---
tag: 48QQ7T98Z9
kind: concept
name: Homotopy category
msc: [18N40, 55U35, 55P10]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopy_category
same_as:
  wikidata: Q14775269
  wikipedia: Homotopy category
---

## Summary

The homotopy category of a [[model category|SDTZVZ6B4Q]] is obtained by making all weak equivalences invertible.
It is the setting in which homotopy theory compares objects.
Two objects are isomorphic in it when they are connected by a chain of weak equivalences.
For topological spaces it can be described by CW complexes and homotopy classes of maps.
Quillen's theorem identifies its morphisms with homotopy classes of maps between suitably replaced objects.

## Explanations

### L1 human

In homotopy theory, two maps are treated as equal when one can be deformed into the other.
The homotopy category records only this information: objects are spaces, and an arrow is a deformation class of maps.
In it, a disc and a point become isomorphic, since a disc can be shrunk to a point.
The circle and a point stay different, because the circle has a loop that cannot be shrunk.