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In this paper we elaborate on a 2-categorical construction of the homotopy category of a Quillen model category. Given any category A and a class of morphisms Σ ⊂ A containing the identities, we construct a 2-category Ho(A) obtained by the addition of 2-cells determined by homotopies. A salient feature here is the use of a novel notion of cylinder introduced in [1]. The inclusion 2-functor A → Ho(A) has a universal property which yields the 2-localization of A at Σ provided that the arrows of Σ become equivalences in Ho(A). This result together with a fibrant-cofibrant replacement is then used to obtain the 2-localization of a model category C at the weak equivalences W. The set of connected components of the hom categories yields a novel proof of Quillen's results. We follow the general lines established in [1], [2] for model bicategories.
@article{TAC_2024_40_a17,
author = {Dubuc E. J. and Girabel J.},
title = {The 2-localization of a model category},
journal = {Theory and applications of categories},
pages = {537--574},
publisher = {mathdoc},
volume = {40},
year = {2024},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2024_40_a17/}
}
Dubuc E. J.; Girabel J. The 2-localization of a model category. Theory and applications of categories, Bunge Festschrift, Tome 40 (2024), pp. 537-574. http://geodesic.mathdoc.fr/item/TAC_2024_40_a17/