Contents

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. Like multicategories, polycategories have both symmetric and non-symmetric variants.

Just as a symmetric multicategory with one object is also called an operad, and so a general symmetric multicategory is sometimes called a “colored operad”, a symmetric polycategory with one object is also called a dioperad and a general symmetric polycategory is sometimes called a “colored dioperad”.

## 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 one approach to star-polycategories; they are the form of “negation” that makes a linearly distributive category into a star-autonomous category.

## Generalizations

Just as multicategories are a special case of generalized multicategories, which can be defined relative to any suitable monad, polycategories are a special case of generalized polycategories?, which can be defined relative to any suitable pseudo-distributive law.

• 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

• Wee Liang Gan, Koszul duality for dioperads, arxiv