Abstract

. We define distributive laws between pseudomonads in a Gray-category A, as the classical two triangles and the two pentagons but commuting only up to isomorphism. These isomorphisms must satisfy nine coherence conditions. We also define the Gray-category PSM(A) of pseudomonads in A, and define a lifting to be a pseudomonad in PSM(A). We define what is a pseudomonad with compatible structure with respect to two given pseudomonads. We show how to obtain a pseudomonad with compatible structure from a distributive law, how to get a lifting from a pseudomonad with compatible structure, and how to obtain a distributive law from a lifting. We show that one triangle suffices to define a distributive law in case that one of the pseudomonads is a (co-)KZ-doctrine and the other a KZ-doctrine. 1. Introduction Distributive laws for monads were introduced by J. Beck in [2]. As pointed out by G. M. Kelly in [7], strict distributive laws for higher dimensional monads are rare. We need then a study ...

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