Geodesic
Detecting model categories among Quillen categories using homotopies
Theory and applications of categories, Tome 38 (2022), pp. 1-26.

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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.
Publié le :
Classification : Primary 18N40, Secondary 55U35
Keywords: Model category, Quillen category, weak equivalence, two out of three property, homotopy 
@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},
     publisher = {mathdoc},
     volume = {38},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2022_38_a0/}
}
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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/