The 2-localization of a model category

TL;DR

This paper develops a 2-dimensional localization framework for model categories using a novel cylinder notion, enabling a refined understanding of homotopy and equivalences in a 2-category setting.

Contribution

It introduces a new 2-localization construction for model categories, utilizing a novel cylinder concept and avoiding issues with non-invertible 2-cells, providing new proofs and simplifications.

Findings

Constructs a 2-category $ ext{Ho}( extbf{A})$ with homotopies as 2-cells.

Establishes the universal property of the inclusion 2-functor as a 2-localization.

Applies the theory to fibrant-cofibrant and general categories, recovering Quillen's results.

Abstract

In this paper we study a 2-dimensional version of Quillen's homotopy category construction. Given a category $\mathscr{A}$ and a class of morphisms $\Sigma \subset \mathscr{A}$ containing the identities, we construct a 2-category $\mathcal{H}o(\mathscr{A})$ 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 \cite{e.d.2}. The inclusion 2-functor $\mathscr{C} \longrightarrow \mathcal{H}o(\mathscr{A})$ has a universal property which implies that it will be the 2-localization of $\mathscr{A}$ at $\Sigma$ as soon as the arrows of $\Sigma$ become equivalences in $\mathcal{H}o(\mathscr{A})$. This is then used to obtain 2-localizations of a model category $\mathscr{A}{C}$, with $\Sigma = \mathcal{W}$, the weak equivalences, and $\mathscr{A} = \mathscr{C}_{fc}$, the full subcategory of fibrant-cofibrant objects, as well as with $\mathscr{A} = \mathscr{C}$. The set of connected components of the hom categories yields Quillen's results. We follow the general lines established in \cite{e.d.2}, \cite{e.d.} for model bicategories. The development here is not just the examination of the general theory in a particular case. It is not concerned with and avoids the problems which arise when dealing with non invertible 2-cells. Also, the use here of functorial factorization adds further simplifications by eliminating the need of pseudofunctors. New proofs are produced which are not a mere simplified adaptation of the ones of the general case.