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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.
Keywords: Model category, Quillen category, weak equivalence, two out of three property, homotopy
Seunghun Lee. Detecting model categories among Quillen categories using homotopies. Theory and applications of categories, Tome 38 (2022), pp. 1-26. http://geodesic.mathdoc.fr/item/TAC_2022_38_a0/
@article{TAC_2022_38_a0,
author = {Seunghun Lee},
title = {Detecting
model categories among {Quillen} categories using homotopies},
journal = {Theory and applications of categories},
pages = {1--26},
year = {2022},
volume = {38},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2022_38_a0/}
}