Distributive laws between monads (triples) were defined by Jon Beck in the 1960s. They were generalized to monads in 2-categories and noticed to be monads in a 2-category of monads. Mixed distributive laws are comonads in the 2-category of monads; if the comonad has a right adjoint monad, the mate of a mixed distributive law is an ordinary distributive law. Particular cases are the entwining operators between algebras and coalgebras. Motivated by work on weak entwining operators, we define and study a weak notion of distributive law for monads. In particular, each weak distributive law determines a wreath product monad (in the terminology of Lack and Street); this gives an advantage over the mixed case.

Keywords: monad, triple, distributive law, weak bialgebra

2000 MSC: 18C15, 18D10, 18D05

*Theory and Applications of Categories,*
Vol. 22, 2009,
No. 12, pp 313-320.

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