The 2-localization of a model category

GIRABEL, JAQUELINE; DUBUC, EDUARDO J.

THEORY AND APPLICATIONS OF CATEGORIES

Mount Allison University

Año: 2023

1201-561X

In this paper we elaborate on a 2-categorical construction of the homotopy category of a Quillen model category. Given any category $sr{A}$ and a class of morphisms $Sigma subset sr{A}$ containing the identities, we construct a 2-category $HAo$ 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 cite{e.d.2}. The inclusion 2-functor $sr{A} mr{} HAo$ has a universal property which yields the 2-localization of $sr{A}$ at $Sigma$ provided that the arrows of $Sigma$ become equivalences in $HAo$. This result together with a fibrant-cofibrant replacement is then used to obtain the 2-localizations of a model category $sr{C}$ at the weak equivalences $cc{W}$. The set of connected components of the hom categories yields a novel proof of Quillen´s results. We follow the general lines established in cite{e.d.2}, cite{e.d.} for model bicategories.