A localization of bicategories via homotopies

TL;DR

This paper introduces a method to construct bicategorical localizations using homotopies related to a specific family of arrows, simplifying the process by avoiding changes to objects or arrows.

Contribution

It provides conditions under which bicategorical localization can be achieved solely through 2-cells defined by homotopies, extending the theory of localizations in bicategories.

Findings

Localization via homotopies is possible under certain conditions.

Homotopies are closely related to Quillen's homotopy in model categories.

Application to the homotopy bicategory of a model bicategory.

Abstract

Given a bicategory C and a family W of arrows of C, we give conditions on the pair (C,W) that allow us to construct the bicategorical localization with respect to W by dealing only with the 2-cells, that is without adding objects or arrows to C. We show that in this case, the 2-cells of the localization can be given by the homotopies with respect to W, a notion defined in this article which is closely related to Quillen's notion of homotopy for model categories but depends only on a single family of arrows. This localization result has a natural application to the construction of the homotopy bicategory of a model bicategory, which we develop elsewhere, as the pair (C_{fc},W) given by the weak equivalences between fibrant-cofibrant objects satisfies the conditions given in the present article.