An étendue (also ‘etendue’, or ‘etendu’; from French ‘étendue’ (fem.)- extent) is a topos $\mathcal{Y}$ that locally looks like the category of sheaves on a space:
Briefly, the slogan is that $\mathcal{Y}$ is locally a topological space. (Lawvere 1976, p.129)
Originally defined by A. Grothendieck in one of the famous ‘excercises’ of 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 étendues are known and the Ur-example of an étendue, 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 $\infinity$-étendues (cf. Carchedi 2013).
Étendues 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 petit toposes (generalized spaces). In this context, Lawvere (1989,1991) interprets the cancellative property of the site as enabling an interpretation of étendues as categories of processes.
A topos $\mathcal{Y}$ is called an étendue 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.^{1}
The first example of an étendue 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 étendue 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. (Lawvere 1976, pp.129-130)
The Sierpinski topos $\mathcal{S}^{\cdot\rightarrow\cdot}$, as the sheaf topos on the Sierpinski space, is an étendue.
The topos $\mathcal{S}^{\cdot\rightrightarrows\cdot}$ of directed graphs (aka quivers; Lawvere calls them irreflexive graphs) is an étendue, 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 reflexive graphs is a paradigmatic example of the distinction between a petit and a gros topos.
The Jónsson-Tarski topos $\mathcal{J}_2$ is an étendue, as it is locally equivalent to the sheaf topos on the Cantor space. It is discussed as a petit topos for labeled graphs in (Lawvere 1989).
Proposition. A Grothendieck topos $\mathcal{Y}$ is an étendue iff there exists a site $(\mathcal{C}, J)$ for $\mathcal{Y}$ such that every morphism of $\mathcal{C}$ is monic.
In particular for small $\mathcal{C}$, the presheaf topos $\mathcal{S}^{\mathcal{C}^{op}}$ is an étendue iff all morphisms in $\mathcal{C}$ are monic. In particular for monoids $\mathcal{C}$ this is sometimes called left cancellative (e.g. each free monoid is left cancellative).
Subtoposes of étendues are étendues. 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 étendues $Sh_j(\mathcal{S}^{\mathcal{C}^{op}})$ via a topology $j$ from a suitable functor $H:\mathcal{C}\to\mathcal{S}$ (for further details see Jónsson-Tarski topos or Rosenthal(1981)).
A Grothendieck topos is a Boolean étendue precisely if it satisfies the internal axiom of choice (Freyd&Scedrov 1990). An example of such a Boolean étendue is $\mathcal{S}^G$, for $G$ a group.
Étendues 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).
One can use the site characterization to show that being an étendue topos is a local property.
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.
It suffices to show that $Sh(\mathcal{C},J)/P$ has an all-monic site presentation. By exercise III.8 in Mac Lane-Moerdijk (1994, p.157) there exists a topology $J'$ on the category of elements $\int_\mathcal{C} P$ such that
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.
Let $\mathcal{E}$ be a Grothendieck topos satisfying the internal axiom of choice (IAC). Then any slice topos $\mathcal{E}/X$ satisfies the internal axiom of choice as well.
By Freyd-Scedrov (1990, p.181) a Grothendieck topos $\mathcal{E}$ satisfies IAC iff $\mathcal{E}$ is a Boolean étendue. But being Boolean is a local property whence by the preceding proposition all slices $\mathcal{E}/X$ are Boolean étendues.
M. Artin, A. Grothendieck, J. L. Verdier, Théorie des Topos et Cohomologie Etale des Schémas (SGA4), Springer LNM vol.269 (1972), pp.479-484.
David Carchedi, Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity Topoi , arXiv1312.2204 (2013). (pdf)
P. J. Freyd, A. Scedrov, Categories, Allegories , North-Holland Amsterdam 1990.
Peter Johnstone, Sketches of an Elephant II, Oxford UP 2002, pp.769-775.
A. Kock, I. Moerdijk, Presentations of Etendues , Cah. Top. Géom. Diff. Cat. XXXII 2 (1991) pp.145-164. (pdf)
A. Kock, I. Moerdijk, Every étendue comes from a local equivalence relation , JPAA 82 (1992) pp.155-174.
M. V. Lawson, B. Steinberg, Ordered groupoids and étendues , Cah. Top. Géom. Diff. Cat. XXXXV 2 (2004) pp.82-108. (pdf)
F. William Lawvere, Variable sets etendu and variable structure in topoi , Lecture notes University of Chicago 1975.
F. William Lawvere, Variable quantities and variable structures in topoi , pp.101-131 in Heller, Tierney (eds.), Algebra, Topology and Category Theory, Academic Press New York 1976.
F. William Lawvere, Categories of Spaces may not be Generalized Spaces as Exemplified by Directed Graphs, Revista Colombiana de Matemáticas XX (1986) pp.179-186. Reprinted with commentary in TAC 9 (2005) pp.1-7. (pdf)
F. William Lawvere, Qualitative Distinctions between some Toposes of Generalized Graphs , Cont. Math. 92 (1989) pp.261-299.
F. William Lawvere, Some Thoughts on the Future of Category Theory , pp.1-13 in Springer LNM vol. 1488 (1991).
F. William Lawvere, Axiomatic Cohesion , TAC 19 no. 3 (2007) pp.41-49. (pdf)
F. William Lawvere, Cohesive Toposes: Combinatorial and Infinitesimal Cases, Como Ms. 2008. (pdf)
Saunders Mac Lane, Ieke Moerdijk, Sheaves in Geometry and Logic , Springer Heidelberg 1994.
Colin McLarty, Every Grothendieck Topos has a One-Way Site , TAC 16 no. 5 (2006) pp.123-126. (pdf)
Dorette A. Pronk, Etendues and stacks as bicategory of fractions , Comp. Math. 102 3 (1996) pp.243-303. (pdf)
Pedro Resende, Groupoid Sheaves as Quantale Modules , arXiv.0807.4848v3 (2011). (pdf)
Kimmo I. Rosenthal, Étendues and Categories with Monic Maps , JPAA 22 (1981) pp.193-212.
Kimmo I. Rosenthal, Sheaves and Local Equivalence Relations , Cah. Top. Géom. Diff. Cat. XXV 2 (1984) pp.179-206. (pdf)
Kimmo I. Rosenthal, Covering Étendues and Freyd’s Theorem , Proc. AMS 99 2 (1987) pp.221-222. (pdf)
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 $\mathcal{Y}/X$ covers $\mathcal{Y}$. $k_\ast$ is an étale geometric morphism. ↩
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