---
tag: FH3C0T643F
kind: concept
name: Groupoid
msc: [20L05, 18B40]
specializes: [A2DE66937D]
sources:
  - url: https://en.wikipedia.org/wiki/Groupoid
same_as:
  wikidata: Q1196038
  wikipedia: Groupoid
---

## Summary

A groupoid is a [[category|A2DE66937D]] in which every morphism is invertible.
A [[group|G5S1M9]] is the same as a groupoid with one object, so groupoids generalize groups to symmetries between several objects.
The [[fundamental groupoid|ZBHGXJF24E]] of a space, the action groupoid of a [[group action|0WCY42EV9V]] and the groupoid of an [[equivalence relation|0G8VEZ3WTK]] are standard examples.
Groupoids model homotopy 1-types, and they appear in topology, geometry and the theory of stacks.

## Explanations

### L1 human

A group describes the symmetries of one object.
A groupoid describes reversible transformations between several objects, where not every pair of objects needs to be connected.
In the fundamental groupoid, the objects are the points of a space and the arrows are paths up to homotopy, which can be run backwards.
Unlike the fundamental group, it does not need a chosen base point.