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Distributive laws for pseudomonads

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Francisco Marmolejo

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.

Keywords: Pseudomonads, distributive laws, KZ-doctrines, Gray-categories.

1991 MSC: 18C15, 18D05, 18D20.

*Theory and Applications of Categories*, Vol. 5, 1999, No. 5, pp 81-147.

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