Generic bicategories

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.