The weak theory of monads

TL;DR

This paper introduces a 'weak' version of the 2-category of monads, explores weak liftings, and characterizes weak bialgebras through these concepts, expanding the theoretical framework of monad interactions.

Contribution

It constructs a new 'weak' 2-category of monads, defines weak liftings in this context, and characterizes weak bialgebras using these structures, advancing monad theory.

Findings

Defined the weak 2-category EM^w(K) of monads.

Established relations between monads in EM^w(K) and pre-monads in K.

Characterized weak bialgebras via weak liftings of monads and comonads.

Abstract

We construct a `weak' version EM^w(K) of Lack & Street's 2-category of monads in a 2-category K, by replacing their compatibility constraint of 1-cells with the units of monads by an additional condition on the 2-cells. A relation between monads in EM^w(K) and composite pre-monads in K is discussed. If K admits Eilenberg-Moore constructions for monads, we define two symmetrical notions of `weak liftings' for monads in K. If moreover idempotent 2-cells in K split, we describe both kinds of a weak lifting via an appropriate pseudo-functor EM^w(K) --> K. Weak entwining structures and partial entwining structures are shown to realize weak liftings of a comonad for a monad in these respective senses. Weak bialgebras are characterized as algebras and coalgebras, such that the corresponding monads weakly lift for the corresponding comonads and also the comonads weakly lift for the monads.