---
tag: THRHFW3SMP
kind: concept
name: Free category
msc: [18A10, 18A40]
specializes: [A2DE66937D]
sources:
  - url: https://en.wikipedia.org/wiki/Free_category
same_as:
  wikidata: Q5500202
  wikipedia: Free category
---

## Summary

The free category on a [[directed graph|0KFYGSHHF5]] has the vertices of the graph as objects and the finite paths of edges as morphisms.
Paths are composed by joining them end to end.
It is the most general category generated by the graph: no equations between paths hold except those forced by the axioms of a category.
It plays for categories the role that the [[free monoid|CV013KHVJG]] plays for monoids.

## Explanations

### L1 human

Draw some dots and arrows between them.
A morphism of the free category is a route you can travel by following arrows one after another, and composing two routes means doing one and then the other.
Doing nothing at a dot is the identity route.
If the graph has one dot and one loop $a$, the routes are $a^0, a^1, a^2, \dots$, and composing them adds the exponents, so the free category is the monoid $(\N, +)$.
If the graph is $x \to y \to z$, there are exactly six routes: three identities, the two edges, and the route from $x$ to $z$ through $y$.