# Ab-categories

-categories

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

An **-category** (also called a

**preadditive**category) is a category

Note that because the composition of morphisms is bilinear, it can also be written using the tensor product (over

It is possible to define an

- 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 groups This 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

If **additive** when every function

## Biproducts

Our main example categories, namely

This situation can happen more generally in any

In a **biproduct diagram** for a pair of objects

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

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

Two objects

In particular, two objects

In the categories **direct sum** of the given objects. Because of this, if the biproduct diagram exists for all

(i.e., as defined for the product functor

(i.e., as defined for the coproduct functor

In other words, the identification of the product functor

We can iterate this process. Given

and the equations

where

Moreover, for given objects

This implies that each morphism **components**