additive and abelian categories
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
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
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