---
tag: BZEJKTQ76T
kind: concept
name: Biproduct
msc: ["18E05", 18A30]
sources:
  - url: https://en.wikipedia.org/wiki/Biproduct
same_as:
  wikidata: Q4915506
  wikipedia: Biproduct
---

## Summary

A biproduct of two objects is a single object that is at the same time their [[product|S5744QZV4N]] and their [[coproduct|4FJ0S71P58]], with compatible projections and inclusions.
It generalizes the direct sum $A \oplus B$ of abelian groups or modules, which has both universal properties.
Additive categories are the preadditive categories with a zero object in which all binary biproducts exist.
In a preadditive category, a finite product is automatically a biproduct.

## Explanations

### L1 human

The direct sum $\R^2 = \R \oplus \R$ can be seen in two ways.
It is built from two copies of $\R$ by taking pairs, with the two coordinate projections.
It also receives the two copies of $\R$ as the axes, so that a pair of maps out of the two copies combines into one map out of $\R^2$.
A biproduct is an object with both these features, where projecting after including the same copy gives the identity and after including the other copy gives zero.