nLab
polycategory

Contents

Idea

A polycategory is like a category or a multicategory, but where both the domain and the codomain of a morphism can be finite lists of objects rather than just single objects.

Properties

Relation to PROPs

A (multicolored) PROP can also be described as a polycategory; what distinguishes a polycategory from a PROP is that in a polycategory, we can only compose along one object at once. That is, we have a composition operation

D:Hom(A,B;C,D,E)×Hom(F,D,G;H)Hom(F,A,B,G;C,H,E)\circ_D \colon Hom(A,B;C,D,E) \times Hom(F,D,G; H) \to Hom(F,A,B,G; C,H,E)

but not an operation such as

B,C:Hom(A;B,C)×Hom(B,C;D)Hom(A,D).\circ_{B,C} \colon Hom(A; B,C) \times Hom(B,C; D) \to Hom(A,D).

Internal logic

Polycategories provide a natural categorical semantics for linear logic.

Representability

A polycategory that is “representable on both sides”, meaning informally that morphisms in Hom(A 1,,A n;B 1,,B m)Hom(A_1,\dots,A_n;B_1,\dots,B_m) correspond to morphisms A 1A nB 1B mA_1 \otimes \cdots\otimes A_n \to B_1 \invamp \cdots \invamp B_m, is a linearly distributive category. A formal definition can be found in (Cockett-Seely).

References

  • M.E. Szabo, Polycategories, Comm. Algebra 3 (1975) 663-689. DOI

  • Robin Cockett, Robert Seely, Weakly Distributive Categories, Journal of Pure and Applied Algebra, 114(1997)2, pp 133-173 (ps.gz)

  • Richard Garner, Polycategories via pseudo-distributive laws, arXiv

Revised on May 20, 2016 11:12:23 by Mike Shulman (128.100.216.55)