Additive category

A category $\mathfrak C$ in which for any two objects $X$ and $Y$ an Abelian group structure is defined on the set of morphisms $\Hom_{\mathfrak C}(X,Y)$, such that the composition of morphisms

$$\Hom_{\mathfrak C}(X,Y)\times\Hom_{\mathfrak C}(Y,Z)\to\Hom_{\mathfrak C}(X,Z)$$

is a bilinear mapping. Another necessary condition is that $\mathfrak C$ includes a null object (zero object, cf. Null object of a category) as well as the product $X\times Y$ of any two objects $X$ and $Y$.

In an additive category the direct sum $X\oplus Y$ of any two objects exists. It is isomorphic to their product $X\times Y$. The dual category to an additive category is also an additive category.

A functor $F\colon\mathfrak C\to\mathfrak C_1$ from an additive category $\mathfrak C$ into an additive category $\mathfrak C_1$ is said to be additive if, for any objects $X$ and $Y$ in $\mathfrak C$, the mapping $F\colon\Hom_{\mathfrak C}(X,Y)\to\Hom_{\mathfrak C_1}(F(X),F(Y))$ is a homomorphism of Abelian groups. An additive category is said to be pre-Abelian if a kernel (cf. Kernel of a morphism in a category) and a cokernel exist for any morphism.

If for a morphism $u\colon X\to Y$ in an additive category there exists an image $\operatorname{Im}(u)$ and a co-image $\operatorname{Coim}(u)$, then there exists a unique morphism $u\colon\operatorname{Coim}(u)\to\operatorname{Im}(u)$ such that the morphism $u$ splits as the composition

$$X\to\operatorname{Coim}(u)\to\operatorname{Im}(u)\to Y.$$

An Abelian category is additive by definition. Examples of non-Abelian additive categories are the category of topological modules over a given topological ring with respect to the morphisms which are continuous linear mappings, and also the category of Abelian groups $\Gamma$ with a filtration $\Gamma=\Gamma_0\supset\Gamma_1\supset\dots\supset\Gamma_n=\{0\}$ with respect to the morphisms which are group homomorphisms preserving the filtration.

References

[1]	I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968)
[2]	A. Grothendieck, "Sur quelques points d'algèbrique homologique" Tohôku Math. J. , 9 (1957) pp. 119–221
[3]	L. Gruson, "Complétion abélienne" Bull. Sci. Math. (2) , 90 (1966) pp. 17–40

Comments

The requirement, in the definition of an additive category $\mathfrak C$ above, that $\mathfrak C$ possesses a null object as well as the product of any two objects in it, is not standard. Most authors drop this requirement, and take an additive category to mean a category satisfying only the requirement in the first sentence of the main text above.