Presheaf of groups

From Groupprops
Jump to: navigation, search
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.