficient condition for an injective or projective local model structure of simplicial presheaves over a [[site]] $C$ to be right proper is that the weak equivalences are precisely the [[stalk]] wise weak equivalences of simplicial sets.

\end{prop}
By prop. \ref{OverSiteWithEnoughPointsWeakEquivalencesDetectedOnStalks} this is true for instance for the injective Jardine model structure when $C$ has [[point of a topos|enough points]].

\begin{proof}
The key is that forming [[stalks]] is, being the [[inverse image]] of a [[point of a topos|geometric morphism]]

\begin{displaymath}
(x^* \dashv x_*)  :=
  Set
  \stackrel{\overset{x^*}{\leftarrow}}{\underset{x_*}{\to}}
  Sh(C)
\end{displaymath}
an operation that preserves [[finite limits]].

Let therefore $f : X \to A$ be a stalkwise weak equivalence of simplicial presheaves and let $g : A \to B$ be a fibration. Notice that in all the model structures (injective, projective, global, local) the fibrations are always \emph{in particular} objectwise fibrations.

Then the pullback $g^* f$ in

\begin{displaymath}
\itexarray{
    g^* X &\to& X
    \\
    \downarrow^{\mathrlap{g^* f}} && \downarrow^{\mathrlap{f}} 
    \\
    A &\stackrel{g}{\to}& B
  }
\end{displaymath}
is a weak equivalence if for all topos points $x$ the stalk $x^* (g^* f)$ is a weak equivalence of simplicial sets. But since stalks preserve finite limits, we have a pullback diagram of simplicial sets

\begin{displaymath}
\itexarray{
    x^*(g^* X) &\to& x^*( X)
    \\
    \downarrow^{\mathrlap{x^*(g^* f)}} && \downarrow^{\mathrlap{x^*(f)}} 
    \\
    x^*(A) &\stackrel{x^*(g)}{\to}& x^*(B)
  }
  \,.
\end{displaymath}
It is now sufficient to observe that $x^* g$ is a [[Kan fibration]], this implies the result then by the fact that the [[classical model structure on simplicial sets]] is right proper.

To see this, notice that $x^*(g)$ is a Kan fibration precisely if for all $1 \leq k$ and $0 \leq i \leq k$ the morphism

\begin{displaymath}
(x^* A)^{\Delta[k]}
  \to
  (x^* A)^{\Lambda[k]^i} \times_{(x^* B)^{\Lambda[k]^i} } (x^* B)^{\Delta[k]}
\end{displaymath}
is an [[epimorphism]] of sets. Since stalks commute with finite limits, this is equivalent to

\begin{displaymath}
x^* 
  \left(
  A^{\Delta[k]}
   \to
  A^{\Lambda[k]^i} \times_{ B^{\Lambda[k]^i}   }  B^{\Delta[k]}  
  \right)
\end{displaymath}
being an epimorphism. Now the morphism in parenthesis is an epimorphism since the fibration $f$ is in particular an objectwise Kan fibration, and [[left adjoint]] functors such as $x^*$ preserve epimorphisms.

\end{proof}
This is mentioned for instance in (\hyperlink{Olsson}{Olsson, remark 4.3}).

\hypertarget{MonoidalStructure}{}\subsection*{{Closed monoidal structure}}\label{MonoidalStructure}

If the underlying site has [[finite products]], then both the injective and the projective, the global and the local model structure on simplicial presheaves becomes a [[monoidal model category]] with respect to the standard [[closed monoidal structure on presheaves]].

See for instance \href{http://www.math.univ-toulouse.fr/~toen/crm-2008.pdf#page=24}{here}.

\begin{lemma}
\label{}\hypertarget{}{}
Let $C$ be a category with [[products]]. Then the [[closed monoidal structure on presheaves]] makes $[C^{op}, sSet]_{proj}$ a [[monoidal model category]].

\end{lemma}
\begin{proof}
It is sufficient to check that the Cartesian product of presheaves

\begin{displaymath}
\otimes : sPSh(C)_{proj} \times sPSh(C)_{proj} \to sPSh(C)_{proj}
\end{displaymath}
is a left [[Quillen bifunctor]]. As discussed at [[Quillen bifunctor]], since $sPSh(C)$ is a [[cofibrantly generated model category]] for that it is sufficient to check that $\otimes$ satisfies the pushout-prodct axiom on generating (acyclic) cofibrations.

As discussed at [[model structure on functors]], these are those morphisms of the form

\begin{displaymath}
Id \times i : U \cdot S \to U \cdot T
\end{displaymath}
for $U \in C$ representable and $i : S \to T$ an (acylic) cofibration in $sSet_{Quillen}$. For these morphisms checking the pushout-product axiom amounts to checking it in $sSet$, where it is evident.

