A monadic approach to polycategories

Jürgen Koslowski

Abstract

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 TS<--ω-->ST in the functor category [X,X] and guarantees essential associativity for a canonical pullback-induced composition of ST-spans over X, identifying them as the 1-cells of a bicategory, whose (internal) monoids then qualify as ``ω-categories''. In case that S and T both are the free monoid monad on set, we construct an w 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 interrest than Szabo's original polycategories. Weakly cartesian monads on X may be accommodated as well by first quotienting the bicategory of X-spans.