Presheaf of groups

Category-theoretic definition
A presheaf of groups on a topological space, is a contravariant functor from the category of open sets of the topological space (under inclusion) to the category of groups.

Hands-on definition
Let $$X$$ be a topological space. A presheaf of groups $$F$$ on $$X$$ is the following data:


 * For every open subset $$U \subset X$$, a group $$F(U)$$
 * For every pair of open subsets $$U \subset V$$, a restriction homomorphism $$res_{VU}: F(V) \to F(U)$$

such that:


 * $$res_{UU}$$ is the identity map for any $$U$$
 * If $$W \subset V \subset U$$ then $$res_{UW} = res_{VW} \circ res_{UV}$$

A particular case of a presheaf of groups is a sheaf of groups.