A generalized polycategory combines the notion of generalized multicategory with that of polycategory (or prop). While a generalized multicategory is parametrized by a monad (on a double category or bicategory or virtual double category) that specifies the arities of domains of morphisms, a generalized polycategory is parametrized by two such monads, specifying the arities of domains and codomains, together with a suitable sort of distributive law between them that specifies the allowable composites.

The theory of generalized polycategories is very underdeveloped in the literature compared to generalized multicategories. A Burroni–Leinster style definition, using a double category of spans only, appears in Koslowski 2005, and is used to define planar (non-symmetric) polycategories. A definition using profunctors is implicit in Garner 2008, and is used to define symmetric polycategories. But no general framework, or other examples, appears to have been written down yet.

References

A Burroni-style definition of generalized polycategory, called “D-categories” is defined in

Elizabeth Burroni?. Algèbres non déterministiques et D-catégories. Notebooks of Categorical Topology and Differential Geometry 14.4 (1973): 417-475. (pdf)

though has a weaker notion of associativity than might be expected. The notion of associativity is strengthened by

Juergen Koslowski, A monadic approach to polycategories, TAC Vol. 14, 2005, No. 7, pp 125-156.

Richard Garner, Polycategories via pseudo-distributive laws, arXiv, Adv. Math. 218 no. 3 (2008)

Last revised on October 25, 2022 at 20:44:55.
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