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\section*{étendue}

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

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{slice_toposes_of_tendues_are_tendues}{Slice toposes of étendues are étendues}\dotfill \pageref*{slice_toposes_of_tendues_are_tendues} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

An \textbf{\'e{}tendue} (also `etendue', or `etendu'; from French `\'e{}tendue' (fem.)- \emph{extent}) is a [[topos]] $\mathcal{Y}$ that locally looks like the category of sheaves on a space:

\begin{quote}%
Briefly, the slogan is that $\mathcal{Y}$ is locally a topological space. (\hyperlink{Lawvere76}{Lawvere 1976}, p.129)


\end{quote}
Originally defined by [[A. Grothendieck]] in one of the famous `excercises' of \hyperlink{SGA4}{SGA4} (ex. 9.8.2) as a [[Grothendieck topos]] $\mathcal{Y}$ that has a well-supported object $X$ such that the [[slice topos]] $\mathcal{Y}/X$ is equivalent to a sheaf topos on a topological space, the definition was generalized by [[Lawvere]] (1975,1976) by dropping the spatiality of the slice and require only that $\mathcal{Y}/X$ is a [[localic topos]].

Several characterizations of \'e{}tendues are known and the \emph{Ur}-example of an \'e{}tendue, the presheaf topos $\mathcal{S}^G$ of group actions, exhibits one in terms of sites rather directly: it has a [[site]] where every morphism is monic. Other characterizations involve (local) [[equivalence relations]] and yield connections to [[orbifolds]], [[foliations]], and [[stacks]], which are instrumental for the generalization to \emph{$\infinity$-\'e{}tendues} (cf. \hyperlink{Carchedi13}{Carchedi 2013}).

\'E{}tendues play an important role in Lawvere's approach to [[cohesion]] and the distinction between [[petit and gros toposes]] where they provide one of the classes of \textbf{petit} toposes (generalized spaces). In this context, Lawvere (1989,1991) interprets the cancellative property of the site as enabling an interpretation of \'e{}tendues as \emph{categories of processes}.

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

A topos $\mathcal{Y}$ is called an \emph{\'e{}tendue} if there is an object $X\in|\mathcal{Y}|$ such that the unique $X\rightarrow 1$ is epic and the [[slice topos]] $\mathcal{Y}/X$ is a [[localic topos]].\footnote{An epic $k:X\rightarrow Y$ induces a [[geometric morphism]] $k_\ast:\mathcal{Y}/X\rightarrow \mathcal{Y}/Y$ whose inverse image part, the change of base functor, $k^\ast:\mathcal {Y}/Y\rightarrow\mathcal{Y}/X$ is faithful, which says by definition that $k_\ast$ is a surjection, and in case $Y=1$, one says that $\mathcal{Y}/X$ \emph{covers} $\mathcal{Y}$. $k_\ast$ is an [[étale geometric morphism]].} 

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

\begin{quote}%
The first example of an \'e{}tendue seems to have been the [[space of moduli]] of algebraic curves, which is prevented from being globally a space due to the action of the [[Galois groups]] within each point. Yes, something vaguely reminiscent of particle spin is going on in such spaces, and the most naked form is that for any group G, the category $\mathcal{S}^G$ is an \'e{}tendue with only one point! This is easily seen from the observations that $\mathcal{S}^G/G\cong\mathcal{S}^G$ and that $G\twoheadrightarrow 1$ where the last two $G$`s denote the regular representation. (\hyperlink{Lawvere76}{Lawvere 1976}, pp.129-130)


\end{quote}
\begin{itemize}%
\item The [[Sierpinski topos]] $\mathcal{S}^{\cdot\rightarrow\cdot}$, as the sheaf topos on the [[Sierpinski space]], is an \'e{}tendue.


\item The topos $\mathcal{S}^{\cdot\rightrightarrows\cdot}$ of \emph{directed [[graphs]]} (aka [[quivers]]; Lawvere calls them \emph{irreflexive graphs}) is an \'e{}tendue, as it is locally equivalent to the sheaf topos on a three point space (Lawvere 1986). The contrast between $\mathcal{S}^{\cdot\rightrightarrows\cdot}$ and the topos $\mathcal{S}^{\Delta_1^{op}}$ of \emph{reflexive graphs} is a paradigmatic example of the distinction between a \emph{petit} and a \emph{gros} topos.


\item The [[Jónsson-Tarski topos]] $\mathcal{J}_2$ is an \'e{}tendue, as it is locally equivalent to the sheaf topos on the [[Cantor space]]. It is discussed as a petit topos for \emph{labeled graphs} in (Lawvere 1989).



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

\begin{itemize}%
\item \textbf{Proposition}. A Grothendieck topos $\mathcal{Y}$ is an \'e{}tendue iff there exists a [[site]] $(\mathcal{C}, J)$ for $\mathcal{Y}$ such that every morphism of $\mathcal{C}$ is monic.


