The 2-Localization of a Quillen's model category

Abstract: In [Homotopical Algebra, Springer LNM 43] Quillen introduces the notion of a model category: a category $\mathcal{C}$ provided with three distinguished classes of maps ${\mathcal{W},\, \mathcal{F},\, co\mathcal{F}}$ (weak equivalences, fibrations, cofibrations), and gives a construction of the localization $\mathcal{C}[\mathcal{W}^{-1}]$ as the quotient of $\mathcal{C}$ by the congruence relation determined by the homotopies on the sets of arrows $\mathcal{C}(X,\,Y)$. We develop here the 2-categorical localization, in which the 2-cells of this 2-localization are given by homotopies, and one can get the Quillen's localization when applying the connected components functor $\pi_0$ on the hom-categories of the 2-localization. Our proof is not just a generalization of the well-known Quillen's one. We work with definitions of cylinders and homotopies introduced in [M.E. Descotte, E.J. Dubuc, M. Szyld; Model bicategories and their homotopy bicategories, arXiv:1805.07749 (2018)] considering only a single family of arrows $\Sigma$. When $\Sigma$ is the class $\mathcal{W}$ of weak equivalences of a model category, we get the Quillen's results.