On the 2-Categories of Weak Distributive Laws

Abstract

A weak mixed distributive law (also called weak entwining structure [8]) in a 2-category consists of a monad and a comonad, together with a 2-cell relating them in a way which generalizes a mixed distributive law due to Beck. We show that a weak mixed distributive law can be described as a compatible pair of a monad and a comonad, in 2-categories extending, respectively, the 2-category of comonads and the 2-category of monads in [13]. Based on this observation, we define a 2-category whose 0-cells are weak mixed distributive laws. In a 2-category 𝒦 which admits Eilenberg–Moore constructions both for monads and comonads, and in which idempotent 2-cells split, we construct a fully faithful 2-functor from this 2-category of weak mixed distributive laws to 𝒦^2×2.

Key Words:

• Arrow 2-category
• Eilenberg–Moore object
• Monad
• Weak distributive law
• Weak lifting

2010 Mathematics Subject Classification:

• 18D05
• 18C15
• 18C20
• 16T05

ACKNOWLEDGMENT

G. B. would like to thank the organizers of the Conference in Hopf algebras and noncommutative algebra in the honor of Mia Cohen, in Sde-Boker, May 2010, for a generous invitation and an unforgettable first time in Israel. She also acknowledges financial support of the Hungarian Scientific Research Fund OTKA, grant no. F67910. All authors are grateful for partial support from the Australian Research Council, project DP0771252.

Notes

Communicated by H.-J. Schneider.

Dedicated to Mia Cohen on the occasion of her retirement.