\end{proof}
\begin{lemma}
\label{}\hypertarget{}{}
Let $C$ be a [[site]] with [[product]]s and let $[C^{op}, sSet]_{proj,cov}$ be the left [[Bousfield localization of model categories|Bousfield localization]] at the [[Čech nerve]] projections.

Then for $X$ any cofibrant object, the [[closed monoidal structure on presheaves]]-adjunction

\begin{displaymath}
(X \times (-) \dashv [X,-]) : [C^{op}, sSet]_{proj,cov}
  \to 
  [C^{op}, sSet]_{proj,cov}
\end{displaymath}
is a [[Quillen adjunction]].

\end{lemma}
\begin{proof}
The above lemma implies that the [[left adjoint]] $X \times (-)$ preserves cofibrations. As discussed in the  at [[Quillen adjunction]] since the adjunction is $sSet$-enriched and since $[C^{op}, sSet]_{proj,cov}$ is a [[proper model category|left proper]] [[simplicial model category]] it suffices to check that $[X,-]$ preserves fibrant objects.

For that let $\{U_i \to U\}$ be a covering family and $C(\{U_i\})$ the corresponding [[Čech nerve]]. We need to check that if $A \in [C^{op}, sSet]_{proj,cov}$ is fibrant, then

\begin{displaymath}
[C^{op}, sSet](U, [X,A]) \to [C^{op},sSet](C(\{U_i\}), [X,A])
\end{displaymath}
is an equivalence of [[Kan complex]]es.

Writing $C(\{U_i\}) = \int^{[n]} \Delta[n] \cdot \coprod U_{i_0, \cdots, i_n}$ and using that the [[hom-functor]] preserves [[end]]s, this is eqivalent to

\begin{displaymath}
[C^{op},sSet]( X \times C(\{U_i\}) \to X \times U , A)
\end{displaymath}
being an equivalence. Now we observe that $X \times C(\{U_i\}) \to X\times U$ is a [[local epimorphism]] in the above sense, namely a morphism such that for every morphism $V \to X \times U$ out of a representable, there is a lift $\sigma$

\begin{displaymath}
\itexarray{
     && X \times C(\{U_i\})
     \\
     & {}^{\mathllap{\sigma}}\nearrow & \downarrow
     \\
     V &\to& X \times U
  }
  \,.
\end{displaymath}
By the above discussion of the Čech-localization of $[C^{op}, sSet]_{proj}$, this is a local morphism, hence does produce an equivalence when hommed into the fibrant object $A$.

\end{proof}
\hypertarget{HomotopyLimits}{}\subsection*{{Homotopy (co)limits}}\label{HomotopyLimits}

Properties of [[homotopy limit]]s and [[homotopy colimit]]s of simplicial presheaves are discussed at

\begin{itemize}%
\item \href{http://ncatlab.org/nlab/show/homotopy+limit#SimpSheaves}{Homotopy (co)limits of simplicial (pre)sheaves}

\end{itemize}
Let $C$ be a [[site]].

\begin{prop}
\label{}\hypertarget{}{}
Let $F : D \to [C^{op}, sSet]$ be a [[finite limit|finite]] diagram.

Write $\mathbb{R}_{glob}\lim_{\leftarrow} F \in [C^{op}, sSet]$ for any representative of the [[homotopy limit]] over $F$ computed in the global model structure $[C^{op}, sSet]_{proj}$, well defined up to [[isomorphism]] in the [[homotopy category]].

Then $\mathbb{R}_{glob}\lim_{\leftarrow} F \in [C^{op},sSet]$ presents also the [[homotopy limit]] of $F$ computed in the local model structure $[C^{op}, sSet]_{proj,loc}$.

\end{prop}
\begin{proof}
By the discussion at [[(∞,1)-limit]] the [[homotopy limit]] $\mathbb{R}\lim_{\leftarrow}$ computes the corresponding [[(∞,1)-limit]] and [[(∞,1)-sheafification]] $L$ is a left [[exact (∞,1)-functor]] and preserves these finite [[(∞,1)-limit]]s:

\begin{displaymath}
\itexarray{
     ([D, [C^{op}, sSet]_{proj, loc}]_{inj})^\circ
     &\stackrel{L_*}{\leftarrow}&
     ([D, [C^{op}, sSet]_{proj}]_{inj})^\circ
     \\
     \downarrow^{\mathrlap{\mathbb{R} \lim_\leftarrow}}
     &&
     \downarrow^{\mathrlap{\mathbb{R} \lim_\leftarrow}}
     \\
     ([C^{op}, sSet]_{proj,loc})^\circ
     &\stackrel{L \simeq \mathbb{L} Id}{\leftarrow}&
     ([C^{op}, sSet]_{proj})^\circ
  }
 \,.
\end{displaymath}
Here $L \simeq \mathbb{L} Id$ is the left [[derived functor]] of the identity for the \hyperlink{PresentationOfTheInfinTopos}{above} left Bousfield localization. Since left Bousfield localization does not change the cofibrations and includes the global weak equivalences into the local weak equivalences, the postcomposition of the diagram $F$ with $\mathbb{L} Id$ is given by cofibrant replacement in the local structure, too. But the [[homotopy limit]] of the diagram is invariant, up to equivalence, under cofibrant replacement, and hence a finite homotopy limit diagram in the global structure is also one in the local structure.

