Is there anything better than an exact equivalence?

It's a basic principle of category theory that any "good" concept should be preserved by equivalence of categories. However, since Abelian categories are by definition $\mathbf{Ab}$-enriched, we require a bit more structure for our functors – we almost always like them to be additive, and it should preserve at least some form of exactness.

But are there any concepts in Abelian categories that are not preserved by exact equivalences? Or is there some way of proving that any good concept will be preserved by exact equivalences?

On another hand, is it enough to require the functor to be simply an equivalence, in order to guarantee that it is additive and exact?

• For the half of the question that hasn't been addressed: it's a bit hard to formalize the idea that all good properties are preserved under (exact) equivalences. For instance, do you have such a statement in mind for ordinary categories? Jan 9, 2017 at 22:15

• If $F\colon \mathcal{A} \to \mathcal{B}$ and $G\colon \mathcal{B}\to \mathcal{A}$ give an equivalence, then $F$ is both left- and right-adjoint to $G$.