Cohomology and homotopy
In higher category theory
objects such that commutes with certain colimits
The notion of compact topos is the generalization from topology to topos theory of the notion of compact topological space.
More generally, over a general base topos, the notion of proper geometric morphism is the generalization to morphisms between toposes of the notion of proper map between topological spaces.
A topos is called strongly compact if commutes even with all filtered colimits.
A geometric morphism is called tidy if it exhibits as a strongly compact topos over .
(MV, p. 53)
This are the first stages of a notion that in (∞,1)-topos theory continue as follows
In this terminology
Stability and closure properties
(VM, I.1, I.2)
The pullback of a proper geometric morphism is again proper.
The pullback of a tidy geometric morphism is again tidy.
(VM, theorem 5.8)
Properness and Beck-Chevalley conditions
A geometric morphism of toposes is said to satisfy the stable (weak) Beck-Chevalley condition if any pullback of satisfies the (weak) Beck-Chevalley condition ((weak)BCC).
A map satisfies the stable weak BCC iff it is proper.
(MV, Corollary I.5.9)
We discuss classes of sites such that their sheaf topos is a compact topos, def. 1 (VM, I.5).
Strongly compact sites
We discuss classes of sites such that their sheaf topos is a strongly compact topos, def. 1 (VM, III.4).
Let be a topos and an object. If
- The slice topos is a compact topos, def. 1.
The terminal object of is the identity in . A subterminal object of is a monomorphism in .
The global section geometric morphism sends an object to its set of sections
Therefore it sends all subterminal object in to the empty set except the terminal object itself, which is sent to the singleton set.
So let now be a compact-topological-space-object and is directed system of subterminals.
If their union does not cover , then . But then also none of the can be itself, and hence also for all and so . On the other hand, if then the form a cover, hence then by assumption there is a finite subset which still covers. By the assumption that the system is a directed set it also contains the union . Therefore is the singleton, as is . So preserves directed unions of subterminals and hence is a compact topos.
Strongly compact toposes
The following propositions say in summary that
the petit topos over a compact topological space that is also Hausdorff is strongly compact.
the gros topos over a compact object is strongly compact.
See also (VM, III.1).
(MV, Examples III.1.1)
Let be a topos over Set and an object. Then the following are equivalent
is a compact object (in the sense that the hom functor preserves filtered colimits)
the slice topos is strongly compact, def. 3.
The direct image of the global section geometric morphism
is given by the hom functor out of the terminal object. The terminal object in is the identity morphism . So the terminal geometric morphism takes any in to the set of sections, given by the pullback of the hom set along the inclusion of the identity
By the discussion at overcategory – limits and colimits we have that colimits in are computed in . So if is a filtered colimit in , then is a filtered colimit in .
If now is a compact object, then this commutes over this colimit and hence
where in the second but last step we used that in the topos Set colimits are preserved by pullback.
This shows that commutes over filtered colimits if is a compact object.
Conversely, assume that commutes over all filtered colimits. For every (filtered) diagram there is the corresponding filtered diagram , where is the projection. As before, the product with preserves forming colimits
Moreover, sections of a trivial bundle are maps into the fiber
So it follows that is a compact object:
An object in a topos is a Kuratowski finite object precisely if the étale geometric morphism
out of the slice topos is a proper geometric morphism. And precisely if is even decidable is this a tidy geometric morphism.
(Moerdijk-Vermeulen, examples III 1.4)
A typical condition on a geometric stack to qualify as an orbifold/Deligne-Mumford stack is that its diagonal be proper. This is equivalent to the corresponding map of toposes being a proper geometric morphism (e.g. Carchedi 12, section 2, Lurie Spectral, section 3).
The theory of proper geometric morphisms is largly due to
Ieke Moerdijk, Jacob Vermeulen, Relative compactness conditions for toposes (pdf)
Ieke Moerdijk, Jacob Vermeulen, Proper maps of toposes , Memoirs of the American Mathematical Society, no. 705 (2000)
based on the localic case discussed in
- Jacob Vermeulen, Proper maps of locales, J. Pure Applied Alg. 92 (1994)
A textbook account is in section C3.2 of
Discussion with relation to properness of geometric stacks includes
Discussion of higher compactness conditions in (∞,1)-topos theory is in section 3 of
and in section 2.3 of
and for the special case of spectral Deligne-Mumford stacks in section 1.4 of