Detecting model categories among Quillen categories using homotopies

Seunghun Lee

Theory and applications of categories, Tome 38 (2022), pp. 1-26

Résumé

A model category has two weak factorizations, a pair of cofibrations and trivial fibrations and a pair of trivial cofibrations and fibrations. Then the class of weak equivalences is the set W consisting of the morphisms that can be decomposed into trivial cofibrations followed by trivial fibrations. One can build a model category out of such two weak factorizations by defining the class of weak equivalences by W as long as it satisfies the two out of three property. In this note we show that given a category with two weak factorizations, if every object is fibrant and cofibrant, W satisfies the two out of three property if and only if W is closed under the homotopies.

Détail

Publié le : 2022-01-06

Classification : Primary 18N40, Secondary 55U35
Keywords: Model category, Quillen category, weak equivalence, two out of three property, homotopy