A monadic approach to polycategories

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 S -T -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 ω utilizing an apparently new classical distributive law linking the free semigroup monad with itself. Our construction then gives rise to socalled “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. 0. Motivation and Outline Lately multicategories have received renewed attention in the field of higher-dimensional category theory, cf., [Lei98] and [Her00], respectively. But in categorical logic, where they were introduced originally by Jim Lambek in the 1960’s [Lam69], without imposing further structure multicategories correspond to a rather simple logical system. Basically the comma separating objects in the domain list is interpreted as a conjunction. Cut-elimination holds for (in hindsight) fairly trivial reasons. On the other hand, polycategories, where lists of objects are allowed not just as domains but also as codomains of morphisms, at least in their planar variant are of much greater interest in categorical logic. It was shown by Robin Cockett and Robert Seely [CS92, CS97] that planar polycategories are closely related to linearly distributive categories. These in turn correspond precisely to the tensor-par fragment of linear logic. There the cut-elimination procedure is highly nontrivial, and has an interesting graph-theoretic representation in terms of proof nets. While small multicategories have been characterized elegantly as monoids in a bicategory of set-spans with free set-monoids as domains, a special case of Albert Burroni’s notion of T -category [Bur71], no such description of small polycategories seems to have been available so far. To keep this paper self-contained, Section 1 reviews the basic set-up for multicategories before addressing Manfred E. Szabo’s original definition of polycatReceived by the editors 2004-01-15 and, in revised form, 2005-01-09. Transmitted by Richard Blute. Published on 2005-05-08. 2000 Mathematics Subject Classification: 18C15; 18D05.