nLab sheaf of abelian groups

Idea

For $C$ a site, a sheaf of abelian groups on $C$ is an abelian group object in the sheaf topos $Sh(C)$.

The category $Ab(Sh(C))$ of sheaves of abelian groups is an abelian category and hence serves as a context for homological algebra “parameterized over $C$”. For the case that $C = *$ is the point, this is just Ab itself.

More generally, for $\mathcal{A}$ an abelian category one can consider $\mathcal{A}$-valued sheaves $Sh(C,\mathcal{A})$: abelian sheaves. For this to have good properties $\mathcal{A}$ has to be a Grothendieck category.

References

A basic textbook introduction begins for instance around Definition 1.6.5 of

• Charles Weibel, An Introduction to Homological Algebra

A detailed textbook discussion is in section 18 of

• Masaki Kashiwara, Pierre Schapira, Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften 332, Springer (2006)

category: sheaf theory