Sheaf on a site

Summary

A sheaf on a site $(\mathcal{C}, J)$ is a presheaf on $\mathcal{C}$ with a gluing property. Compatible local data on every covering sieve glue to a unique global datum. It generalizes a sheaf on a topological space, which is the case of the site of open sets. Sheaves on a small site form a Grothendieck topos, and every presheaf has a best approximation by a sheaf, its sheafification. Sheaves on the étale site of a scheme give étale cohomology.