Cohomology and homotopy
In higher category theory
The condition that a presheaf be a sheaf may be seen as a condition of unique existence. A presheaf is separated if it satisfies the uniqueness part.
Let be a site.
Recall that a sheaf on is a presheaf such that for all local isomorphisms the induced morphism (under the hom-functor ) is an isomorphism. (For an arbitrary class of morphisms , the corresponding condition is called being a local object.)
It is sufficient to check this on the dense monomorphisms instead of all local isomorphisms. This is equivalent to checking covering sieves.
A presheaf is called separated (or a monopresheaf) if for all local isomorphisms the induced morphism is a monomorphism.
More generally, for a class of arrows in a category , an object is -separated if for all morphisms in , the induced morphism is a monomorphism.
See also at locally connected site.
The full subcategory
of separated presheaves in a presheaf category is
Being a reflective subcategory means that there is a left adjoint functor to the inclusion
For the separafication of is the presheaf that assigns equivalence classes
where is the equivalence relation that relates two elements iff there exists a covering such that for all .
This construction extends in the evident way to a functor
This functor is indeed a left adjoint to the inclusion .
Let and . We need to show that morphisms in are in natural bijection with morphisms in .
For such a morphism and its component over any object , consider any covering , let be the corresponding sieve and consider the commuting diagram
obtained from the naturality of .
If for two elements that are not equal their restrictions to the cover become equal in that , then also and since the right vertical morphism is monic there is a unique mapping to the latter. The commutativity of the diagram then demands that .
Since this argument applies to all covers of , we have that factors uniquely through the projection map onto the quotient. Since this is true for every object we have that factors uniquely through .
Often one is interested in separated presheaves with respect to one coverage that are sheaves with respect to another coverage. These are called biseparated presheaves .
This typically arises if a reflective subcategory
of a sheaf topos is given. This is the localization at a set of morphisms in , with the full subcategory of all local objects : objects such that is an isomorphism for all . A -separated object is then called a biseparated presheaf on and their collection factors the reflective inclusion as
A bisite is a small category equipped with two coverages: and such that .
A presheaf is called -biseparated if it is
for the full subcategory on biseparated presheaves.
Biseparated presheaves form a reflective subcategory of all sheaves
See quasitopos for the proof.
The general theory of biseparated presheaves and Grothendieck quasitoposes is in section C.2.2 of
A concrete description of separafication appears on page 43 of
- Angelo Vistoli, Notes on Grothendieck topologies, fibered categories and descent theory (pdf)