Sheaf of groups

Definition

Let $\mathcal C$ be a site. A sheaf of groups on $\mathcal C$ is a sheaf $G$ such that every set of sections $G(U)$ is a group and every restriction map associated to a morphism $V\to U$,

is a group homomorphism. Equivalently, $G$ is a group object in the category of sheaves on $\mathcal C$: multiplication, identity, and inverse are morphisms of sheaves satisfying the group axioms.

A right action of $G$ on a sheaf $P$ is a morphism of sheaves

whose maps on sections are right group actions and commute with restriction. This is the type of action used in a $G$-torsor on a site.