Presheaves and sheaves of groups, rings, and modules

Let $X$ be a topological space. Fix a ring $R$ when modules are under discussion.

A presheaf of groups on $X$ is a presheaf $\mathcal{F}$ such that every $\mathcal{F}(U)$ is a group and every restriction map ${\rho }_V^U:\mathcal{F}(U)\to \mathcal{F}(V)$ is a group homomorphism (Groups and group homomorphisms form the large locally small category $\mathbf{G}\mathbf{r}\mathbf{p}$).

A presheaf of rings on $X$ is defined the same way with rings and ring homomorphisms (Unital rings and unit-preserving ring homomorphisms form the large locally small category $\mathbf{R}\mathbf{i}\mathbf{n}\mathbf{g}$).

A presheaf of left $R$-modules on $X$ is defined the same way with left $R$-modules and $R$-linear maps (Left modules over a fixed ring and module homomorphisms form the large locally small category $R\text{-}\mathbf{M}\mathbf{o}\mathbf{d}$).

A sheaf of groups, sheaf of rings, or sheaf of left $R$-modules is such a presheaf whose underlying set-valued presheaf is a sheaf.

In particular, each sheaf of groups has a distinguished identity section on every open set; in additive notation this section is written $0$, and all restrictions preserve the algebraic operations.