category theory

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

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

but not an operation such as

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

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 $Hom(A_1,\dots,A_n;B_1,\dots,B_m)$ correspond to morphisms $A_1 \otimes \cdots\otimes A_n \to B_1 \invamp \cdots \invamp B_m$, is a linearly distributive category. A formal definition can be found in (Cockett-Seely).

Internal structures

Some categorical structures that are normally defined in a monoidal category can instead be defined in a polycategory, including Frobenius algebras and dual objects. Dual objects, in particular, are a form of “negation” that make a linearly distributive category into a star-autonomous category.

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 August 1, 2016 12:48:00 by Mike Shulman (24.43.127.99)