\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*{infinity-stack}

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

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos theory}}\label{topos_theory}

[[!include (infinity,1)-topos - contents]]

\hypertarget{locality_and_descent}{}\paragraph*{{Locality and descent}}\label{locality_and_descent}

[[!include descent and locality - 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{derived_stacks}{Derived $\infty$-stacks}\dotfill \pageref*{derived_stacks} \linebreak
\noindent\hyperlink{higher_stacks}{Higher $\infty$-stacks}\dotfill \pageref*{higher_stacks} \linebreak
\noindent\hyperlink{quasicoherent_stacks}{Quasicoherent $\infty$-stacks}\dotfill \pageref*{quasicoherent_stacks} \linebreak
\noindent\hyperlink{affine_stacks}{Affine $\infty$-stacks}\dotfill \pageref*{affine_stacks} \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}

The notion of, equivalently

\begin{itemize}%
\item \emph{$\infty$-stack},

\end{itemize}
and specifically of

\begin{itemize}%
\item \emph{[[(∞,1)-sheaf]]},


\item \emph{[[geometric homotopy type]]}



\end{itemize}
is the $\infty$-[[vertical categorification|categorification]] of the notion of, equivalently

\begin{itemize}%
\item [[sheaf]] and [[stack]]


\item [[geometric homotopy type]].



\end{itemize}
Where a sheaf is a [[presheaf]] with values in [[Set]] that satisfies the sheaf condition, an [[higher category theory|∞-category]]-valued ([[pseudofunctor|pseudo]])[[presheaf]] is an \emph{$\infty$-stack} if it ``satisfies descent'' in that its assignment to a space $X$ is equivalent to its [[descent]] data for any [[cover]] or [[hypercover]] $Y^\bullet \to X$: if the canonical morphism

\begin{displaymath}
\mathbf{A}(X) \to
  Desc(Y^\bullet, \mathbf{A})
\end{displaymath}
is an equivalence. This is the \emph{descent condition}.

One important motivation for $\infty$-stacks is that they generalize the notion of [[Grothendieck topos]] from [[category theory|1-categorical]] to [[higher category theory|higher categorical context]].

This is a central [[motivation for sheaves, cohomology and higher stacks|motivation for considering higher stacks]]. They may also be thought of as [[internal ∞-groupoid]]s in a [[Grothendieck topos|sheaf topos]].

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

A well developed theory exists for $\infty$-stacks that are sheaves with values in [[∞-groupoids]]. Given that ordinary sheaves may be thought of as sheaves of [[0-category|0-categories]] and that $\infty$-groupoid-values sheaves may be thought of as sheaves of [[(infinity,0)-category|(∞,0)-categories]], these may be called [[(infinity,1)-sheaf|(∞,1)-sheaves]]. In the case that these $\infty$-groupoids have vanishing [[homotopy groups]] above some degree $n$, these are sometimes also called [[sheaves of n-types|sheaf of n-types]].

The currently most complete picture of [[(infinity,1)-sheaf|(∞,1)-sheaves]] appears in

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

\end{itemize}
but is based on a long development by other authors, some of which is indicated in the list of references below.

With the general machinery of [[(∞,1)-category]] theory in place, the definition of the [[(infinity,1)-category of (infinity,1)-sheaves|(∞,1)-category of ∞-stacks]] is literally the same as that of a [[category of sheaves]]: it is a [[reflective (∞,1)-subcategory]]

\begin{displaymath}
\infty Stacks(C) \simeq Sh_\infty(C)
  \stackrel{\stackrel{\bar{(\cdot)}}{\leftarrow}}{\to}
  PSh_\infty(C)
\end{displaymath}
of the [[(infinity,1)-category of (infinity,1)-functors|(∞,1)-category of (∞,1)-presheaves]] with values in [[∞Grpd]], such that the left adjoint [[(∞,1)-functor]] $\bar {(\cdot)}$ -- the [[∞-stackification]] operation -- is left exact.

