Universal properties of bicategories of polynomials

TL;DR

This paper explores the fundamental universal properties of bicategories of polynomials, spans, and spans with invertible 2-cells, providing new proofs and leveraging generic bicategory properties to simplify complex coherence conditions.

Contribution

It offers a novel proof approach for the universal properties of bicategories of polynomials and spans, enhancing understanding of their foundational categorical structures.

Findings

Universal properties of bicategories of polynomials established

New proof of universal properties of bicategory of spans

Universal properties of spans with invertible 2-cells demonstrated

Abstract

We establish the universal properties of the bicategory of polynomials, considering both cartesian and general morphisms between these polynomials. A direct proof of these universal properties would be impractical due to the complicated coherence conditions arising from polynomial composition; however, in this paper we avoid most of these coherence conditions using the properties of generic bicategories. In addition, we give a new proof of the universal properties of the bicategory of spans, and also establish the universal properties of the bicategory of spans with invertible 2-cells; showing how these properties may be used to describe the universal properties of polynomials.