nLab relative pseudomonad

Idea

The notion of relative pseudomonad generalizes that of pseudomonad, to be relative to a 2-functor, analogous to the notion of relative monad in the one-dimensional setting. This enables the construction of Kleisli bicategories of pseudomonads on Prof (e.g. for defining generalized multicategories) by means of pseudo-distributive laws as if the presheaf category construction were a monad with Prof as its Kleisli bicategory, even though it is not for size reasons.

An example is the free cocompletion construction on a small category.

References

• Marcelo Fiore, Nicola Gambino, Martin Hyland, Glynn Winskel, Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures, arxiv

• Nathanael Arkor, Philip Saville?, Andrew Slattery, Bicategories of algebras for relative pseudomonads, arXiv:2501.12510

An early observation that the bicategory of profunctors should be a Kleisli category for a relative monad is Appendice I.1(b) of:

• René Guitart. Monades involutives complémentées, Cahiers de topologie et géométrie différentielle catégoriques 16 1 (1975) 17-101 [numdam:CTGDC_1975__16_1_17_0]