---
tag: QBDT694BAP
kind: concept
name: Free object
msc: [18A40, 08B20]
sources:
  - url: https://en.wikipedia.org/wiki/Free_object
same_as:
  wikidata: Q5500272
  wikipedia: Free object
---

## Summary

A free object on a set $X$ is an object generated by $X$ with no relations other than those forced by the axioms.
It is defined by a [[universal property|EN4DD2FAJH]]: every map from $X$ into the underlying set of an object extends uniquely to a morphism.
[[Free groups|0KTW26C80J]], [[free modules|0DZ9DKAKZR]] and [[free monoids|CV013KHVJG]] are the main examples.
When free objects exist on every set, they form a left [[adjoint|TZHF6ZM92S]] of the forgetful functor.

## Explanations

### L1 human

A free object is the most general object you can build from a set of letters.
In the free group on $\set{a, b}$, the elements are words such as $a b a^{-1} b^{-1}$, and no two different reduced words are equal.
Because nothing extra holds, you can send $a$ and $b$ to any two elements of any group, and this gives exactly one homomorphism.
A vector space with basis $X$ is the free vector space on $X$ in the same sense.