Grothendieck topology

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

1. A category, denoted $cat(T)$
2. A set of coverings $\{U_{i}\to U\}$, denoted $cov(T)$, 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

1. 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.
2. 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)

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.