Generic bicategories

TL;DR

This paper generalizes the correspondence between oplax functors and comonads from terminal bicategories to a broader class called generic bicategories, simplifying their description and applying it to well-known bicategories.

Contribution

It introduces the concept of generic bicategories and shows how oplax functors can be described similarly to comonads without involving composition directly.

Findings

Generalization of the oplax functor-comonad correspondence to generic bicategories.

Simplified description of oplax functors in terms of comonad-like data.

Application to bicategories of spans, polynomials, and cartesian monoidal categories.

Abstract

It is well known that to give an oplax functor of bicategories $\mathbf{1}\to\mathscr{C}$ is to give a comonad in $\mathscr{C}$. Here we generalize this fact, replacing the terminal bicategory by any bicategory $\mathscr{A}$ for which the composition functor admits generic factorisations. We call bicategories with this property generic, and show that for generic bicategories $\mathscr{A}$ one may express the data of an oplax functor $\mathscr{A}\to\mathscr{C}$ much like the data of a comonad; the main advantage of this description being that it does not directly involve composition in $\mathscr{A}$. We then go on to apply this result to some well known bicategories, such as cartesian monoidal categories (seen as one object bicategories), bicategories of spans, and bicategories of polynomials with cartesian 2-cells.