The 2-category of weak entwining structures

TL;DR

This paper introduces the 2-category of weak entwining structures, generalizing mixed distributive laws, and explores their properties and relationships with Eilenberg-Moore constructions in a 2-category setting.

Contribution

It defines the 2-category of weak entwining structures and establishes conditions for pseudo-functors to relate these structures to monads and comonads.

Findings

Weak entwining structures are characterized as compatible pairs of monads and comonads.

Pseudo-functors relate weak entwining structures to monads and comonads via weak liftings.

Eilenberg-Moore objects of lifted monads and comonads are shown to be equivalent.

Abstract

A weak entwining structure in a 2-category K consists of a monad t and a comonad c, together with a 2-cell relating both structures in a way that generalizes a mixed distributive law.A weak entwining structure can be characterized as a compatible pair of a monad and a comonad, in 2-categories generalizing the 2-category of comonads and the 2-category of monads in K, respectively. This observation is used to define a 2-category Entw^w(K) of weak entwining structures in K. If the 2-category K admits Eilenberg-Moore constructions for both monads and comonads and idempotent 2-cells in K split, then there are pseudo-functors from Entw^w(K) to the 2-category of monads and to the 2-category of comonads in K, taking a weak entwining structure (t,c) to a `weak lifting' of t for c and a `weak lifting' of c for t, respectively. The Eilenberg-Moore objects of the lifted monad and the lifted comonad are shown to be equivalent. If K is the 2-category of functors induced by bimodules, then these Eilenberg-Moore objects are isomorphic to the usual category of weak entwined modules.