---
tag: W5WS5EY8ZM
kind: concept
name: Cofibration
msc: [55P05]
specializes: [0VGFHTM75Y]
sources:
  - url: https://en.wikipedia.org/wiki/Cofibration
same_as:
  wikidata: Q1048949
  wikipedia: Cofibration
---

## Summary

A cofibration is a continuous map $i \colon A \to X$ along which homotopies extend: it has the [[homotopy extension property|D42X0F1EG1]] for every target space.
Inclusions of subcomplexes of CW complexes are the main examples.
Cofibrations are well behaved for gluing, and for a cofibration the quotient $X / A$ has the homotopy type of the mapping cone.
They are dual, in the sense of [[Eckmann–Hilton duality|1EK3355WMC]], to fibrations.

## Explanations

### L1 human

A cofibration is a nicely attached subspace.
Whenever a map on the whole space is given and you deform it on the subspace, the deformation can be carried along on the whole space.
The boundary circle of a disk is attached in this nice way.
An open interval inside a closed interval is not, because its closure adds new points.