|WikiProject Mathematics||(Rated Start-class, Low-importance)|
I have removed the Elementary characterisation. There's an elementary characterisation of Abelian categories, which is pretty slick, and which I intend to describe on that page, and there's an elementary characterisation of pre-Abelian categories too. So I wanted to describe the elementary characterisation of additive categories, the only problem being that it was pretty messy. But I thought that I'd at least indicate its basic idea.
Well, upon further review, it turns out that the elementary characterisation of pre-Abelian categories starts out "Suppose that you have an additive category. ...", which I didn't notice at first. So that wasn't going to be anything interesting. In the light of that, I'm just going to forget the whole thing. (I wrote it, after all.) The text is below if you want to see it; some of it will probably be cannibalised on Abelian_category later too. — Toby 21:45 Jul 20, 2002 (PDT)
Elementary characterisation of additive categories
Additive categories can be characterised entirely in terms of elementary category theory, without any reference to the category Ab. One may consider a category with a zero object and all finite products and coproducts. The existence of the zero object will define a notion of zero morphism, as the unique morphism between a given pair of objects that factors through the zero object. Then the zero morphisms and identity morphisms can be used to construct a morphism from the coproduct A + B to the product A × B, modelled after the 2-by-2 identity matrix, which one requires to be an isomorphism. Then using this isomorphism, one can construct a method of adding morphisms, which turns out to be associative and commutative, so that the hom-sets form Abelian monoids. Finally, one requires the existence of a morphsim −: A → A for each object A such that − satisfies a commutative diagram that establishes it as an analogue of multiplication by the integer −1. Then this morphism can be used to prove that the hom-sets are in fact Abelian groups.
We will not spell out this definition in more detail in this article, because it's messy, but it's useful to know that there exists such an elementary characterisation, similar to the (simpler) elementary characterisations of pre-Abelian categories and Abelian categories.
First Condition Necessary?
The last line of the definition states
"Also, since the empty biproduct is a zero object in the category, we may omit the first condition. If we do this, however, we need to presuppose that the category C has zero morphisms, or equivalently that C is enriched over the category of pointed sets."