# Pseudo-distributive laws

## Idea

A pseudo-distributive law is a vertical categorification of a distributive law, relating two pseudomonads on a bicategory. There are various different levels of weakness that such a thing can exist at. As with ordinary distributive laws, a pseudo-distributive law governs the lifting of one pseudomonad to the Eilenberg-Moore and Kleisli bicategories of the other.

## Definition

See any of the references, particularly (CHP)

## Examples

• If $T$ is a pseudo-commutative? 2-monad on Cat, then there is a pseudo-distributive law between $T$ and the 2-monad whose algebras are symmetric monoidal categories; see (Kelly).

• Grothendieck fibrations and opfibrations on a category $C$ (or more generally an object of a suitable 2-category) are the algebras for a pair of pseudomonads. If $C$ has pullbacks, there is a pseudo-distributive law between these pseudomonads, whose joint algebras are the bifibrations satisfying the Beck-Chevalley condition; see (von Glehn).

• generalized polycategories? are naturally defined relative to a pseudo-distributive law on a Prof-like bicategory; see (Garner) for the canonical example of ordinary (symmetric) polycategories.

• Pseudo-distributive laws involving lax-idempotent 2-monads have an especially nice form; see (Marmolejo) and (Walker).

## References

• Max Kelly, Coherence theorems for lax algebras and for distributive laws, Proceedings of the Sydney Category Seminar 1972-73.
• Francisco Marmolejo, Distributive laws for pseudomonads, TAC

Last revised on July 4, 2017 at 11:49:13. See the history of this page for a list of all contributions to it.