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The 2-localization of a model category

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Dubuc E. J., Girabel J.

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.

Keywords: localization, 2-category, homotopy

2020 MSC:
18N10, 18N40, 18N55

*Theory and Applications of Categories,*
Vol. 40, 2024,
No. 18, pp 537-574.

Published 2024-06-06.

http://www.tac.mta.ca/tac/volumes/40/18/40-18.pdf

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