One of the main theorems of [[Higher Topos Theory]] says that the old [[model structure on simplicial presheaves|model structures on simplicial presheaves]] are the canonical

\begin{itemize}%
\item [[models for ∞-stack (∞,1)-toposes]].

\end{itemize}
This allows to regard various old technical results in a new conceptual light and provides powerful tools for actually handling $\infty$-stacks.

In particular this implies that the old definition of [[abelian sheaf cohomology]] is secretly the computation of [[∞-stackification]] for $\infty$-stacks that are in the image of the [[Dold-Kan correspondence|Dold-Kan embedding]] of [[chain complex]]es of sheaves into [[simplicial presheaves|simplicial sheaves]].

\hypertarget{derived_stacks}{}\subsubsection*{{Derived $\infty$-stacks}}\label{derived_stacks}

Notice that an $\infty$-stack is a [[(∞,1)-presheaf]] for which not only the codomain is an [[(∞,1)-category]], but where also the domain, the [[site]], may be an [[(∞,1)-category]].

To emphasize that one considers $\infty$-stacks on higher categorical sites one speaks of [[derived stacks]].

\hypertarget{higher_stacks}{}\subsubsection*{{Higher $\infty$-stacks}}\label{higher_stacks}

The above concerns $\infty$-stacks with values in [[∞-groupoids]], i.e, [[(∞,0)-category|(∞,0)-categories]]. More generally there should be notions of $\infty$-stacks with values in [[(n,r)-category|(n,r)-categories]]. These are expected to be modeled by the [[model structure on homotopical presheaves]] with values in the category of [[Theta spaces]].

\hypertarget{quasicoherent_stacks}{}\subsubsection*{{Quasicoherent $\infty$-stacks}}\label{quasicoherent_stacks}

An archetypical class of examples of $\infty$-stacks are [[quasicoherent ∞-stack]]s of [[module]]s, being the [[vertical categorification|categorification]] of the notion of [[quasicoherent sheaf]]. By their nature these are really $(\infty,1)$-stacks in that they take values not in [[∞-groupoid]]s but in [[(∞,1)-categories]], but often only their [[∞-groupoid]]al [[core]] is considered.

\hypertarget{affine_stacks}{}\subsubsection*{{Affine $\infty$-stacks}}\label{affine_stacks}

In

\begin{itemize}%
\item [[Bertrand Toen]], \emph{Affine stacks (Champs affines)} (\href{http://arxiv.org/abs/math/0012219}{arXiv:math/0012219})

\end{itemize}
for the [[site]] $C = Alg_k^{op}$ with a suitable topology a [[Quillen adjunction]]

\begin{displaymath}
\mathcal{O} : sPSh(C)_{loc} \stackrel{\leftarrow}{\to}
  [\Delta^{op},Alg_k] \simeq dgAlg_k^{+}
  : 
  Spec
\end{displaymath}
is presented, where $\mathcal{O}$ sends and $\infty$-stack to its global [[dg-algebra]] of functions and $Spec$ constructs the simplicial presheaf ``represented'' degreewise by a cosimplicial algebra (under the [[monoidal Dold-Kan correspondence]] these are equivalent to dg-algebras).

An $\infty$-stack in the image of $Spec : dgAlg_k^+ \to sPSh(C)$ is an \textbf{affine $\infty$-stack}. The image of an arbitrary $\infty$-stack under the composite

\begin{displaymath}
Aff : sPSh(C) \stackrel{\mathcal{O}}{\to} dgAlg_k^+ \stackrel{Spec}{\to} sPSh(C)
\end{displaymath}
is its \textbf{affinization}.

