# 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? – Kevin Carlson Jan 9 '17 at 22:15

## 2 Answers

Equivalences between abelian categories are automatically both additive and exact. This is a good exercise; for a steadily larger series of hints read meditation on semiadditive categories.

To see that any equivalence of abelian categories is additive and exact, use the following observations:

• 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 $G$.

• Any left adjoint functor is right exact (as it preserves all small colimits), and any right adjoint functor is left exact (as it preserves all small limits).

• In particular, any left/right adjoint functor preserves binary coproducts/products (which coincide in an additive category), hence it is additive.