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.
Theory and Applications of Categories, Vol. 5, 1999, No. 5, pp 91-147 http://www.tac.mta.ca/tac/volumes/1999/n5/n5.dvi http://www.tac.mta.ca/tac/volumes/1999/n5/n5.ps ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1999/n5/n5.dvi ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1999/n5/n5.psTAC Home