A "nice" category $\mathcal{C}$ should be (for the purposes of this question) locally presentable at a minimum, and maybe a bit more. One might require $\mathcal{C}$ to be (in order of increasing restrictiveness)

• ABn for some $n$.

• Grothendieck

• locally finitely presentable

• etc.

For instance: if the category of quasicoherent sheaves on a variety has enough projectives, is that variety affine?