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 $T$ consists of

A category, denoted $cat(T)$
A set of coverings , denoted , such that
1. $\{id:U\mapsto U\}\in cov(T)$ for each object $U$ of $cat(T)$
2. If $\{U_i\to U\}\in cov(T)$ , and $V\to U$ is any morphism in $cat(T)$ , then the canonical morphisms of the fiber products determine a covering $\{U_i\times_U V\to V\}\in cov(T)$
3. If $\{U_i\to U\}\in cov(T)$ and $\{V_{i,j}\to U_i\}\in cov(T)$ , then $\{V_{i,j}\to U_i\to U\}\in cov(T)$

Examples

A standard topological space $X$ becomes a category $op(X)$ when you regard the open subsets of $X$ as objects, and morphisms are inclusions. An open covering of open subsets $U$ clearly verify the axioms above for coverings in a site. Notice that a presheaf of rings is just a contravariant functor from the category $op(X)$ into the category of rings.
The Small Étale Site Let $S$ be a scheme. Then the category of étale schemes over $S$ (i.e., $S$ -schemes $X$ over $S$ 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 $\{U_i\to U\}\in cov(T)$ , the diagram $0\to F(U)\to \prod F(U_i)\to \prod F(U_i\times_U U_j)$ is exact.