An $\mathbf{Ab}$-category is a category enriched over the category of abelian groups. What is an example of a category that can be enriched over abelian groups in more than one way?


An abelian category is a very special type of $\mathbf{Ab}$-category. So can a category be given the structure of an abelian category in more than one way, or is being abelian unique and hence a property of the category?



Can a category be enriched over abelian groups in more than one way?
You can easily find examples among categories with one element: a category with one element is a (multiplicative) monoid, and $Ab$-enrichment over it is a choice of an addition which turns it into a ring. And there can be multiple such additive structures: you can for instance consider pullback along a permutation which preserves multiplication but not addition.


More interestingly, if we instead work with additive categories rather than preadditive ones (meaning we require all finite products to exist), then it turns out $Ab$-enrichment is necessarily unique. This in particular applies to abelian categories. Indeed, given two morphisms $f,g:A\to B$, we can characterize $f+g$ as the composition
$$A\to A\oplus A\to B\oplus B\to B,$$
where the first map is the diagonal map, the last map is the codiagonal, and the middle map is the map which you can think of being $f,g$ on the components. See [Wikipedia](https://en.wikipedia.org/wiki/Additive_category#Internal_characterisation_of_the_addition_law) for some more details.


