\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{local 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{contents}{Contents}\dotfill \pageref*{contents} \linebreak
\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{local_model_structures}{Local model structures}\dotfill \pageref*{local_model_structures} \linebreak
\noindent\hyperlink{local_weak_equivalences}{Local weak equivalences}\dotfill \pageref*{local_weak_equivalences} \linebreak
\noindent\hyperlink{in_terms_of_sheaves_of_homotopy_groups}{In terms of sheaves of homotopy groups}\dotfill \pageref*{in_terms_of_sheaves_of_homotopy_groups} \linebreak
\noindent\hyperlink{in_terms_of_local_liftings}{In terms of local liftings}\dotfill \pageref*{in_terms_of_local_liftings} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{local_injective_model_structure}{Local injective model structure}\dotfill \pageref*{local_injective_model_structure} \linebreak
\noindent\hyperlink{local_projective_model_structure}{Local projective model structure}\dotfill \pageref*{local_projective_model_structure} \linebreak
\noindent\hyperlink{intermediate_model_structures}{Intermediate model structures}\dotfill \pageref*{intermediate_model_structures} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

See [[model structure on simplicial presheaves]].

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

There are many [[model category]] structures on the category of [[simplicial presheaf|simplicial presheaves]] derived from the [[model structure on simplicial sets]].

The \emph{local} such model structures are of interest in that they model [[infinity-stack homotopically|infinity-stacks]] so that they are a [[presentable (infinity,1)-category|presentation]] of the [[(infinity,1)-category of (infinity,1)-sheaves]] on the given [[site]].

They can be thought of as being obtained from global model structures, of which there are two:

\begin{itemize}%
\item the \textbf{[[global model structure on simplicial sheaves|global projective]]} model structure has weak equivalences and fibrations being \emph{objectwise} those of [[simplicial set]]s;


\item the \textbf{[[global model structure on simplicial sheaves|global injective]]} model structure has weak equivalences and cofibrations being \emph{objectwise} those of [[simplicial set]]s;



\end{itemize}
These two model structures are Quillen equivalent (\emph{DHI04} \href{http://www.math.uiuc.edu/K-theory/0563/spre.pdf#page=5}{p. 5} with the Quillen equivalence given by the identity functor). They can be defined on any domain category $S$, not necessarily a [[site]]. If we do have a structure of a [[site]] on $S$ then there is a notion of \emph{local weak equivalences} of simplicial presheaves on $S$, defined below. One gets \emph{local} projective and \emph{local} injective model structures by applying left [[Bousfield localization]] of the above model structures at local weak equivalences (see \href{http://arxiv.org/PS_cache/math/pdf/0205/0205027v2.pdf#page=6}{p. 6} of \emph{DHI04})

\begin{itemize}%
\item the \textbf{local projective} model structure (weak equivalences are locally (usually stalkwise) and cofibrations are those that have the left [[lifting property]] against objectwise acyclic fibrations);


\item the \textbf{local injective} (weak equivalences are locally (usually stalkwise) and fibrations are those that have the right [[lifting property]] against the objectwise acyclic cofibrations).



\end{itemize}
\textbf{Warning} Since the (homotopy classes) of weak equivalences do not form a [[small set]], the general existence theorem recalled at [[Bousfield localization of model categories]] does not apply. The existence of the Bousfield localization has to be shown by hand. For the injective structure this is what Joyal and Jardine accomplished.

Again, the injective and projective local model structures are Quillen equivalent by the identity functors between the underlying categories and hence provide projective and injective versions of the corresponding homotopy theory of [[infinity-stack homotopically|infinity-stacks]].

In the local injective structure all objects are cofibrant, so that the [[opposite category]] of simplicial presheaves with the local injective model structure is a [[category of fibrant objects]].

Both local model structures are proper [[enriched category|simplicially enriched]] categories (\emph{DHI04} \href{http://www.math.uiuc.edu/K-theory/0563/spre.pdf#page=6}{p. 5}).

The \emph{local injective} model structure on simplicial presheaves is originally due to Jardine, following the construction of the Quillen equivalent local [[model structure on simplicial sheaves]] by Joyal. It was only later realized in \emph{DHI04} as a left [[Bousfield localization]] of the global injective model structure.

In between the injective and the projective model structures there are many other model structures obtained by varying the class of generating global cofibrations.

In the following let $S$ be a small [[site]] and denote by $SimpPr(S)$ be the category of [[simplicial presheaf|simplicial presheaves]] on $S$.

\hypertarget{local_model_structures}{}\subsection*{{Local model structures}}\label{local_model_structures}

One usually says that a \textbf{local model structure} on a category of [[presheaf|presheaves]] is one whose weak equivalences are not defined objectwise but on [[cover]]s and/or on [[stalk]]s.

