\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*{coherent (infinity,1)-topos}

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

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

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

\hypertarget{compact_objects}{}\paragraph*{{Compact objects}}\label{compact_objects}

[[!include compact object - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{commutativity_with_filtered_colimits}{Commutativity with filtered colimits}\dotfill \pageref*{commutativity_with_filtered_colimits} \linebreak
\noindent\hyperlink{in_terms_of_sites}{In terms of sites}\dotfill \pageref*{in_terms_of_sites} \linebreak
\noindent\hyperlink{delignelurie_completeness_theorem}{Deligne-Lurie completeness theorem}\dotfill \pageref*{delignelurie_completeness_theorem} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The concept of \emph{coherent $(\infty,1)$-topos} is a notion of \emph{[[compact topos]]} in the context of [[(∞,1)-topos theory]] ([[Spectral Schemes|Lurie VII, def. 3.1]]).

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\begin{defn}
\label{}\hypertarget{}{}
An (∞,1)-topos $\mathbf{H}$ is called \textbf{[[quasicompact morphism|quasi-compact]]} if, for every [[effective epimorphism]]

\begin{displaymath}
\coprod_{i\in I} U_i\to *
\end{displaymath}
there exists a [[finite set|finite]] [[subset]] $J\subset I$ such that $\coprod_{i\in J} U_i\to *$ is an effective epimorphism. An object $X\in\mathbf{H}$ is called quasi-compact if the [[slice (∞,1)-topos]] $\mathbf{H}_{/X}$ is quasi-compact.

\end{defn}
We then define $n$-coherence by [[induction]] on $n$.

\begin{defn}
\label{nCoherentTopos}\hypertarget{nCoherentTopos}{}
Let $\mathbf{H}$ be an [[(∞,1)-topos]]. We say that $\mathbf{H}$ is 0-coherent if it is quasi-compact. If $n\geq 1$, we say that $\mathbf{H}$ is \textbf{n-coherent} if

\begin{enumerate}%
\item it is \textbf{locally $(n-1)$-coherent}, i.e., for every $X\in\mathbf{H}$ there exists an effective epimorphism $\coprod_{i\in I} U_i\to X$ such that each $U_i$ is (n-1)-coherent;
\item the [[sub-(∞,1)-category]] of (n-1)-coherent objects in $\mathbf{H}$ is closed under [[finite products]].

\end{enumerate}
We say that $\mathbf{H}$ is \textbf{coherent} if it is $n$-coherent for every $n\geq 0$, and \textbf{locally coherent} if for every $X\in\mathbf{H}$ there exists an effective epimorphism $\coprod_{i\in I} U_i\to X$ such that each $U_i$ is coherent.

\end{defn}
([[Spectral Schemes|Lurie SpecSch, def. 3.1, def. 3.12]])

\begin{remark}
\label{}\hypertarget{}{}
This terminology differs from the one in [[SGA]]4: a [[topos]] is a [[coherent topos]] in the sense of [[SGA]]4 if and only if it is 2-coherent according to the above definition.

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
An [[object]] $X \in \mathcal{X}$ in an [[(∞,1)-topos]] is a \emph{[[n-coherent object]]} if the [[slice (∞,1)-topos]] $\mathcal{X}_{/X}$ is $n$-coherent according to def. \ref{nCoherentTopos}.

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

\hypertarget{commutativity_with_filtered_colimits}{}\subsubsection*{{Commutativity with filtered colimits}}\label{commutativity_with_filtered_colimits}

Notice that a [[compact object in an (∞,1)-category]] is one that distributes over [[filtered (∞,1)-colimits]].

In an $n$-coherent $\infty$-topos the [[global section geometric morphism]] (given by homming out of the [[terminal object]]) preserves [[filtered (∞,1)-colimits]] of [[truncated object in an (infinity,1)-topos|(n-1)-truncated objects]].

\hypertarget{in_terms_of_sites}{}\subsubsection*{{In terms of sites}}\label{in_terms_of_sites}

An [[(∞,1)-site]] is \emph{finitary} if every [[covering]] [[sieve]] is generated by a finite family of morphisms. If $C$ is a finitary (∞,1)-site with [[finite (∞,1)-limits]], then the [[(∞,1)-topos of (∞,1)-sheaves]] on $C$ is coherent and locally coherent.

\hypertarget{delignelurie_completeness_theorem}{}\subsubsection*{{Deligne-Lurie completeness theorem}}\label{delignelurie_completeness_theorem}

The following generalizes the [[Deligne completeness theorem]] from [[topos theory]] to [[(∞,1)-topos theory]].

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{[[Deligne-Lurie completeness theorem]]}

An [[hypercomplete (∞,1)-topos]] which is locally coherent has [[enough points]].

\end{theorem}
([[Spectral Schemes|Lurie SpecSchm, theorem 4.1]]).

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

\begin{example}
\label{}\hypertarget{}{}
[[∞Grpd]] is coherent and locally coherent. An [[object]] $X$, hence an [[∞-groupoid]], is an [[n-coherent object]] if all its [[homotopy groups]] in degree $k \leq n$ are [[finite set|finite]]. Hence the fully coherent objects here are the [[homotopy types with finite homotopy groups]].

\end{example}
([[Spectral Schemes|Lurie SpecSchm, example 3.13]])

\begin{example}
\label{}\hypertarget{}{}
Let $X$ be a [[scheme]] and let $Sh_\infty(X_{Zar})$ be the [[(∞,1)-topos of (∞,1)-sheaves]] on the [[small site|small]] [[Zariski site]] of $X$. Then the following assertions are equivalent:

\begin{enumerate}%
\item $Sh_\infty(X_{Zar})$ is coherent;
\item $Sh_\infty(X_{Zar})$ is 1-coherent;
\item $X$ is quasi-compact and quasi-separated.

\end{enumerate}
\end{example}
\begin{example}
\label{}\hypertarget{}{}
A [[spectral scheme]] or [[spectral Deligne-Mumford stack]], regarded as a [[structured (∞,1)-topos]] is locally coherent.

\end{example}
([[Quasi-Coherent Sheaves and Tannaka Duality Theorems|Lurie QCoh, cor. 1.4.3]])

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

\begin{itemize}%
\item [[Jacob Lurie]], section 3 of \emph{[[Spectral Schemes]]}


\item [[Jacob Lurie]], section 2.3 of \emph{[[Rational and p-adic Homotopy Theory]]}


\item [[Jacob Lurie]], \emph{[[Quasi-Coherent Sheaves and Tannaka Duality Theorems]]}



\end{itemize}
[[!redirects coherent (infinity,1)-toposes]]

[[!redirects coherent (∞,1)-topos]] [[!redirects coherent (∞,1)-toposes]] [[!redirects n-coherent (∞,1)-topos]] [[!redirects n-coherent (infinity,1)-topos]] [[!redirects n-coherent (∞,1)-toposes]] [[!redirects n-coherent (infinity,1)-toposes]]

[[!redirects locally coherent (∞,1)-topos]] [[!redirects locally coherent (∞,1)-toposes]] [[!redirects locally n-coherent (∞,1)-topos]] [[!redirects locally n-coherent (infinity,1)-topos]] [[!redirects locally n-coherent (∞,1)-toposes]] [[!redirects locally n-coherent (infinity,1)-toposes]]



\end{document}