Cohomology and homotopy
In higher category theory
A sheaf on (the site of open subsets of) a topological space corresponds to an étalé space . This space has itself a sheaf topos associated to it, and the map induces a geometric morphism of sheaf toposes
Due to the special nature of , the topos on the left is equivalent to the slice topos , and the projection morphism above factors through a canonical standard geometric morphism
And conversely, every local homeomorphism of topological spaces corresponds to a geometric morphism of sheaf toposes of this form.
This motivates calling a geometric morphism
a local homeomorphism of toposes or étale geometric morphism if it factors as an equivalence followed by a projection out of an overcategory topos.
If the topos is a locally ringed topos, or more generally a structured (∞,1)-topos, it makes sense to require additionally that the local homeomorphism is compatible with the extra structure.
For a topos (or (∞,1)-topos, etc.) and for an object, the overcategory is also a topos (-topos, etc), the slice topos (slice (∞,1)-topos, …).
The canonical projection is part of an essential (in fact, locally connected/ locally ∞-connected) geometric morphism:
This is the base change geometric morphism for the terminal morphism .
A geometric morphism is called a local homeomorphism of toposes, or an étale geometric morphism, if it is equivalent to such a projection— in other words, if it factors by geometric morphisms as for some .
For structured toposes
If the (∞,1)-toposes in question are structured (∞,1)-toposes, then this is refined to the following
A morphism of structured (∞,1)-toposes is an étale morphism if
the underlying morphism of -toposes is an étale geometric morphism;
the induced map is an equivalence.
This is StSp, Def. 2.3.1.
If is a localic topos over a topological space we have that corresponds to an étalé space over and to an étale map.
If is a geometry (for structured (∞,1)-toposes) then for an admissible morphism in , the induced morphism of structured (∞,1)-toposes
is an étale geometric morphism of structured -toposes.
This is StrSp, example 2.3.8.
See at cartesian closed functor for proof.
(recognition of étale geometric morphisms)
A geometric morphism is étale precisely if
it is essential;
is a conservative functor;
For every diagram in the induced diagram
is a pullback diagram.
For (∞,1)-toposes this is HTT, prop. 220.127.116.11.
(Recovering a topos from its etale overcategory)
For an -topos we have
where is the full sub-(∞,1)-category of the over-(∞,1)-category on the etale geometric morphisms .
This is HTT, remark 18.104.22.168.
The notion of local homeomorphisms of toposes is page 651 (chapter C3.3) of
The notion of étale geometric morphisms between (∞,1)-toposes is introduced in section 6.3.5 of
Discussion of the refinement to structured (∞,1)-toposes is in section 2.3 of