---
tag: EN4DD2FAJH
kind: concept
name: Universal property
msc: [18A05, 18A40]
sources:
  - url: https://en.wikipedia.org/wiki/Universal_property
same_as:
  wikidata: Q1417809
  wikipedia: Universal property
---

## Summary

A universal property describes an object with a morphism by a factorisation rule.
Every other morphism of the same type factors through it in exactly one way.
Products, quotients, free groups, tensor products, limits and colimits are all defined by universal properties.
An object with a universal property is unique up to a unique isomorphism, so the property pins it down without choosing a construction.

## Explanations

### L1 human

Instead of saying what an object is made of, a universal property says how it relates to everything else.
For example, the [[free group|0KTW26C80J]] on a set $S$ is a group that contains $S$.
Any function from $S$ to any group extends to exactly one group homomorphism on it.
Any two objects with the same universal property are matched by a unique isomorphism, so it does not matter how the object was built.
This is why mathematicians often say "the" product or "the" free group.