---
tag: B2VP96NJSK
kind: concept
name: Additive category
msc: ["18E05"]
specializes: [SC6FD5F4J0]
sources:
  - url: https://en.wikipedia.org/wiki/Additive_category
same_as:
  wikidata: Q4681343
  wikipedia: Additive category
---

## Summary

An additive category is a [[preadditive category|SC6FD5F4J0]] with a zero object in which any two objects have a [[biproduct|BZEJKTQ76T]].
Its morphisms between finite direct sums can be written as matrices of morphisms, composed by matrix multiplication.
Categories of modules, of free modules and of vector bundles are additive.
Additive categories are the natural home of chain complexes and of the homotopy category of chain complexes.

## Explanations

### L1 human

In an additive category one can add maps, and one can form direct sums of objects.
A map from $A \oplus B$ to $C \oplus D$ is then a $2 \times 2$ matrix whose entries are maps from $A$ or $B$ to $C$ or $D$.
This is exactly how one computes with linear maps between spaces split into pieces.
For example, finite-dimensional vector spaces over a field form an additive category, and a linear map $K^2 \to K^3$ is a $3 \times 2$ matrix.