\item In particular for small $\mathcal{C}$, the presheaf topos $\mathcal{S}^{\mathcal{C}^{op}}$ is an \'e{}tendue iff all morphisms in $\mathcal{C}$ are monic. In particular for [[monoid]]s $\mathcal{C}$ this is sometimes called [[left cancellative category|left cancellative]] (e.g. each free monoid is left cancellative).


\item [[Subtoposes]] of \'e{}tendues are \'e{}tendues. K. Rosenthal uses this together with the preceding remark on $\mathcal{S}^{\mathcal{C}^{op}}$ for all-monic $\mathcal{C}$ in order to construct further \'e{}tendues $Sh_j(\mathcal{S}^{\mathcal{C}^{op}})$ via a topology $j$ from a suitable functor $H:\mathcal{C}\to\mathcal{S}$ (for further details see [[Jónsson-Tarski topos]] or Rosenthal(1981)).


\item A Grothendieck topos is a \emph{[[boolean topos|Boolean]] \'e{}tendue} precisely if it satisfies the \emph{internal [[axiom of choice]]} (Freyd\&Scedrov 1990). An example of such a Boolean \'e{}tendue is $\mathcal{S}^G$, for $G$ a group.


\item \'E{}tendues are `locally co-decidable' in the sense that for a small $\mathcal{C}$ the functor category $[\mathcal{C},Set]$ is a [[locally decidable topos]] precisely if $[\mathcal{C}^{op},Set]$ is an [[étendue]]. Also the all-monic-site property is dual to the all-epic-site property of locally decidable toposes. Both concepts are subsumed under the notion of having a (sub canonical) site representation with no (non-trivial) [[idempotents]] (McLarty 2006, Lawvere 2007).



\end{itemize}
\hypertarget{slice_toposes_of_tendues_are_tendues}{}\subsection*{{Slice toposes of étendues are étendues}}\label{slice_toposes_of_tendues_are_tendues}

One can use the site characterization to show that being an étendue topos is a \emph{local property}.

\begin{prop}
\label{etendue_slices}\hypertarget{etendue_slices}{}
Let $Sh(\mathcal{C},J)$ be an étendue topos with $(\mathcal{C}, J)$ being an all-monic-site presentation and $P:C^{op}\to Set$ be a presheaf on $\mathcal{C}$ that is a $J$-sheaf. Then $Sh(\mathcal{C},J)/P$ is an étendue topos.

\end{prop}
\begin{proof}
It suffices to show that $Sh(\mathcal{C},J)/P$ has an all-monic site presentation. By exercise III.8 in \hyperlink{MacLaneMoerdijk}{Mac Lane-Moerdijk (1994, p.157)} there exists a topology $J'$ on the [[category of elements]] $\int_\mathcal{C} P$ such that

\begin{displaymath}
Sh(\mathcal{C},J)/P\simeq Sh(\int_\mathcal{C} P,J')\; .
\end{displaymath}
Whence it suffices to show that $\int_\mathcal{C} P$ is all-monic: Let $f:(C,x)\to (D,y)$ be a morphism in $\int_\mathcal{C} P$ and $g,h:(B,z)\rightrightarrows (C,x)$ such that $f\cdot g=f\cdot h$ then $g=h$ since composition in $\int_\mathcal{C} P$ is inherited from $\mathcal{C}$ and the latter was all-monic by assumption.

\end{proof}
\begin{prop}
\label{IAC_local}\hypertarget{IAC_local}{}
Let $\mathcal{E}$ be a Grothendieck topos satisfying the [[axiom of choice|internal axiom of choice]] (IAC). Then any [[slice topos]] $\mathcal{E}/X$ satisfies the internal axiom of choice as well.

\end{prop}
\begin{proof}
By Freyd-Scedrov (\hyperlink{FreydScedrov90}{1990}, p.181) a Grothendieck topos $\mathcal{E}$ satisfies IAC iff $\mathcal{E}$ is a Boolean étendue. But being [[Boolean category|Boolean]] is a local property whence by the preceding proposition all slices $\mathcal{E}/X$ are Boolean étendues.

\end{proof}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[localic topos]]


\item [[petit topos]]


\item [[locally decidable topos]]


\item [[Deligne-Mumford stack]]


\item [[bicategory of fractions]]



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

\begin{itemize}%
\item [[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], \emph{Th\'e{}orie des Topos et Cohomologie Etale des Sch\'e{}mas ([[SGA4]])}, Springer LNM vol.269 (1972), pp.479-484.


