# Presheaf of groups

From Groupprops

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 be a topological space. A **presheaf of groups** on is the following data:

- For every open subset , a group
- For every pair of open subsets , a
*restriction homomorphism*

such that:

- is the identity map for any
- If then

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