---
tag: 4FJ0S71P58
kind: concept
name: Coproduct
msc: [18A30]
specializes: [BJS1CGTKQ5]
sources:
  - url: https://en.wikipedia.org/wiki/Coproduct
same_as:
  wikidata: Q692689
  wikipedia: Coproduct
---

## Summary

The coproduct of a family of objects in a [[category|A2DE66937D]] is an object with an injection from each member of the family.
Any family of morphisms out of the members factors uniquely through it.
It is the notion dual to the [[product|S5744QZV4N]], and it is the [[colimit|BJS1CGTKQ5]] of a diagram with no non-identity arrows.
It is the disjoint union for sets and topological spaces, the direct sum for abelian groups and modules, and the free product for groups.

## Explanations

### L1 human

A map out of a coproduct is the same thing as a list of maps, one out of each piece.
For sets, the coproduct of $X$ and $Y$ is their disjoint union.
A function on it is given by saying what it does on $X$ and what it does on $Y$, with no further constraint.
For abelian groups the answer is different: the coproduct of $A$ and $B$ is the [[direct sum|0S7W6EG32V]] $A \oplus B$.
This works because homomorphisms out of $A$ and out of $B$ can be added together.
In a partially ordered set seen as a category, the coproduct of two elements is their least upper bound.