additive and abelian categories
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
see also sheaf of abelian groups
An abelian sheaf is a sheaf with values in an abelian category which usually is itself, or is taken to be embedded in, a category of complexes in an abelian category.
In light of the Dold-Kan correspondence this means that abelian sheaves can usefully be regarded as special cases of simplicial presheaves and in particular the corresponding derived category of abelian sheaves, traditionally mainly investigated in terms of sheaf cohomology, is analogous to the homotopy category of abelian infinity-stacks. In this way, via Dold-Kan, plain abelian sheaves already go a long way towards (abelian) Higher Topos Theory, which is one way of understanding the relevance of the concept of abelian sheaves.
For instance Deligne cohomology, which classifies higher abelian gerbes (certain infinity-stacks) with connection), is the sheaf cohomology of a certain class of sheaves with values in abelian complexes. This is understood conceptually by realizing that after embedding complexes of abelian sheaves – via Dold-Kan – into general simplicial sheaves, a complex of abelian sheaves becomes an abelian $\infty$-prestack and the computation of its sheaf cohomology corresponds to passing to its infinity-stackification.
See at projective object the section Existence of enough projectives.
an abelian sheaf of torsion groups is called a torsion sheaf
A basic textbook introduction begins for instance around Definition 1.5.6 of
A detailed textbook discussion is in section 18 of