---
tag: 4Q3WY1RX91
kind: concept
name: Weak homotopy equivalence
msc: [55P10, 55Q05]
sources:
  - url: https://en.wikipedia.org/wiki/Weak_equivalence_(homotopy_theory)
---

## Summary

A weak homotopy equivalence is a continuous map that induces a bijection on path components and isomorphisms on all [[homotopy groups|P976WYEFMB]], at every base point.
Every [[homotopy equivalence|XZG6MTSWCN]] is a weak homotopy equivalence, and between CW complexes the converse holds by the [[Whitehead theorem|6P1E2FJPSD]].
Every space is weakly equivalent to a CW complex, so weak equivalences are the natural notion of sameness in homotopy theory.
They are the model for weak equivalences in abstract homotopy theory, such as in model categories.

## Explanations

### L1 human

Homotopy groups only see a space through the spheres that map into it.
A weak homotopy equivalence is a map that looks like an equivalence to every sphere.
For nice spaces this is the same as a true equivalence up to deformation.
For strange spaces, such as the Warsaw circle, a map to a point can fool every sphere and still not be an equivalence.