Cohomology and homotopy
In higher category theory
The notion of site may be internalized in any topos to yield a notion of internal site.
The definition of internal site is obvious and straightforward.
For a topos, an internal site in is an internal category equipped with an internal coverage.
Spelled out in components, this means the following (as in (Johnstone), we shall only define sifted coverages). First, we define the subobject of sieves, where a subobject is a sieve if the composite
factors through . Also recall the usual membership relation .
An internal sifted coverage is given by a span subject to:
commutes, where the pullback in the top left corner is of the map along .
If we define the subobject as
(in the internal language), the composite is required to be an epimorphism.
We can additionally ask that more saturation conditions (as discussed at coverage) hold.
We discuss how to every internal site there is a corresponding external site such that the internal sheaf topos on the former agrees with the external sheaf topos on the latter.
Let be a small category and let be its presheaf topos. Let be an internal site. Regarded, by the Yoneda lemma, as a functor , this induces via the Grothendieck construction a fibered category which we denote
This is reviewed for instance in (Johnstone, p. 596).
The notation is motivated from the following example.
Let be a group (in Set, hence a discrete group) and let be its delooping groupoid. Then
is the topos of permutation representations of . Let be a group object. This is equivalently a group in equipped with a -action. Its internal delooping gives the internal groupoid in .
In this case we have that
is the delooping gorupoid of the semidirect product group of the -action on .
Generally we have
We have an equivalence of categories
between the category of internal presheaves in over the internal category , and external presheaves over the semidirect product site .
This appears as (Johnstone, lemma C2.5.3).
This result generalizes straightforwardly to an analogous statement for internal sheaves.
If is equipped with a coverage and is equipped with an internal coverage , define a coverage on by declaring that a sieve on an object is -covering if there exists an element with ,
Let be a sheaf topos and an internal site in . Then with def. 4 we have an equivalence of categories
between internal sheaves in on and external sheaves on the semidirect product site .
Moreover, the projection functor is cover-reflecting and induces a geometric morphism
This appears as (Johnstone, prop. C2.5.4).
Section C2.4 and C2.5 of
The semidirect product externalization of internal sites is due to
- Ieke Moerdijk, Continuous fibrations and inverse limits of toposes, Composition Math. 68 (1986)