The notion of a Grothendieck topology or site' captures the essential properties necessary for constructing a robust theory of cohomology of sheaves. The theory of Grothendieck topologies was developed by Alexander Grothendieck and Michael Artin.
A Grothendieck topology consists of
- A category, denoted
- A set of coverings , denoted , such that
- for each object of
- If , and is any morphism in , then the canonical morphisms of the fiber products determine a covering
- If and , then
- A standard topological space becomes a category when you regard the open subsets of as objects, and morphisms are inclusions. An open covering of open subsets clearly verify the axioms above for coverings in a site. Notice that a presheaf of rings is just a contravariant functor from the category into the category of rings.
- The Small Étale Site Let be a scheme. Then the category of étale schemes over (i.e., -schemes over whose structural morphisms are étale) becomes a site if we require that coverings are jointly surjective; that is,
Sheaves on Sites
In analogy with the situation for topological spaces, a presheaf may be defined as a contravariant functor such that for all coverings , the diagram is exact.