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

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)

Created on August 23, 2020 at 10:45:28.
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