In the quest for an elegant formulation of the notion of ``polycategory'' we develop a more symmetric counterpart to Burroni's notion of ``T- category'', where T is a cartesian monad on a category X with pullbacks. Our approach involves two such monads, S and T, that are linked by a suitable generalization of a distributive law in the sense of Beck. This takes the form of a span omega : TS <--> ST in the functor category [X,X] and guarantees essential associativity for a canonical pullback-induced composition of S-T-spans over X, identifying them as the 1-cells of a bicategory, whose (internal) monoids then qualify as ``omega-categories''. In case that S and T both are the free monoid monad on set, we construct an omega utilizing an apparently new classical distributive law linking the free semigroup monad with itself. Our construction then gives rise to so-called ``planar polycategories'', which nowadays seem to be of more intrinsic interest than Szabo's original polycategories. Weakly cartesian monads on X may be accommodated as well by first quotienting the bicategory of X-spans.
Keywords: cartesian monad, S-T-span, (cartesian) distributive law, multicategory, (planar) polycategory, fc-polycategory, associative double semigroup
2000 MSC: 18C15, 18D05
Theory and Applications of Categories,
Vol. 14, 2005,
No. 7, pp 125-156.