\hypertarget{local_weak_equivalences}{}\subsubsection*{{Local weak equivalences}}\label{local_weak_equivalences}

There are different equivalent ways to define local weak equivalences of simplicial presheaves on a site $S$.

\hypertarget{in_terms_of_sheaves_of_homotopy_groups}{}\subsubsection*{{In terms of sheaves of homotopy groups}}\label{in_terms_of_sheaves_of_homotopy_groups}

(see section 2 of \emph{Jardine07})

We want to say that a local weak equivalence of simplicial presheaves is one which is ``over each point'' an isomorphism of homotopy groups.

we need the following terminology about sheaves of simplicial homtopy groups:

\begin{itemize}%
\item for $X$ a [[simplicial set]], write $\pi_0(X)$ for its set of connected components and $\pi_n(X,x)$, $n \geq 1$, $x \in X_0$, for its $n$th simplicial homotopy group at $x$ (the homotopy group of its [[geometric realization]]), $\pi_n(X,x) = \pi_n(|X|, x)$. This yields functors $\pi_0 : SimpSet \to Set$ and $\pi_n : SimpSet \to Grps$.


\item By postcomposition these functors induce functors $\pi_0 : SimpSet^{S^{op}} \to Set^{S^{op}}$ and $\pi_n : SimpSet^{S^{op}} \to Grps^{S^{op}}$.


\item By postcomposition with the sheafification functor this yields functors $\tilde \pi_0 : SimpSet^{S^{op}} \to Sh(S)$ and $\tilde \pi_n : SimpSet^{S^{op}} \to Sh(S,Grps)$.



\end{itemize}
\begin{udefn}
A \textbf{local weak equivalence} of simplicial presheaves is a morphism $f : X \to Y$ such that

\begin{enumerate}%
\item the morphism $\tilde \pi_0(f) : \tilde \pi_0 X \to \tilde \pi_0 Y$ is an isomorphism in $Sh(S)$;


\item the diagrams

\begin{displaymath}
\itexarray{
  \tilde \pi_n X &\to& \tilde \pi_n Y
  \\
  \downarrow && \downarrow 
  \\
  \tilde X_0
  &\to&
  \tilde Y_0
}
\end{displaymath}
are pullback diagrams in $Sh(S,SimpSet)$, for all $n \geq 1$, where $\tilde X_0$ denotes the sheaf associated to $X_0$.



\end{enumerate}
\end{udefn}
Equivalently a morphism $f : X \to Y$ of simplicial presheaves is, equivalently, a local weak equivalence if all induced morphisms of sheaves

\begin{displaymath}
\tilde \pi_n (X|_U, x) \to \tilde \pi_n Y|_U, f(x)
\end{displaymath}
are isomorphisms for all $U \in S$, for $X|_U, Y|_U$ the pullbacks to the over-category site $S/U$, for all $x \in X_0(U)$ and all $n \geq 0$.

If the site $S$ \emph{has enough points} then this condition is equivalent to saying that $f$ is a weak equivalence in the [[model structure on simplicial sets]] over every stalk (see \href{http://www.intlpress.com/HHA/v3/n2/a5/v3n2a5.pdf#page=3}{p. 363} of \emph{Jardine01}).

\hypertarget{in_terms_of_local_liftings}{}\subsubsection*{{In terms of local liftings}}\label{in_terms_of_local_liftings}

(see \emph{DI02}, i.e. Dugger and Isaksen, Weak equivalences of simplicial presheaves )

If $X$ and $Y$ are local fibrations there is a characterisation in terms of local homotopy liftings. Write $P$ for the pushout of the diagram $\partial \Delta^n \leftarrow \partial \Delta^n\times \Delta^1 \rightarrow \Delta^n\times \Delta^1$. Then there are two maps $\Delta^n\rightarrow P$ by restriction of $\Delta^n\times \Delta^1\rightarrow P$ along the cofaces.

Then a local weak equivalence is a morphism $f : X \to Y$ such that for all commuting diagrams

\begin{displaymath}
\itexarray{
    \partial \Delta^n \otimes U &\to& X
    \\
    \downarrow^{i_U} && \downarrow^f
    \\
    \Delta^n \otimes U &\to& Y
  }
\end{displaymath}
with $U$ simplicially constantly representable there exists a covering sieve $R$ of $U$ such that for every $V\in R$ there are morphisms $g:\Delta^n \otimes V\rightarrow X$ and $h:P\otimes V\rightarrow Y$ for which $g\circ i_V=\partial\Delta^n \otimes V\rightarrow \partial\Delta^n \otimes U \rightarrow X$ and $\Delta^n\otimes V \rightarrow \Delta^n \otimes U\rightarrow Y= \Delta^n\otimes V\rightarrow P\otimes V\rightarrow Y$ and in addition the square

\begin{displaymath}
\itexarray{
    \Delta^n\otimes V &\to& X
    \\
    \downarrow && \downarrow^f
    \\
    P \otimes U &\to& Y
  }
\end{displaymath}
commutes.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item Every object-wise weak equivalence is in particular a local weak equivalence.

