Uniqueness of pre-abelian structures on categories For a pre-additive cateory we need the structure of an abelian group on the Hom sets of a category, with respect to which composition is bilinear. Is this structure, when it exists, unique? More generally, does every category come endowed with at most one abelian category structure?
 A: As Zhen Lin says, for a pre-additive category with one object you're asking whether the addition on a ring is uniquely determined (not up to isomorphism but as a function) by its multiplication. This is false; to prove this it suffices to exhibit a bijection from a ring to itself which preserves multiplication but not addition, and for example we can consider multiplicative bijections $\mathbb{Z} \to \mathbb{Z}$ which permute the primes.
(A harder and also interesting question is to ask if you can find two rings with isomorphic multiplicative monoids but nonisomorphic additive groups. We can do this with $\mathbb{Q}$ and $\mathbb{Q}(x)$; their additive groups can be distinguished by dimension over $\mathbb{Q}$, but their multiplicative monoids consist of their multiplicative groups together with a zero element, and their multiplicative groups are both $C_2 = \{ \pm 1 \}$ times the free abelian group on countably many generators.)
However, the following structures on a category are all unique:

*

*semiadditive

*additive

*abelian

These all follow more or less from the first one; the key point is that biproducts are on the one hand a limit and a colimit, and on the other hand uniquely determine the addition of morphisms. There is a slight subtlety here involving zero objects and morphisms which is needed to make this sketch really work out; we need to consider the nullary case first and separately and as far as I know this is unavoidable. This is all done in detail in my blog post A meditation on semiadditive categories.
