nLab
étendue

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Idea

An étendue (also ‘etendue’, or ‘etendu’; from French ‘étendue’ (fem.)- extent) is a topos 𝒴\mathcal{Y} that locally looks like the category of sheaves on a space:

Briefly, the slogan is that 𝒴\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 𝒴\mathcal{Y} that has a well-supported object XX such that the slice topos 𝒴/X\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 𝒴/X\mathcal{Y}/X is a localic topos.

Several characterizations of étendues are known and the Ur-example of an étendue, the presheaf topos 𝒮 G\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.

Definition

A topos 𝒴\mathcal{Y} is called an étendue if there is an object X|𝒴|X\in|\mathcal{Y}| such that the unique X1X\rightarrow 1 is epic and the slice topos 𝒴/X\mathcal{Y}/X is a localic topos.1

Examples

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 𝒮 G\mathcal{S}^G is an étendue with only one point! This is easily seen from the observations that 𝒮 G/G𝒮 G\mathcal{S}^G/G\cong\mathcal{S}^G and that G1G\twoheadrightarrow 1 where the last two GG‘s denote the regular representation. (Lawvere 1976, pp.129-130)

  • The Sierpinski topos 𝒮 \mathcal{S}^{\cdot\rightarrow\cdot}, as the sheaf topos on the Sierpinski space, is an étendue.

  • The topos 𝒮 \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 𝒮 \mathcal{S}^{\cdot\rightrightarrows\cdot} and the topos 𝒮 Δ 1 op\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 𝒥 2\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).

Properties

  • Proposition. A Grothendieck topos 𝒴\mathcal{Y} is an étendue iff there exists a site (𝒞,J)(\mathcal{C}, J) for 𝒴\mathcal{Y} such that every morphism of 𝒞\mathcal{C} is monic.

  • In particular for small 𝒞\mathcal{C}, the presheaf topos 𝒮 𝒞 op\mathcal{S}^{\mathcal{C}^{op}} is an étendue iff all morphisms in 𝒞\mathcal{C} are monic. In particular for monoids 𝒞\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 𝒮 𝒞 op\mathcal{S}^{\mathcal{C}^{op}} for all-monic 𝒞\mathcal{C} in order to construct further étendues Sh j(𝒮 𝒞 op)Sh_j(\mathcal{S}^{\mathcal{C}^{op}}) via a topology jj from a suitable functor H:𝒞𝒮H:\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 𝒮 G\mathcal{S}^G, for GG a group.

  • Étendues are ‘locally co-decidable’ in the sense that for a small 𝒞\mathcal{C} the functor category [𝒞,Set][\mathcal{C},Set] is a locally decidable topos precisely if [𝒞 op,Set][\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).

Slice toposes of étendues are étendues

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

Proposition

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

Proof

It suffices to show that Sh(𝒞,J)/PSh(\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 JJ' on the category of elements 𝒞P\int_\mathcal{C} P such that

Sh(𝒞,J)/PSh( 𝒞P,J).Sh(\mathcal{C},J)/P\simeq Sh(\int_\mathcal{C} P,J')\; .

Whence it suffices to show that 𝒞P\int_\mathcal{C} P is all-monic: Let f:(C,x)(D,y)f:(C,x)\to (D,y) be a morphism in 𝒞P\int_\mathcal{C} P and g,h:(B,z)(C,x)g,h:(B,z)\rightrightarrows (C,x) such that fg=fhf\cdot g=f\cdot h then g=hg=h since composition in 𝒞P\int_\mathcal{C} P is inherited from 𝒞\mathcal{C} and the latter was all-monic by assumption.

Corollary

Let \mathcal{E} be a Grothendieck topos satisfying the internal axiom of choice (IAC). Then any slice topos /X\mathcal{E}/X satisfies the internal axiom of choice as well.

Proof

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 /X\mathcal{E}/X are Boolean étendues.

References


  1. An epic k:XYk:X\rightarrow Y induces a geometric morphism k *:𝒴/X𝒴/Yk_\ast:\mathcal{Y}/X\rightarrow \mathcal{Y}/Y whose inverse image part, the change of base functor, k *:𝒴/Y𝒴/Xk^\ast:\mathcal {Y}/Y\rightarrow\mathcal{Y}/X is faithful, which says by definition that k *k_\ast is a surjection, and in case Y=1Y=1, one says that 𝒴/X\mathcal{Y}/X covers 𝒴\mathcal{Y}. k *k_\ast is an étale geometric morphism.

Last revised on October 22, 2018 at 08:30:10. See the history of this page for a list of all contributions to it.