Presheaf of groups

This article defines the notion of group object in the category of set-valued presheafs|View other types of group objects

Definition

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.