There are many familiar categories in which the hom-sets have additional structure:
In the category of vector spaces over a field , each hom-set has the structure of an -vector space. Addition is defined by addition of outputs; i.e., for every . The additive identity is the zero map.
In the category of left -modules over a fixed ring , each hom-set has the structure of an abelian group. Once again, addition of morphisms is defined via addition of outputs.
When is commutative, the hom-sets in have the structure of -modules.
In the category of abelian groups, each hom-set has the structure of an abelian group, via addition of outputs.
We are going to focus on the examples of the categories and , in which the hom-sets have the structure of abelian groups.
Definition of -category
An -category (also called a preadditive category) is a category in which each hom-set is an abelian group and for which the composition is bilinear. In other words, for morphisms and we have
Note that because the composition of morphisms is bilinear, it can also be written using the tensor product (over ) as a linear map:
It is possible to define an -category directly (without first defining a category), as given by the following data:
A set of objects.
A function that assigns to each ordered pair of objects an abelian group .
For each ordered triple of objects a morphism of abelian groupsThis morphism is called "composition" and written .
For each object a group morphism . (This is the analogue of each object in a category having a unique identity arrow, which corresponds to a set map .)
These data are required to satisfy the usual associative and unit laws for composition. This is a definition of -category completely analogous to the definition of a conventional category, with:
replaced by
Cartesian product of sets replaced by tensor product in
the one-point set replaced by
This suggests a generalization to a concept called an enriched category, but we will not explore that for now.
If we are dealing with -categories, we will probably want to restrict our functors to those that respect that morphism addition:
Definition of additive functors
If and are -categories, a functor is said to be additive when every function is a group morphism; i.e., when for all parallel morphisms .
Biproducts
Our main example categories, namely and have additional properties beyond the hom-sets being abelian groups. One of those properties is that products and coproducts always exist for every pair of objects, and they're always isomorphic. In other words, the product functor and coproduct functor are naturally isomorphic. Because of this, there is usually some confusion as to which type of product to use, and you often see textbooks using coproduct (usually called the direct sum and denoted ) as if it's a product; e.g., by referring to projection maps and .
This situation can happen more generally in any -category. We first introduce a new type of product:
Biproducts
In a -category , a biproduct diagram for a pair of objects is a diagram
with arrows that satisfy the identities
Note that this definition is entirely "internal" in that it refers only to morphisms to and from the objects and , as opposed to the usual product or coproduct , which are defined as limits and colimits and hence involve morphisms to and from every object in the category.
In other words, to verify you have a biproduct diagram, you just need to directly verify the three identity relations above. On the other hand, to verify is a product you need to verify it satisfies the appropriate universal property, which involves comparing (with its projections) to every other object (that also has projections to and ).
Products, coproducts, and biproducts
Two objects and in an -category have a product in if and only if they have a biproduct in . In the biproduct diagram above, the object with the morphisms and is a product of and , while the object with the morphisms and is a coproduct of and .
In particular, two objects and have a product in exactly when they have a coproduct in .
In the categories and the biproduct is usually called the direct sum of the given objects. Because of this, if the biproduct diagram exists for all in a given -category , the object is usually written . This defines a functor , with defined for morphisms and either by the equations
(i.e., as defined for the product functor ), or by the equations
(i.e., as defined for the coproduct functor ). By the properties of the biproduct diagram, either choice implies the other.
In other words, the identification of the product functor with the coproduct functor is a natural isomorphism.
We can iterate this process. Given we can form a product characterized (up to isomorphism) by the diagram
and the equations
where if and .
Moreover, for given objects with and there is an isomorphism of abelian groups
This implies that each morphism is determined by the matrix of its components. Composition of morphisms is then given by the usual matrix product of the matrices of components.