Additive Category

An additive category is a category enriched over abelian groups: each hom-set carries an abelian group structure compatible with composition. This is the first step toward the axiomatics of homological algebra, sitting between general categories and abelian categories.

Definition

Definition3.1Preadditive Category

A category $\mathcal{C}$ is preadditive (or Ab-enriched) if:

For every pair of objects $A, B$ , the set $\mathrm{Hom}(A, B)$ is an abelian group.
Composition is bilinear: for $f, f' : A \to B$ and $g, g' : B \to C$ ,

$g \circ (f + f') = g \circ f + g \circ f', \qquad (g + g') \circ f = g \circ f + g' \circ f$

Definition3.2Additive Category

A preadditive category $\mathcal{C}$ is additive if:

It has a zero object $0$ (both initial and terminal).
It has all finite biproducts: for any $A, B \in \mathcal{C}$ , the biproduct $A \oplus B$ exists.

Definition3.3Biproduct

A biproduct of $A$ and $B$ in a preadditive category is an object $A \oplus B$ equipped with morphisms

$\iota_A : A \to A \oplus B, \quad \iota_B : B \to A \oplus B, \quad \pi_A : A \oplus B \to A, \quad \pi_B : A \oplus B \to B$

satisfying:

$\pi_A \circ \iota_A = \mathrm{id}_A, \quad \pi_B \circ \iota_B = \mathrm{id}_B, \quad \pi_A \circ \iota_B = 0, \quad \pi_B \circ \iota_A = 0$

$\iota_A \circ \pi_A + \iota_B \circ \pi_B = \mathrm{id}_{A \oplus B}$

The biproduct is simultaneously a product (via $\pi_A, \pi_B$ ) and a coproduct (via $\iota_A, \iota_B$ ).

Examples

ExampleAb is additive

$\mathbf{Ab}$ is additive: $\mathrm{Hom}(A, B)$ is an abelian group under pointwise addition $(f + g)(a) = f(a) + g(a)$ . The zero object is $\{0\}$ , and the biproduct is the direct sum $A \oplus B$ .

ExampleR-Mod is additive

$R\text{-}\mathbf{Mod}$ is additive with the same structure: pointwise addition of homomorphisms, zero module as zero object, and direct sum as biproduct.

ExampleVect_k is additive

$\mathbf{Vect}_k$ is additive. The direct sum $V \oplus W$ is the biproduct. In matrix terms, a morphism $V \oplus W \to V' \oplus W'$ corresponds to a $2 \times 2$ block matrix.

ExampleCh(A) is additive

If $\mathcal{A}$ is additive, then the category of chain complexes $\mathbf{Ch}(\mathcal{A})$ is additive. Addition of chain maps is defined degreewise, the zero complex is the zero object, and the biproduct is the direct sum of complexes.

ExampleMat_k is additive

The matrix category $\mathbf{Mat}_k$ (objects: natural numbers, morphisms: matrices) is additive: matrix addition gives the abelian group structure on $\mathrm{Hom}(m, n) = M_{n \times m}(k)$ , the zero object is $0$ , and the biproduct is $m \oplus n = m + n$ .

ExampleSet is not additive

$\mathbf{Set}$ is not additive: there is no natural abelian group structure on $\mathrm{Hom}(A, B)$ compatible with composition. The product and coproduct of two nonempty sets differ ( $|A \times B| \neq |A \sqcup B|$ in general), so there is no biproduct.

ExampleGrp is not additive

$\mathbf{Grp}$ has a zero object (the trivial group), but finite products and coproducts differ: $G \times H \neq G * H$ in general. There is no biproduct structure, so $\mathbf{Grp}$ is not additive.

ExampleTop is not preadditive

$\mathbf{Top}$ is not preadditive: there is no natural way to add two continuous maps $f, g : X \to Y$ to get another continuous map. Without a group structure on $Y$ , pointwise addition is not defined.

ExampleCategory of finite abelian groups

The category $\mathbf{FinAb}$ of finite abelian groups is additive. The biproduct is $A \oplus B$ , the zero object is $\{0\}$ , and all hom-sets are finite abelian groups.

ExampleAdditive functors

A functor $F : \mathcal{A} \to \mathcal{B}$ between additive categories is additive if $F(f + g) = F(f) + F(g)$ for all $f, g$ . Equivalently, $F$ preserves finite biproducts. The forgetful functor $R\text{-}\mathbf{Mod} \to \mathbf{Ab}$ is additive.

ExamplePreadditive category from a ring

A ring $R$ defines a preadditive category with one object: $\mathrm{Hom}(*,*) = R$ with addition from the ring structure and composition from multiplication. This is additive iff $R$ has the zero element (which it always does as a ring).

ExampleEndomorphism ring

In an additive category, $\mathrm{End}(A) = \mathrm{Hom}(A, A)$ is a ring: addition comes from the abelian group structure, and multiplication is composition. For $V \in \mathbf{Vect}_k$ , $\mathrm{End}(V) = M_n(k)$ .

Properties

Theorem3.1Biproduct = product = coproduct

In a preadditive category, the following are equivalent for an object $P$ with maps to/from $A$ and $B$ :

$P$ is a biproduct of $A$ and $B$ .
$P$ is a product of $A$ and $B$ (and the canonical map $A \sqcup B \to A \times B$ is an isomorphism).
$P$ is a coproduct of $A$ and $B$ (and the canonical map $A \sqcup B \to A \times B$ is an isomorphism).

RemarkLooking ahead

An additive category becomes abelian when every morphism has a kernel and cokernel, and every monomorphism is a kernel and every epimorphism is a cokernel. The additive structure ensures that exact sequences can be defined and manipulated, laying the groundwork for homological algebra.