\item [[David Carchedi]], \emph{Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity Topoi} , arXiv1312.2204 (2013). (\href{http://arxiv.org/pdf/1312.2204v1.pdf}{pdf})


\item [[Peter Freyd|P. J. Freyd]], A. Scedrov, \emph{[[Categories, Allegories]]} , North-Holland Amsterdam 1990.


\item [[Peter Johnstone]], \emph{Sketches of an [[Elephant]] II}, Oxford UP 2002, pp.769-775.


\item [[A. Kock]], [[I. Moerdijk]], \emph{Presentations of Etendues} , Cah. Top. G\'e{}om. Diff. Cat. \textbf{XXXII} 2 (1991) pp.145-164. (\href{http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1991__32_2/CTGDC_1991__32_2_145_0/CTGDC_1991__32_2_145_0.pdf}{pdf})


\item [[A. Kock]], [[I. Moerdijk]], \emph{Every \'e{}tendue comes from a local equivalence relation} , JPAA \textbf{82} (1992) pp.155-174.


\item [[M. V. Lawson]], [[B. Steinberg]], \emph{Ordered groupoids and \'e{}tendues} , Cah. Top. G\'e{}om. Diff. Cat. \textbf{XXXXV} 2 (2004) pp.82-108. (\href{http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_2004__45_2/CTGDC_2004__45_2_82_0/CTGDC_2004__45_2_82_0.pdf}{pdf})


\item [[F. William Lawvere]], \emph{Variable sets etendu and variable structure in topoi} , Lecture notes University of Chicago 1975.


\item [[F. William Lawvere]], \emph{Variable quantities and variable structures in topoi} , pp.101-131 in Heller, Tierney (eds.), Algebra, Topology and Category Theory, Academic Press New York 1976.


\item [[F. William Lawvere]], \emph{Categories of Spaces may not be Generalized Spaces as Exemplified by Directed Graphs}, Revista Colombiana de Matem\'a{}ticas \textbf{XX} (1986) pp.179-186. Reprinted with commentary in TAC \textbf{9} (2005) pp.1-7. (\href{http://www.tac.mta.ca/tac/reprints/articles/9/tr9.pdf}{pdf})


\item [[F. William Lawvere]], \emph{Qualitative Distinctions between some Toposes of Generalized Graphs} , Cont. Math. \textbf{92} (1989) pp.261-299.


\item [[F. William Lawvere]], \emph{Some Thoughts on the Future of Category Theory} , pp.1-13 in Springer LNM vol. 1488 (1991).


\item [[F. William Lawvere]], \emph{Axiomatic Cohesion} , TAC \textbf{19} no. 3 (2007) pp.41-49. (\href{http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf}{pdf})


\item [[F. William Lawvere]], \emph{Cohesive Toposes: Combinatorial and Infinitesimal Cases}, Como Ms. 2008. (\href{http://comocategoryarchive.com/Archive/temporary_new_material/FWLawvere-Cohesive-Toposes-Como-January-2008.pdf}{pdf})


\item [[Saunders Mac Lane]], [[Ieke Moerdijk]], \emph{[[Sheaves in Geometry and Logic]]} , Springer Heidelberg 1994.


\item [[Colin McLarty]], \emph{Every Grothendieck Topos has a One-Way Site} , TAC \textbf{16} no. 5 (2006) pp.123-126. (\href{http://www.tac.mta.ca/tac/volumes/16/5/16-05.pdf}{pdf})


\item [[Dorette A. Pronk]], \emph{Etendues and stacks as bicategory of fractions} , Comp. Math. \textbf{102} 3 (1996) pp.243-303. (\href{http://archive.numdam.org/ARCHIVE/CM/CM_1996__102_3/CM_1996__102_3_243_0/CM_1996__102_3_243_0.pdf}{pdf})


\item [[Pedro Resende]], \emph{Groupoid Sheaves as Quantale Modules} , arXiv.0807.4848v3 (2011). (\href{http://arxiv.org/pdf/0807.4848v3.pdf}{pdf})


\item [[Kimmo I. Rosenthal]], \emph{\'E{}tendues and Categories with Monic Maps} , JPAA \textbf{22} (1981) pp.193-212.


\item [[Kimmo I. Rosenthal]], \emph{Sheaves and Local Equivalence Relations} , Cah. Top. G\'e{}om. Diff. Cat. \textbf{XXV} 2 (1984) pp.179-206. (\href{http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1984__25_2/CTGDC_1984__25_2_179_0/CTGDC_1984__25_2_179_0.pdf}{pdf})


\item [[Kimmo I. Rosenthal]], \emph{Covering \'E{}tendues and Freyd's Theorem} , Proc. AMS \textbf{99} 2 (1987) pp.221-222. (\href{http://www.ams.org/journals/proc/1987-099-02/S0002-9939-1987-0870775-X/S0002-9939-1987-0870775-X.pdf}{pdf})



\end{itemize}


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