sheaf of abelian groups
Additive and abelian categories
Cohomology and homotopy
In higher category theory
For a site, a sheaf of abelian groups on is an abelian group object in the sheaf topos .
The category of sheaves of abelian groups is an abelian category and hence serves as a context for homological algebra “parameterized over ”. For the case that is the point, this is just Ab itself.
More generally, for an abelian category one can consider -valued sheaves : abelian sheaves. For this to have good properties has to be a Grothendieck category.
A basic textbook introduction begins for instance around Definition 1.6.5 of
A detailed textbook discussion is in section 18 of