lo.logic - What is a topos?

Let us concentrate on Grothendieck topoi. As mentioned in earlier posts, these are those topoi which arise as the category of sheaves for a category equipped with a Grothendieck topology. These are those topoi which "behave the most like sheaves of sets on a topological space". Let me try to explain to what extent Grothendieck topoi are topological in nature.

First, given a continuous map f:X->Y, it produces a geometric morphism Sh(X)->Sh(Y). If X and Y are sober, then there is a bijection between Hom(X,Y) and Hom(Sh(X),Sh(Y)) (where the later is again geometric morphisms). This means, if we restrict to sober spaces, we get a fully faithful functor Sh:SobTop->topoi. (Recall via stone duality that the category of sober spaces is equivalent to the category of locales with enough points).

More generally, if G is a topological groupoid (a groupoid object in Top), we can construct its classifying topos. This can be defined as follows: Take the enriched nerve of G to obtain a simplicial space, applying the functor "Sh" (viewing the nerve as a diagram of space) and obtain a simplicial topos. Now take the (weak) colimit of the diagram to obtain a topos BG. This topos can be described concretely as equivariant sheaves over G_0.

Geometrically, BG is a model for the topos of "small sheaves" over the topological stack associated to G. In fact, on etale topological stacks* (this include all orbifolds), we also have an equivalence between maps of stacks and geometric morphisms between their categories of sheaves, so, there is a subcategory (sub-2-category) of Grothendieck topoi which is equivalent to etale topological stacks. These Grothendieck topoi are called topological etendue.

*(over sober spaces)

It turns out that a large class of topoi can be obtained as BG for some topological groupoid. In fact, every Grothendick topos "with enough points" is equivalent to BG for some topological groupoid. The more general statement is that EVERY Grothendieck topos is equivalent to BG for some localic groupoid (a groupoid object in locales). Since locales are a model for "pointless topology", we see in some sense, every Grothendieck topos is "topological". You can make sense of the statement that every Grothendieck topos is equivalent to the category of small sheaves on a localic stack.