Grothendieck topology

Summary

A Grothendieck topology on a category says which families of morphisms into an object count as coverings of that object. It is stated with sieves and abstracts the open covers of a topological space. A category with a Grothendieck topology is a site, and one can define sheaves and sheaf cohomology on it. Grothendieck introduced it to define étale cohomology of schemes, where the "open sets" are étale maps rather than subsets.