nLab
additive category

Context

Enriched category theory

Additive and abelian categories

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Definition

Definition

An additive category is a category which is

  1. an Ab-enriched category;

    (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 AbAb-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 CMonCMon-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 AbAb-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)