\end{proof}
\hypertarget{InclusionOfChainComplexes}{}\subsection*{{Inclusion of chain complexes of sheaves}}\label{InclusionOfChainComplexes}

We discuss how [[chain complex]]es of presheaves of [[abelian group]]s embed into the model structure on simplicial presheaves. Under passing to the intrinsic [[cohomology]] of the [[(∞,1)-topos]] \hyperlink{PresentationOfInfiniToposes}{presented by} by $[C^{op}, sSet]_{loc}$, this realizes traditional [[abelian sheaf cohomology]] over $C$ and generalizes it to general base objects.

Observe from the discussion at [[model structure on simplicial abelian groups]] that the degreewise [[free functor]]-[[forgetful functor]] [[adjunction]] $(F \dashv U) : Ab \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} Set$ (see [[algebra over a Lawvere theory]] for details) induces a [[Quillen adjunction]]

\begin{displaymath}
(F \dashv U) : 
 sAb_{Quillen } 
   \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} 
  sSet_{Quillen}
\end{displaymath}
between the [[model structure on simplicial abelian groups]] and the [[classical model structure on simplicial sets]], which exhibits $sAb_{Quillen}$ as the corresponding [[transferred model structure]].

Moreover, the [[Dold-Kan correspondence]] constitutes in particular a [[Quillen equivalence]]

\begin{displaymath}
(N_\bullet \dashv \Gamma) 
   : 
  Ch_\bullet^+_{proj} 
   \stackrel{\overset{N_\bullet}{\leftarrow}}{\underset{\Gamma}{\to}} 
  sAb_{Quillen}
\end{displaymath}
between the projective [[model structure on chain complexes]] of [[abelian group]]s in non-negative degree and simplicial abelian groups.

We write

\begin{displaymath}
(N_\bullet F \dashv \Xi) 
  : 
  Ch_\bullet^+_{proj} 
   \stackrel{\overset{N_\bullet}{\leftarrow}}{\underset{\Gamma}{\to}} 
  sAb_{Quillen}
   \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} 
  sSet_{Quillen}
\end{displaymath}
for the composite [[Quillen adjunction]]. For $C$ any [[category]], postcomposition with $\Xi$ induces a Quillen adjunction

\begin{displaymath}
(N_\bullet F \dashv \Xi)
  : 
  [C^{op}, Ch_\bullet^+_{proj}]_{proj}
   \stackrel{\overset{N_\bullet F}{\leftarrow}}{\underset{\Xi}{\to}}
  [C^{op}, sSet]_{proj}
\end{displaymath}
between the projective [[model structure on functors]] $[C^{op}, Ch_\bullet^+_{proj}]_{proj}$ and the global projective model structure on simplicial presheaves, which by convenient abuse of notation we denote by the same symbols.

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

\begin{itemize}%
\item [[local fibration]]

\end{itemize}
[[model topos]]

\begin{itemize}%
\item \textbf{model structure on simplicial presheaves}


\item [[model structure on simplicial sheaves]]


\item [[model structure on sSet-enriched presheaves]]


\item [[model structure on cubical presheaves]]


\item [[model structure on presheaves of spectra]]


\item [[model structure for (2,1)-sheaves]]



\end{itemize}
[[!include locally presentable categories - table]]

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

A nice introduction and survey is provided in the notes

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

\end{itemize}
Detailed discussion of the injective model structures on simplicial presheaves is in

\begin{itemize}%
\item [[John F. Jardine]], \emph{Simplicial presheaves} Journal of Pure and Applied Algebra 47 (1987), 35-87 (\href{http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf}{pdf})


\item [[John F. Jardine]], \emph{Stacks and the homotopy theory of simplicial sheaves}, Homology, homotopy and applications, vol. 3 (2), 2001, pp.361--384


\item [[John F. Jardine]], \emph{Boolean localization, in practice} (\href{http://www.math.uni-bielefeld.de/documenta/vol-01/13.html}{web})


\item [[John F. Jardine]], \emph{[[Local homotopy theory]]} (2011) (\href{http://www.math.uwo.ca/~jardine/papers/preprints/book.pdf}{pdf})



