(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
An additive category is a category which is
A pre-abelian category is an additive category which also has kernels and cokernels. Equivalently, it is an Ab-enriched category with all finite limits and finite colimits. An especially important sort of additive category is an abelian category, which is a pre-abelian one satisfying the extra exactness property that all monomorphisms are kernels and all epimorphisms are cokernels. See at additive and abelian categories for more.
The Ab-enrichment of an additive category does not have to be given a priori. Every semiadditive category (a category with finite biproducts) is automatically enriched over commutative monoids (as described at biproduct), so an additive category may be defined as a category with finite biproducts whose hom-monoids happen to be groups. (The requirement that the hom-monoids be groups can even be stated in elementary terms without discussing enrichment at all, but to do so is not very enlightening.) Note that the entire -enriched structure follows automatically for abelian categories.
Some authors use additive category to simply mean an Ab-enriched category, with no further assumptions. It can also be used to mean a -enriched (commutative monoid enriched) category, with or without assumptions of products.
More precisely, for a finite set of objects in an Ab-enriched category, the unique morphism
Consider first the zero-ary case. Given an initial object and a terminal object , observe that since the hom-sets and by definition contain a single element, this element has to be the zero element in the abelian group structure. But it also has to be the identity morphism, and hence and . It follows that the 0-element in is a left and right inverse to the unique element in , and so this is an isomorphism
Consider now the case of binary (co-)products. Using the existence of the zero object, hence of zero morphisms, then in addition to its canonical projection maps , any binary product also receives “injection” maps , and dually for the coproduct:
Observe some basic compatibility of the -enrichment with the product:
First, for then
(using that the projections and are linear and by the universal property of the porduct).
Second, and are two projections on whose sum is the identity:
Now observe that for any two morphisms, the sum
gives a morphism of cocones
Moreover, this is unique: suppose is another morphism filling this diagram, then, by using equation , we get
The coincidence of products with biproducts in prop. 1 does not extend to infinite products and coproducts.) In fact, an Ab-enriched category is Cauchy complete just when it is additive and moreover its idempotents split.
Given an additive category according to def. 1, then the enrichement in commutative monoids which is induced on it via prop. 1 and prop. 2 from its underlying semiadditive category structure coincides with the original enrichment.
Prop. 3 says that being an additive category is an extra property on a category, not extra structure. We may ask whether a given category is additive or not, without specifying with respect to which abelian group structure on the hom-sets.
Discussion of model category structures on additive categories is around def. 4.3 of