# 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.