On the 2-Categories of Weak Distributive Laws

A weak mixed distributive law (also called weak entwining structure [8 Caenepeel , S. , De Groot , E. ( 2000 ). Modules over weak entwining structures . In: Andruskiewitsch , N. , Ferrer Santos , W. R. , Schneider , H.-J. , eds. New Trends in Hopf Algebra Theory . Contemp. Math. 267. AMS Providence , pp. 4701 – 4735 .[Crossref] , [Google Scholar]]) 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 Street , R. ( 1972 ). The formal theory of monads . J. Pure and Applied Algebra 2 : 149 – 168 .[Crossref] , [Google Scholar]]. 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 𝒦2×2.