---
tag: N3457SJ63D
kind: definition
name: Preadditive category
msc: ["18E05"]
defines: SC6FD5F4J0
uses: [A2DE66937D, 082GHNPJ53, 0B1261B5ED, 0NZ8JA25WK, BPH9SJ30Y0]
sources:
  - url: https://en.wikipedia.org/wiki/Preadditive_category
---

## Definition

A *preadditive category* is a [[category|A2DE66937D]] $\mathcal{C}$ with an [[abelian group|082GHNPJ53]] structure, written additively, on each set $\Hom(X, Y)$.
Composition must be bilinear:
$$h \circ (f + g) = h \circ f + h \circ g, \qquad (f + g) \circ k = f \circ k + g \circ k$$
for all $f, g \colon X \to Y$, $h \colon Y \to Z$ and $k \colon W \to X$.

## Remarks

Equivalently, a preadditive category is a category [[enriched|BPH9SJ30Y0]] over the monoidal category of abelian groups with the tensor product.
The zero element of $\Hom(X, Y)$ is the zero morphism $0_{XY}$, and bilinearity gives $h \circ 0 = 0$ and $0 \circ k = 0$.
Some authors call these categories $\mathbf{Ab}$-categories.

Example: for a [[ring|0B1261B5ED]] $R$, the category of left [[$R$-modules|0NZ8JA25WK]] with pointwise addition $(f + g)(x) = f(x) + g(x)$ is preadditive.
The sum of homomorphisms is a homomorphism, and $h(f(x) + g(x)) = h(f(x)) + h(g(x))$ because $h$ is additive, which gives the first identity; the second holds by definition of the sum.

Example: a ring $R$ is a preadditive category with one object $\ast$, with $\Hom(\ast, \ast) = R$ and composition the multiplication of $R$.

Non-example: the category of sets is not preadditive, since $\Hom(\set{0}, \emptyset)$ is empty and an abelian group is never empty.