Преаддитивные категории и аддитивные категории: определения и свойства

Preadditive category that admits all finitary products
the weaker sense of the term "additive category" (without biproducts)
In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts.

There are two equivalent definitions of an additive category: One as a category equipped with additional structure, and another as a category equipped with no extra structure but whose objects and morphisms satisfy certain equations.

A category C is preadditive if all its hom sets are abelian groups and composition of morphisms is bilinear; in other words, C is enriched over the monoidal category of abelian groups. In a preadditive category, every finitary product (including the empty product, i. e., a final object) is necessarily a coproduct (or initial object in the case of an empty diagram), and hence a biproduct, and conversely every finitary coproduct is necessarily a product (this is a consequence of the definition, not a part of it). Thus an additive category is equivalently described as a preadditive category admitting all finitary products, or a preadditive category admitting all finitary coproducts.

We give an alternative definition. Define a semiadditive category to be a category (note: not a preadditive category) which admits a zero object and all binary biproducts. It is then a remarkable theorem that the Hom sets naturally admit an abelian monoid structure. A proof of this fact is given below. An additive category may then be defined as a semiadditive category in which every morphism has an additive inverse. This then gives the Hom sets an abelian group structure instead of merely an abelian monoid structure.

More generally, one also considers additive R linear categories for a commutative ring R. These are categories enriched over the monoidal category of R modules and admitting all finitary biproducts.

The original example of an additive category is the category of abelian groups Ab. The zero object is the trivial group, the addition of morphisms is given pointwise, and biproducts are given by direct sums. More generally, every module category over a ring R is additive, and so in particular, the category of vector spaces over a field K is additive. The algebra of matrices over a ring, thought of as a category as described below, is also additive.

A functor F: 'C' → 'D' between preadditive categories is additive if it is an abelian group homomorphism on each hom set in C. If the categories are additive, then a functor is additive if and only if it preserves all biproduct diagrams. That is, if B is a biproduct of A1,   , An in C with projection morphisms pk and injection morphisms ij, then F(B) should be a biproduct of F(A1),   , F(An) in D with projection morphisms F(pj) and injection morphisms F(ij). Almost all functors studied between additive categories are additive. In fact, it is a theorem that all adjoint functors between additive categories must be additive functors (see here). Most of the interesting functors studied in category theory are adjoints.

When considering functors between R linear additive categories, one usually restricts to R linear functors, so those functors giving an R module homomorphism on each hom set.

A pre abelian category is an additive category in which every morphism has a kernel and a cokernel. An abelian category is a pre abelian category such that every monomorphism and epimorphism is normal. Many commonly studied additive categories are in fact abelian categories; for example, Ab is an abelian category. The free abelian groups provide an example of a category that is additive but not abelian.