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.
TAC Home