Geodesic
Distributive laws for pseudomonads
Theory and applications of categories, Tome 5 (1999), pp. 81-147.

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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.
Classification : 18C15, 18D05, 18D20.
Keywords: Pseudomonads, distributive laws, KZ-doctrines, Gray-categories. 
@article{TAC_1999_5_a4,
     author = {Francisco Marmolejo},
     title = {Distributive laws for pseudomonads},
     journal = {Theory and applications of categories},
     pages = {81--147},
     publisher = {mathdoc},
     volume = {5},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_1999_5_a4/}
}
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Francisco Marmolejo. Distributive laws for pseudomonads. Theory and applications of categories, Tome 5 (1999), pp. 81-147. http://geodesic.mathdoc.fr/item/TAC_1999_5_a4/