Definition
A Quillen model structure on a category consists of three classes of morphisms:
These classes are required to satisfy axioms that make it possible to do homotopy theory inside . A category equipped with such data is a model category.
Idea
The weak equivalences specify which morphisms should count as equivalences up to homotopy, while cofibrations and fibrations control how maps factor and how lifting arguments work.
Examples
- The standard Quillen model structure on Simplicial Set
- The classical Quillen model structure on topological spaces