# nLab additive category

### Context

#### Enriched category theory

enriched category theory

## Extra stuff, structure, property

### Homotopical enrichment

#### Additive and abelian categories

additive and abelian categories

## Derived categories

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Definition

###### Definition

An additive category is a category which is

1. (sometimes called a pre-additive category–this means that each hom-set is an abelian group and composition is bilinear)

2. which admits finite coproducts

(and hence, by prop. 1 below, finite products).

The natural morphisms between additive categories are additive functors.

###### Remark

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.

###### Remark

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 $Ab$-enriched structure follows automatically for abelian categories.

###### Remark

Some authors use additive category to simply mean an Ab-enriched category, with no further assumptions. It can also be used to mean a $CMon$-enriched (commutative monoid enriched) category, with or without assumptions of products.

## Properties

###### Proposition

In any Ab-enriched category, any finite product is also a coproduct, and dually. This includes the zero-ary case: any terminal object is also an initial object, hence a zero object (and dually), hence additive category have a zero object.

###### Remark

Such products which are also coproducts are sometimes called biproducts and sometimes direct sums; they are absolute limits for $Ab$-enrichment.

###### Remark

The coincidence of products with biproducts 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.

## References

Discussion of model category structures on additive categories is around def. 4.3 of

• Apostolos Beligiannis, Homotopy theory of modules and Gorenstein rings, Math. Scand. 89 (2001) (pdf)

Revised on January 31, 2014 02:23:41 by Urs Schreiber (89.204.139.167)