Bicategories of Lax Fractions

TL;DR

This paper extends the calculus of fractions to 2-categories by introducing lax fractions, enabling formal inversion of morphisms into adjoint right inverses and transforming pseudo-commutative squares into Beck-Chevalley squares.

Contribution

It introduces a calculus of lax fractions for 2-categories, generalizing previous work to include adjoint right inverses and Beck-Chevalley squares.

Findings

Defines lax fractions in 2-categories.

Provides a formal method for inverting morphisms as adjoint right inverses.

Establishes conditions for pseudo-commutative squares to become Beck-Chevalley squares.

Abstract

The well-known calculus of fractions of Gabriel and Zisman provides a convenient way to formally invert morphisms in a category. This was generalised to bicategories by Pronk. We extend these constructions by presenting a calculus of lax fractions for 2-categories that formally turns given morphisms into left adjoint right inverses and given pseudo-commutative squares into Beck-Chevalley squares.