Model category

Summary

A model category is a category with three chosen classes of morphisms, called weak equivalences, fibrations and cofibrations, that satisfy axioms modelled on topological spaces. Quillen introduced model categories in 1967 to do homotopy theory in settings such as simplicial sets and chain complexes. The weak equivalences are the maps to be treated as isomorphisms, and the fibrations and cofibrations give the tools to compute with them. The homotopy category of a model category, obtained by inverting the weak equivalences, has a concrete description in terms of homotopy classes of maps.