additive category


Enriched category theory

Additive and abelian categories

Homological algebra

homological algebra


nonabelian homological algebra


Basic definitions

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diagram chasing

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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.


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.


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.


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.



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 every additive category has a zero object.


Use that

X Y (id,0) (0,id) id X X×Y id Y p X p Y X Y,X Y i X i X id X XY id Y (id,0) (0,id) X Y. \array{ X && && Y \\ & \searrow^{\mathrlap{(id,0)}} && {}^{\mathllap{(0,id)}}\swarrow \\ {}^{\mathllap{id_X}}\downarrow && X \times Y && \downarrow^{\mathrlap{id_Y}} \\ & \swarrow_{\mathrlap{p_X}} && {}_{\mathllap{p_Y}}\searrow \\ X && && Y } \;\;\;\;\;\;\;\;\;\;\;\;\,,\;\;\;\;\;\;\;\;\;\;\;\; \array{ X && && Y \\ & \searrow^{\mathrlap{i_X}} && {}^{\mathllap{i_X}}\swarrow \\ {}^{\mathllap{id_X}}\downarrow && X \sqcup Y && \downarrow^{\mathrlap{id_Y}} \\ & \swarrow_{\mathrlap{(id,0)}} && {}_{\mathllap{(0,id)}}\searrow \\ X && && Y } \,.

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


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.


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 April 27, 2016 14:44:11 by Urs Schreiber (