Dear Colleagues, Belated best wishes for 2004! A substantially revised (and improved ;-) version of my CTCS02 paper "A monadic approach to polycategories" is available on my WEB-page: http://www.iti.cs.tu-bs.de/TI-INFO/koslowj/RESEARCH/Moapp.ps.gz The abstact is appended below. -- Juergen Koslowski %% Abstract for: A monadic approach to polycategories Polycategories should form a rather natural generalization of multicategories: besides the domains also the codomains of morphisms are allowed to be strings of objects. But while small multicategories can be elegantly characterized as monoids in a bicategory of set-spans with free set-monoids as domains, no such description of small polycategories seems to have been known so far. To address this problem, we first investigate distributive laws in the sense of Beck between cartesian monads S and T on a category X with pullbacks. Some ot these can serve as tools for constructing new bicategories of S-T-spans over X, and monoids in such bicategories should be viewed as S-T-categories. We identify a class of ``super-cartesian'' distributive laws that indeed produce such bicategories in a straightforward manner. However, if S and T coincide with the free monoid monad (_)^* on set, there is no distributive law to get this construction off the ground. Since polycategories display a high degree of symmetry, a more symmetric substitute for distributive laws would be desirable. If we decompose (_)^* into the free semigroup monad and the exception monad, a relation on (_)^{**} can be defined by means of three super-cartesian distributive laws that still allows us to construct a bicategory of (_)^*-}(_)^*-spans. Its monoids turn out to be precisely the small planar polycategories. General polycategories, as introduced by Szabo, require a different construction and a span instead of a relation. However, only the notion of planar polycategory admits a 2-dimensional generalization, where objects are replaced by typed 1-cells. The resulting ``fc polycategories'' have essentially the same characterization as planar polycategories, but over the base grph rather than set. In view of other shortcomings of Szabo's general concept this suggests that the planar variant of polycategories may be the ``correct'' generalization of multicategories. One of the three distributive laws used in the construction above behaves like a ``complementation'' on the free semigroup monad and seems to be new. We identify its algebras as associative double semigroups. In fact, the free such structure on a set B can be extended from B^{++} to an associative double monoid structure on B^{**}. We then turn to the fundamental question, which spans between TS and ST correctly generalize (super-cartesian) distributive laws and provide an essentially associative composition for S-T-spans over X with canonical units. This contrasts with Elisabeth Burroni's approach [BurroniE73], who weakened the notion of associativity for her notion of D-categories in order to encompass the non-cartesian power-set monad. Our definition of (super-cartesian) generalized distributive law is best formulated in the fc-multicategory of spans and morphisms in [X,X]; this clarifies the notion of (super-cartesian) distributive law and justifies 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 this applies to the free commutative monoid monad, which fails to be cartesian. We than adapt the construction for planar polycategories to obtain symmetric polycategories in a similar fashion. --=20 Juergen Koslowski If I don't see you no more on this world ITI, TU Braunschweig I'll meet you on the next one koslowj@iti.cs.tu-bs.de and don't be late! http://www.iti.cs.tu-bs.de/~koslowj Jimi Hendrix (Voodoo Child, SR)