---
tag: ZBHGXJF24E
kind: concept
name: Fundamental groupoid
msc: [55Q05, 18B40]
sources:
  - url: https://en.wikipedia.org/wiki/Fundamental_groupoid
same_as:
  wikidata: Q85174068
  wikipedia: Fundamental groupoid
---

## Summary

The fundamental groupoid $\Pi_1(X)$ of a topological space $X$ is a groupoid.
Its objects are the points of $X$, and its morphisms are paths up to [[path homotopy|CCNKA8C711]].
It records the [[fundamental group|S1XK3HXHFX]] at every base point at once, together with the paths that compare them.
It needs no choice of base point, which makes some results, such as versions of the Seifert–van Kampen theorem, easier to state and prove.

## Explanations

### L1 human

The fundamental group looks only at loops from one chosen point.
The fundamental groupoid looks at all paths between all pairs of points, up to deformation with fixed ends.
Paths can be composed when the end of one is the start of the next, and every path can be run backwards.
For a space made of two separate pieces, the groupoid sees both pieces, while a fundamental group sees only the piece of its base point.