---
tag: CCNKA8C711
kind: concept
name: Path homotopy
msc: [55Q05, 57M05]
specializes: [ADDS6HE5SV]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopy
---

## Summary

A path homotopy is a deformation of one path into another with both end points held fixed.
It is a [[homotopy relative|ADDS6HE5SV]] to the end points $\set{0, 1}$ of the interval $[0, 1]$.
Classes of paths up to path homotopy form the [[fundamental group|S1XK3HXHFX]] and the [[fundamental groupoid|ZBHGXJF24E]].

## Explanations

### L1 human

Think of a path as a string from a point $p$ to a point $q$, with both ends nailed down.
A path homotopy moves the string around without moving the nails and without cutting it.
In the plane, any two paths from $p$ to $q$ are path homotopic.
In the plane without the origin, a path that passes above the origin and one that passes below it are not.