Additive category

An additive category is a category in which morphisms can be added and finite direct sums exist.

Definition

A category $\mathcal A$ is preadditive if:

1. For all objects $A,B$, the set $\operatorname{Hom}_{\mathcal A}(A,B)$ is an abelian group (written additively).
2. Composition is bilinear: for morphisms $f,f':A\to B$ and $g,g':B\to C$, $g\circ (f+f') = g\circ f + g\circ f', \qquad (g+g')\circ f = g\circ f + g'\circ f.$

A preadditive category $\mathcal A$ is additive if, in addition: 3. $\mathcal A$ has a zero object $0$ (both initial and terminal), hence a distinguished zero morphism $0_{A,B}:A\to B$ for all $A,B$. 4. $\mathcal A$ has binary biproducts: for all objects $A,B$ there exists an object $A\oplus B$ with morphisms

such that $(A\oplus B, p_A,p_B)$ is a product of $A,B$, and $(A\oplus B, i_A,i_B)$ is a coproduct of $A,B$, and the following identities hold:

Equivalently: an additive category is a preadditive category with all finite biproducts (including the empty biproduct, i.e. a zero object). In an additive category, finite products and coproducts agree (up to canonical isomorphism).

Examples

1. $\mathbf{Ab}$. The category of abelian groups is additive: hom-sets are abelian groups under pointwise addition of homomorphisms, and $A\oplus B$ is the usual direct sum/product.

2. $R$$-$$\mathbf{Mod}$. For a ring $R$, the category of (left) $R$-modules is additive, with biproduct given by the direct sum $M\oplus N$.

3. Chain complexes. For any additive category $\mathcal A$ (e.g. $\mathbf{Ab}$ or $R$$-$$\mathbf{Mod}$), the category $\mathbf{Ch}(\mathcal A)$ of chain complexes in $\mathcal A$ is additive, with biproduct defined degreewise.