A monadic approach to polycategories

Abstract

Polycategories form a rather natural generalization of multicategories. Besides the domains also the codomains of morphisms are allowed to be strings of objects. Small multicategories can be characterized elegantly as monads in a suitable bicategory of special spans with free monoids as domains. To find a similar description of small polycategories, we first investigate distributive laws in the sense of Beck between cartesian monads S and T on a category X with pullbacks as tools for constructing new bicategories of S-T-spans. We identify a class of “super-cartesian” distributive laws that indeed produce such bicategories in a straightforward manner.If we decompose the free monoid monad (_)∗ into the free semigroup monad and the exception monad, a relation on (_)∗∗ can be defined by means of three super-cartesian distributive laws such that the resulting bicategory of (_)∗-(_)∗-spans has precisely the small planar polycategories as monads. General polycategories require a different construction and a span instead of a relation. However, only the notion of planar polycategory can be generalized to 2-dimensional structures, where objects are replaced by typed 1-cells. “ fc-polycategories” have essentially the same characterization as planar polycategories, but over the base grph rather than set.One of the distributive laws used above, complementation on the free semigroup monad, seems to be new. We identify its algebras as associative double semi-groups.Then we address the question, which spans between TS and ST correctly generalize (super-cartesian) distributive laws and provide an associative composition for S-T-spans with canonical units. We obtain four simple sufficient conditions, best formulated in the fc-multicategory of spans and morphisms in [ X,X], that clarify the notion of super-cartesian distributive law and justify the added generality.Finally, we show how by first quotienting the bicategory X-spn the constructions outlined above can be used even for weakly cartesian monads, in particular the free commutative monoid monad, which fails to be cartesian.