additive category


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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 carries the structure of an abelian group and composition is bilinear)

  2. which admits finite coproducts

    (and hence, by prop. 1 below, finite products which coincide with the coproducts, hence finite biproducts).

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 an Ab-enriched category, a finite product is also a coproduct, and dually.

This statement 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.

More precisely, for {X i} iI\{X_i\}_{i \in I} a finite set of objects in an Ab-enriched category, then the unique morphism

iIX ijIX j \underset{i \in I}{\coprod} X_i \longrightarrow \underset{j \in I}{\prod} X_j

whose components are identities for i=ji = j and are zero otherwise is an isomorphism.


Consider first the zero-ary case. Given an initial object \emptyset and a terminal object *\ast, observe that since the hom-sets Hom(,)Hom(\emptyset,\emptyset) and Hom(*,*)Hom(\ast,\ast) by definition contain a single element, this element has to be the zero element in the abelian group structure. But it also has to be the identity morphism, and hence id =0id_\emptyset = 0 and id *=0id_{\ast} = 0. It follows that the 0-element in Hom(*,)Hom(\ast, \emptyset) is a left and right inverse to the unique element in Hom(,*)Hom(\emptyset,\ast), and so this is an isomorphism

0:*. 0 \;\colon\; \emptyset \overset{\simeq}{\longrightarrow} \ast \,.

Consider now the case of binary (co-)products. Using the existence of the zero object, hence of zero morphisms, then in addition to its canonical projection maps p i:X 1×X 2X ip_i \colon X_1 \times X_2 \to X_i, any binary product also receives “injection” maps X iX 1×X 2X_i \to X_1 \times X_2, and dually for the coproduct:

X 1 X 2 (id,0) (0,id) id X 1 X 1×X 2 id X 2 p X 1 p X 2 X 1 X 2,X 1 X 2 i X 1 i X 2 id X 1 X 1X 2 id X 2 (id,0) (0,id) X 1 X 2. \array{ X_1 && && X_2 \\ & \searrow^{\mathrlap{(id,0)}} && {}^{\mathllap{(0,id)}}\swarrow \\ {}^{\mathllap{id_{X_1}}}\downarrow && X_1 \times X_2 && \downarrow^{\mathrlap{id_{X_2}}} \\ & \swarrow_{\mathrlap{p_{X_1}}} && {}_{\mathllap{p_{X_2}}}\searrow \\ X_1 && && X_2 } \;\;\;\;\;\;\;\;\;\;\;\;\,,\;\;\;\;\;\;\;\;\;\;\;\; \array{ X_1 && && X_2 \\ & \searrow^{\mathrlap{i_{X_1}}} && {}^{\mathllap{i_{X_2}}}\swarrow \\ {}^{\mathllap{id_{X_1}}}\downarrow && X_1 \sqcup X_2 && \downarrow^{\mathrlap{id_{X_2}}} \\ & \swarrow_{\mathrlap{(id,0)}} && {}_{\mathllap{(0,id)}}\searrow \\ X_1 && && X_2 } \,.

Then using the additivity of morphisms, it follows that for f i:X iQf_i \;\colon\; X_i \to Q any two morphisms, the sum

ϕ:f 1p 1+f 2p 2 \phi \;\colon\; f_1 \circ p_1 + f_2 \circ p_2

gives a morphism of cocones

X 1 X 2 (id,0) (0,id) id X 1 X 1×X 2 id X 2 X 1 ϕ X 2 f 1 f 2 Q. \array{ X_1 && && X_2 \\ & \searrow^{\mathrlap{(id,0)}} && {}^{\mathllap{(0,id)}}\swarrow \\ {}^{\mathllap{id_{X_1}}}\downarrow && X_1 \times X_2 && \downarrow^{\mathrlap{id_{X_2}}} \\ & && \\ X_1 && \downarrow^{\mathrlap{\phi}} && X_2 \\ & {}_{\mathllap{f_1}}\searrow && \swarrow_{\mathrlap{f_2}} \\ && Q } \,.

Moreover, this is in fact unique: suppose ϕ\phi' is another morphism filling this diagram, then

(ϕϕ) =(ϕϕ)id X 1×X 2 =(ϕϕ)((id X 1,0)p 1+(0,id X 2)p 2) =(ϕϕ)(id X 1,0)=0p 1+(ϕϕ)(0,id X 2)=0p 2 =0 \begin{aligned} (\phi-\phi') & = (\phi - \phi') \circ id_{X_1 \times X_2} \\ &= (\phi - \phi') \circ ( (id_{X_1},0) \circ p_1 + (0,id_{X_2})\circ p_2 ) \\ & = \underset{ = 0}{\underbrace{(\phi - \phi') \circ (id_{X_1}, 0)}} \circ p_1 + \underset{ = 0}{\underbrace{(\phi - \phi') \circ (0, id_{X_2})}} \circ p_2 \\ & = 0 \end{aligned}

and hence ϕ=ϕ\phi = \phi'. This means that X 1×X 2X_1\times X_2 satisfies the universal property of a coproduct.

By a dual argument, the binary coproduct X 1X 2X_1 \sqcup X_2 is seen to also satisfy the universal property of the binary product. By induction, this implies the statement for all finite (co-)products.


Such products which are also coproducts as in prop. 1 are sometimes called biproducts or direct sums; they are absolute limits for Ab-enrichment.


The coincidence of products with biproducts in prop. 1 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.



A semiadditive category is a category that has all finite products which, moreover, are biproducts in that they coincide with finite coproducts as in def. 1.


In a semiadditive category, def. 2, the hom-sets acquire the structure of commutative monoids by defining the sum of two morphisms f,g:XYf,g \;\colon\; X \longrightarrow Y to be

f+gXΔ XX×XXXfgYYYY XY. f + g \;\coloneqq\; X \overset{\Delta_X}{\to} X \times X \simeq X \oplus X \overset{f \oplus g}{\longrightarrow} Y \oplus Y \simeq Y \sqcup Y \overset{\nabla_X}{\to} Y \,.

With respect to this operation, composition is bilinear.


The associativity and commutativity of ++ follows directly from the corresponding properties of \oplus. Bilinearity of composition follows from naturality of the diagonal Δ X\Delta_X and codiagonal X\nabla_X.


Given an additive category according to def. 1, then the enrichement in commutative monoids which is induced on it via prop. 1 and prop. 2 from its underlying semiadditive category structure coincides with the original enrichment.


We may write the formula for the addition of two morphisms induced by semiadditve structure equivalently as

f+g:XΔ XX×X(f,g)Y×Yp 1+p 2Y f+g \;\colon\; X \overset{\Delta_X}{\to} X \times X \overset{(f,g)}{\longrightarrow} Y \times Y \overset{p_1 + p_2}{\longrightarrow} Y

where the last morphism is identified as the sum of the two projections as in the proof of prop. 1. This implies the claim.


Prop. 3 says that being an additive category is an extra property on a category, not extra structure. We may ask whether a given category is additive or not, without specifying with respect to which abelian group structure on the hom-sets.


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 May 18, 2016 06:42:59 by Urs Schreiber (