Existence of projective resolutions in abelian categories

Question

It is a standard result of elementary homological algebra that to every R-module $A$ there exists a projective resolution. It is often said that the category of R-modules has "enough projectives." In which other categories is this also true? In particular is it true for abelian categories?

Answer 1

Among the standard examples of abelian categories without enough projectives, there are

1. the categories of sheaves of abelian groups on a topological space (as VA said), or sheaves of modules over a ringed space, or quasi-coherent sheaves on a non-affine scheme;
2. the categories of comodules over a coalgebra or coring.

No abelian category where the functors of infinite product are not exact can have enough projectives. In Grothendieck categories (i.e. abelian categories with exact functors of small filtered colimits and a set of generators) there are always enough injectives, but may be not enough projectives.

Among the standard examples of abelian categories with enough projectives, there are

1. the category of functors from a small category to an abelian category with enough projectives (as VA said), or the category of additive functors from any small additive category to the category of abelian groups (this class of examples includes the categories of modules over any rings);
2. the category of pseudo-compact modules over a pseudo-compact ring (see Gabriel's dissertation);
3. the category of contramodules over a coalgebra or coring (see Eilenberg-Moore, "Foundations of relative homological algebra").

Answer 2

A simple example of an abelian category having not enough projectives is the category of finite abelian groups. In fact, it contains neither non-trivial projective objects nor non-trivial injective objects.

Answer 3

Way too optimistic: in many abelian categories there are not enough projectives (and in the dual category there are not enough injectives).

The most standard example is sheaves of abelian groups on a topological space X. For most X, this category does not have enough projectives. See for example this question where this was discussed: When are there enough projective sheaves on a space X?

On the positive side: if A has enough projectives and I is a small category then the category $A^I$ (i.e. the category of functors $F:I\to A$) has enough projectives (assuming arbitrary sums exist in A). In particular, the category of complexes in A has enough projectives (taking $I=\mathbb Z$ with arrows $d_n:n\to n+1$ satisfying $d_{n+1}\circ d_n=0$). See this question: How to construct pair of adjoint functors from category A to category A_D(category of diagrams)

All of these are abelian categories.