Subgroupoids and Quotient Theories

Conversely, any equivariant sheaf topos Sh G 1 (G 0 ) classifies a geometric theory with enough models, and G can be regarded as consisting of T-models and isomorphisms. Considering the displayed equivalence (1), there is on the ’theory’ side a correspondence between subtoposes of Set[T] and quotient theories of T (see [2,Theorem 3.6]). On the groupoid side, it is known to specialists (Moerdijk in particular) that a subgroupoid of an open topological groupoid induces a subtopos of equivariant sheaves, but this fact appears not to have been published. As a first outline of the connection between subgroupoids of G and quotient theories of T, this paper first fills in a proof of that fact and points out the resulting Galois connection between subgroupoids of G and subtoposes of Sh G 1 (G 0 ) (and thus quotient theories of T), and then characterizes the subgroupoids of G that are definable by quotient theories. The whole investigation is carried out using Moerdijk’s site description for equivariant sheaf toposes given in [3], and a brief introduction to that construction is given first.

2 Subgroupoids and subtoposes

This section briefly recalls from [4], [3], [5] the topos of equivariant sheaves on a topological groupoid and Moerdijk’s site description for such toposes (written out here for topological rather than localic groupoids and writing out a few additional details, cf. especially [3, §6], a more detailed and selfcontained presentation can be found in the online note [6]). Let G be a topological groupoid, fully written out as a groupoid object in the category Sp of topological spaces and continuous maps as

with m the composition, e the mapping to identities, and i the mapping to inverses. This notation will be mixed with the usual notation g

and composition defines a continuous action on d -1 (U)/ ∼ N , so that we have an equivariant sheaf denoted G , U, N . Objects of the form G , U, N form a generating set for Sh G 1 (G 0 ). Briefly, given an equivariant sheaf r : R → G 0 , ρ and a continuous section t : U → R, we get an open set of arrows

(by pullback of the open set t(U) along an appropriate continuous map) which is closed under composition and inverse, and such that d(N) = c(N) = U. There is a canonical continuous section e : U → d -1 (U)/ ∼ N defined by x → [1 x ] ∼ N t , and the section t lifts to a morphism, t :

). One easily sees that t is 1-1. For reference:

is the join of its subobjects of the form G , U, N A for open subgroupoids (U, N).

The full subcategory of Sh G 1 (G 0 ) consisting of objects of the form G , U, N is, accordingly, a site for Sh G 1 (G 0 ) when equipped with the canonical coverage. Refer to this as the Moerdijk site for Sh G 1 (G 0 ), and denote it S G / / Sh G 1 (G 0 ). Moerdijk sites are closed under subobjects. For consider an object G , U, N and let V ⊆ U be an open subset closed under N, that is, such that x ∈ V and f : x → y in N implies y ∈ V . Then

is an open subset closed under the action, and so a subobject. All subobjects are of this form:

defines an isomorphism between the frame of open subsets of U that are closed under N and the frame of subobjects of G , U, N .

Proof The inverse is given by pulling back along the canonical section e :

The morphisms in the Moerdijk site can be described in a manner similar to the objects in it. Consider a morphism t : G , U, N → G , V, M . It is easily seen that such a morphism determines and is determined by a section t : U → d -1 (V )/ ∼ M with the property that for any f : x → y in N, we have that f • t(x) = t(y). And such a section can be described as an open set:

Moreover, t can be thought of as ‘precomposing with T ‘, in the sense that

Proof Straightforward.

The following corollary will be useful.

Proof c(T ) is closed under N by condition (iv), and the rest is straightforward. ⊣

For a morphism f : H / / G of open topological groupoids, the induced inverse image f * does not necessarily restrict to a functor between the respective Moerdijk-sites. The following condition (somewhat simplified from [3], cf. Lemma 6.2 there, so a proof is included here) ensures that it does.

If the component continuous functions of f are, moreover, subspace inclusions, then we say that H is a replete subgroupoid of G and that f is a replete subgroupoid inclusion.

Proof Consider the diagram

where t is the section obtained by pulling back the canonical section e :

and so

The second claim is a similar computation using Lemma 2.1.3. ⊣

Let G be an open topologic