---
tag: SY38F954FE
kind: concept
name: Quillen adjunction
msc: [18N40, 55U35]
specializes: [TZHF6ZM92S]
sources:
  - url: https://en.wikipedia.org/wiki/Quillen_adjunction
same_as:
  wikidata: Q7271995
  wikipedia: Quillen adjunction
---

## Summary

A Quillen adjunction is a pair of [[adjoint functors|TZHF6ZM92S]] between [[model categories|SDTZVZ6B4Q]] that respects the model structures.
The left adjoint preserves cofibrations and trivial cofibrations, or equivalently the right adjoint preserves fibrations and trivial fibrations.
Such a pair induces an adjunction between the homotopy categories, through the derived functors.
When this derived adjunction is an equivalence, the pair is a [[Quillen equivalence|KZCXHTEYK7]], and the two model categories present the same homotopy theory.

## Explanations

### L1 human

An adjunction is a precise way in which two constructions between categories mirror each other.
In homotopy theory one also wants the constructions to respect weak equivalences, and most do not do so on the nose.
A Quillen adjunction asks for less: the functors must respect the good maps of the model structures.
That is enough to give well-defined functors after one inverts the weak equivalences.
Geometric realization of simplicial sets and the singular simplicial set functor form the basic example.