Bicategories of fractions revisited: towards small homs and canonical 2-cells

TL;DR

This paper refines the theory of bicategories of fractions by weakening conditions for inverting arrows, enabling small hom-categories, and introduces canonical forms for 2-cells to simplify their manipulation, with applications to orbispaces.

Contribution

It introduces weaker conditions for bicategories of fractions, allowing smaller hom-categories and canonical 2-cell representatives, simplifying their composition and comparison.

Findings

Weaker conditions enable bicategories of fractions with small hom-categories.

Canonical representatives for 2-cells are identified under certain conditions.

Simplified pasting of 2-cells is achieved using pseudo pullbacks.

Abstract

This paper adresses two issues in dealing with bicategories of fractions. The first is to introduce a set of conditions on a class of arrows in a bicategory which is weaker than the one given in Pronk, Etendues and stacks as bicategories of fractions, but still allows a bicalculus of fractions. These conditions allow us to invert a smaller collection of arrows so that in some cases we may obtain a bicategory of fractions with small hom-categories. We adapt the construction of the bicategory of fractions to work with the weaker conditions. The second issue is the difficulty in dealing with 2-cells, which are defined by equivalence classes. We discuss conditions under which there are canonical representatives for 2-cells, and how pasting of 2-cells can be simplified in the presence of certain pseudo pullbacks. We also discuss how both of these improvements apply in the category of orbispaces.

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