\end{itemize}
The projective model structure is discussed in

\begin{itemize}%
\item [[Dan Dugger]], \emph{[[Universal Homotopy Theories]]}

\end{itemize}
See also

\begin{itemize}%
\item Benjamin Blander, \emph{Local projective model structures on simplicial presheaves}, K-Theory, Volume 24, Number 3, November 2001 , pp. 283--301(19) (\href{http://www.ingentaconnect.com/content/klu/kthe/2001/00000024/00000003/00384649?crawler=true}{journal})

\end{itemize}
A brief review in the context of [[nonabelian Hodge theory]] is in section 4 of

\begin{itemize}%
\item Martin Olsson, \emph{Towards non-abelian $p$-adic Hodge theory in the good reduction case} (\href{http://math.berkeley.edu/~molsson/PHT3-24-08.pdf}{pdf})

\end{itemize}
A detailed study of [[descent]] for simplicial presheaves is given in

\begin{itemize}%
\item [[Daniel Dugger]], [[Sharon Hollander]], [[Daniel Isaksen]], \emph{Hypercovers and simplicial presheaves} , Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 1, 9--51 (\href{http://www.math.uiuc.edu/K-theory/0563/}{web})


\item [[Daniel Dugger]], [[Daniel Isaksen]], \emph{Weak equivalences of simplicial presheaves} (\href{http://arxiv.org/abs/math/0205025}{arXiv})



\end{itemize}
A survey of many of the model structures together with a treatment of the left local projective one is in

\begin{itemize}%
\item [[Benjamin Blander]], \emph{Local projective model structure on simplicial presheaves} (\href{http://www.math.uiuc.edu/K-theory/0462/combination2.pdf}{pdf})

\end{itemize}
See also

\begin{itemize}%
\item [[Daniel Isaksen]], \emph{Flasque model structure for simplicial presheaves} (\href{http://www.math.uiuc.edu/K-theory/0679/}{web}, \href{http://www.math.uiuc.edu/K-theory/0679/flasque.pdf}{pdf})

\end{itemize}
The characterization of the model category of simplicial presheaves as the canonical [[presentable (infinity,1)-category|presentation]] of the (hypercompletion of) the [[(∞,1)-category of (∞,1)-sheaves]] on a site is in

\begin{itemize}%
\item \href{http://www.math.harvard.edu/~lurie/papers/highertopoi.pdf#page=528}{proposition 6.5.2.1} of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]}

\end{itemize}


\end{itemize}
A set of lecture notes on simplicial presheaves with an eye towrads algebraic sites and [[derived algebraic geometry]] is

\begin{itemize}%
\item [[Bertrand Toen]], \emph{Simplicial presheaves and derived algebraic geometry} , lecture at \href{http://www.crm.es/HigherCategories/}{Simplicial methofs in higher categories} (\href{http://www.crm.cat/HigherCategories/hc1.pdf}{pdf})

\end{itemize}
Last not least, it is noteworthy that the idea of localizing simplicial sheaves at stalkwise weak equivalences is already described and applied in

\begin{itemize}%
\item [[Kenneth Brown]], \emph{[[BrownAHT|Abstract Homotopy Theory and Generalized Sheaf cohomology]]} ,

\end{itemize}
using instead of a full [[model category]] structure the more lightweight one of a Brown [[category of fibrant objects]].

A comparison between Brown-Gersten and Joyal-Jardine approach:

\begin{itemize}%
\item V. Voevodsky, \emph{Homotopy theory of simplicial presheaves in completely decomposable topologies}, \href{http://arxiv.org/abs/0805.4578}{arxiv/0805.4578}

\end{itemize}
The proposal for descent objects for strict $\infty$-groupoid-valued presheaves discussed in \hyperlink{DescentForStrictInf}{Descent for strict infinity-groupoids} appeared in

\begin{itemize}%
\item [[Ross Street]], \emph{Categorical and combinatorial aspects of descent theory} (\href{http://arxiv.org/abs/math/0303175}{arXiv})

\end{itemize}
The relation to the general descent condition is discussed in

\begin{itemize}%
\item [[Dominic Verity]], \emph{[[Verity on descent for strict omega-groupoid valued presheaves|Relating descent notions]]}

\end{itemize}
A useful collection of facts is in

\begin{itemize}%
\item [[Zhen Lin Low]], \emph{[[Notes on homotopical algebra]]}

\end{itemize}
[[!redirects model structures on simplicial presheaves]]

[[!redirects model category of simplicial presheaves]] [[!redirects model categories of simplicial presheaves]]

[[!redirects projective model structure on simplicial presheaves]] [[!redirects projective model structures on simplicial presheaves]]

[[!redirects injective model structure on simplicial presheaves]] [[!redirects injective model structures on simplicial presheaves]]



\end{document}