\end{itemize}
\hypertarget{local_injective_model_structure}{}\subsection*{{Local injective model structure}}\label{local_injective_model_structure}

The \textbf{local injective model structure} on simplicial presheaves on a [[site]] $C$ is the \textbf{left} [[Bousfield localization]] $SPr(C)_{loc inj}$ of the [[global model structure on simplicial presheaves|injective global model structure]] $SPr(C)_{inj}$ at the class of local weak equivalences described above.

So

\begin{itemize}%
\item cofibrations are precisely the objectwise cofibrations of simplicial sets, i.e. the monomorphisms in $SPr(S)$;


\item weak equivalences are the \emph{local weak equivalences} from above.



\end{itemize}
\begin{utheorem}
The inclusion of sheaves into simplicial presheaves $SimpSh(S) \hookrightarrow SimpPr(S)$ and the [[sheafification]] functor $SimpPr(S) \to SimpSh(S)$ constitute a [[Quillen equivalence]] with respect to the above \emph{local injective model structure} on $SimpPr(S)$ ans the local [[model structure on simplicial sheaves]].

\end{utheorem}
\begin{proof}
See \emph{Jardine07}, theorem 5.

\end{proof}
\begin{utheorem}
The fibrant objects in the local injective model structure $SPr(C)_{loc inj}$ are those simplicial presheaves that

\begin{enumerate}%
\item are fibrant in the global injective model structure;


\item satisfy [[descent]] for all [[hypercover]]s.



\end{enumerate}
\end{utheorem}
\begin{proof}
\emph{DHI04}, theorem 1.1

\end{proof}
\hypertarget{local_projective_model_structure}{}\subsection*{{Local projective model structure}}\label{local_projective_model_structure}

The \textbf{local projective model structure} on simplicial presheaves on a [[site]] $C$ is the \textbf{left} [[Bousfield localization]] $SPr(C)_{loc proj}$ of the [[global model structure on simplicial presheaves|projective global model structure]] $SPr(C)_{proj}$ at the class of local weak equivalences described above.

So

\begin{itemize}%
\item cofibrations are precisely the cofibrations in the global projective structure (defined by left lifting property with respect to global Kan fibrations)


\item weak equivalences are the \emph{local weak equivalences} from above.



\end{itemize}
\textbf{Remark}. Notice that this is still using \textbf{left} Bousfield localization. If we used right Bousfield localization the local projective fibrations would simply be the global Kan fibrations. Instead we have the following.

\begin{utheorem}
The local injective model structure $SPr(C)_{loc inj}$ is [[Quillen equivalence|Quillen equivalent]] to the ``universal homtopy thepory'' $U C/S$ constructed by

\begin{enumerate}%
\item formally adding [[homotopy limit|homotopy colimits]] to the category $C$ to create the category $U C$.


\item imposing relations requiring that for every [[hypercover]] $U \to X$, the morphism $hocolim_n U_n \to X$ is a weak equivalence.



\end{enumerate}
\end{utheorem}
\begin{proof}
\emph{DHI04}, theorem 1.2

\end{proof}
In $U C/S$ the fibrant objects have a simpler description than in $SPr(C)_{loc inj}$: they still need to satisfy [[descent]] but the implicit fibrancy condition with respect to the global injective structure is replaced by the fibrancy condition with respect to the global projective structure

\begin{utheorem}
The fibrant objects in $U C/S$ are those simplicial presheaves $A$ that

\begin{enumerate}%
\item are objectwise fibrant (i.e. take values in [[Kan complex]]es)


\item satisfy [[descent]] for all [[hypercover]]s.



\end{enumerate}
\end{utheorem}
\begin{proof}
\emph{DHI04}, theorem 1.3

\end{proof}
\hypertarget{intermediate_model_structures}{}\subsection*{{Intermediate model structures}}\label{intermediate_model_structures}

One can regard the projective and the injective model structure as two extrema of a poset of model structures on simplicial presheaves; see [[intermediate model structure]].

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

See [[model structure on simplicial presheaves]].

The local projective model structure on simplicial presheaves appears as theorem 1.6 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}
Its analog for sheaves, theorem 2.1 there, is due to

\begin{itemize}%
\item K. Brown, Gersten, \ldots{}

\end{itemize}
That the local projective model structure (directly defined) is indeed the left Bousfield localization of the global projective model structure is lemma 4.3 there.

[[!redirects local model structures on simplicial presheaves]]

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

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



\end{document}