Grothendieck topology

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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.

Definition

A Grothendieck topology consists of

  1. A category, denoted
  2. A set of coverings , denoted , such that
    1. for each object of
    2. If , and is any morphism in , then the canonical morphisms of the fiber products determine a covering
    3. If and , then

Examples

  1. 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.
  2. 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.