This notion was considered in the full [[(∞,1)-category]] picture in

\begin{itemize}%
\item [[David Ben-Zvi]], [[David Nadler]], \emph{Loop Spaces and Connections} (\href{http://arxiv.org/abs/1002.3636}{arXiv:math/1002.3636})

\end{itemize}
where it is also generalized to [[derived stack]]s, i.e. to the [[(∞,1)-site]] $dgAlg_k^-$ of cochain [[dg-algebra]]s in non-positive degree, where the pair of [[adjoint (∞,1)-functor]]s is

\begin{displaymath}
\mathcal{O} : 
  Sh_{(\infty,1)}((dgAlg_k^-)^{op})
  \stackrel{\leftarrow}{\to}
  [\Delta^{op},dgAlg_k^-] 
  \simeq
  dgAlg_k
  :
  Spec
\end{displaymath}
with $\mathcal{O}$ taking values in \emph{unbounded} dg-algebras.

In detail, $\mathcal{O}$ acts as follows: every [[∞-stack]] $X$ may be written as a ([[homotopy colimit|colimit]]) over [[representable functor|representable]] $Spec A_i \in dgAlg_i$

\begin{displaymath}
X \simeq \lim_{\to^i} Y(Spec A_i)
  \,,
\end{displaymath}
where $Y : (dgAlg^-)^{op} \to \mathbf{H}$ is the [[(∞,1)-Yoneda embedding]].

The functor $\mathcal{O}$ takes any such colimit-description, and simply reinterprets the colimit in $dgAlg^{op}$, i.e. the limit in $dgAlg$:

\begin{displaymath}
\mathcal{O}(X) = \lim_{\leftarrow^i} A_i
  \,.
\end{displaymath}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[sheaf]]


\item [[2-sheaf]] / [[stack]]


\item [[(∞,1)-sheaf]] / \textbf{$\infty$-stack},

\begin{itemize}%
\item [[sheaf of spectra]]


\item [[sheaf of L-∞ algebras]]



\end{itemize}

\item [[(∞,2)-sheaf]]


\item [[(∞,n)-sheaf]]



\end{itemize}
[[!include homotopy n-types - table]]

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

The study of $\infty$-stacks is known in parts as the study of [[nonabelian cohomology]]. See there for further references.

The search for $\infty$-stacks probably began with [[Alexander Grothendieck]] in \emph{[[Pursuing Stacks]]}.

The notion of $\infty$-stacks can be set up in various notions of $\infty$-categories. [[Andre Joyal]], Jardine, [[Bertrand Toen]] and others have developed the theory of $\infty$-stacks in the context of [[simplicial presheaf|simplicial presheaves]] and also in [[Segal category|Segal categories]].

\begin{itemize}%
\item [[Bertrand Toën]], [[Gabriele Vezzosi]]; \emph{Homotopical algebraic geometry. I. Topos theory}, Adv. Math. 193 (2005), no. 2, 257--372, \href{http://dx.doi.org/10.1016/j.aim.2004.05.004}{doi}, \emph{Homotopical Algebraic Geometry II: geometric stacks and applications}, \href{http://front.math.ucdavis.edu/0404.5373}{math.AG/0404373}


\item [[Bertrand Toën]], [[Gabriele Vezzosi]]; \emph{Segal topoi and stacks over Segal categories}, \href{http://arxiv.org/abs/math.AG/0212330}{math.AG/0212330}.


\item [[Bertrand Toën]]; \emph{Higher and derived stacks: a global overview} (\href{http://arxiv.org/abs/math.AG/0604504}{arXiv}).



\end{itemize}
This concerns $\infty$-stacks with values in [[∞-groupoids]], i.e. $(\infty,0)$-categories. More generally [[descent]] conditions for $n$-stacks and $(\infty,n)$-stacks with values in [[(infinity,n)-category|(∞,n)-categories]] have been earlier discussed in

\begin{itemize}%
\item [[Andre Hirschowitz]], [[Carlos Simpson]]; \emph{Descente pour les $n$-champs} (\href{http://arxiv.org/abs/math/9807049}{arXiv})

\end{itemize}
All this has been embedded into a coherent global theory in the setting of [[quasicategory|quasicategories]] in

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

\end{itemize}
[[!redirects infinity-stacks]] [[!redirects ∞-stack]] [[!redirects ∞-stacks]] [[!redirects infinity stack]]



\end{document}