Grothendieck topology

A Grothendieck topology is a structure defined on an arbitrary category C which allows the definition of sheaves? on C, and with that the definition of general cohomology theories. This tool is mainly used in algebraic geometry, for instance to define [étale cohomology]?.

The motivating example is the following: start with a topological space X and consider the sheaf of all continuous real-valued functions defined on X. This associates to every open set U in X the set F(U) of real-valued continuous functions defined on U. Whenver U is a subset of V, we have a "restriction map" from F(V) to F(U). If we interpret the topological space X as a category, with the open sets being the objects and a morphism from U to V if and only if U is a subset of V, then F is revealed as a contravariant functor from this category into the category of sets. In general, every contravariant functor from a category C to the category of sets is therefore called a pre-sheaf of sets on C. Our functor F has a special property: if you have an open covering (V_i) of the set U, then the elements of F(U) all arise from piecing together mutually compatible elements of F(V_i). This turns F into a sheaf, and a Grothendieck topology on C is an attempt to capture the essense of what is needed to define sheaves on C.

Formally, to be continued...