The weak theory of monads

We construct a 'weak' version EMw(K) of Lack and 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 EMw(K) and composite pre-monads in K is discussed If K admits Eilenberg-Moore constructions for monads, we define two symmetrical notions of 'weak fittings' for monads in K If moreover idempoient 2-cells in K split, we describe both kinds of weak lifting via an appropriate pseudo-functor EMw(K) -> K Weak entwining structures and partial entwining structures are shown to realize weak littings 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 (C) 2010 Elsevier Inc. All rights reserved