Poly-categories via spans and monads

Jürgen Koslowski – 13 March 2002

Multi-categories, introduced by Lambek in 1969, are known to admit a very elegant global description, where monads are employed at two levels: first the free monoid monad is used to suitably ``skew'' the bicategory of spans nad then monads in the resulting bicategory are considered. A very natural generalization of multi-categories was introduced by Szabo in 1975 poly-categories, however by means of a local definition. We provide the missing global characterization, thereby extending the notion of T-category for a cartesian monad T to situatons where both sides of spans are to be subjected to (possibly different) monads.