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.

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.

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 of functors $A^I$ has enough projectives (assuming arbitrary sums exist in A). In particular, the category of complexes in A has enough projectives. See this question: How to construct pair of adjoint functors from category A to category A_D(category of diagrams)

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.