nLab polycategory

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 (much like for properads, but see below). 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 properads and PROPs

(Coloured) properads and PROPs are both similar to polycategories: what distinguishes polycategories from both properads and PROPs 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

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

In properads and PROPs, we allow composition along multiple objects at once. This is analogous to the definition of (coloured) operads in terms of partial composition or simultaneous composition. However, though both kinds of operads are equivalent, there exist coloured properads that are not equivalent to polycategories. For instance, if we have polymorphisms $f : A \to B, B$ and $g : B, B \to C$, we can form the composite $g \circ f : A \to C$ in a properad, but not a polycategory.

PROPs generalise properads further by, in addition to having the multiple composition operator of properads, also having a tensoring operator (given by the action of $\otimes$ on morphisms) that allows for composition along zero objects.

Internal logic

Polycategories provide a natural categorical semantics for classical 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.

Symmetry

The relation between symmetric and non-symmetric (a.k.a. planar) polycategories is more subtle than that between symmetric and non-symmetric multicategories. To obtain a truly “planar” notion of polycategory, it is not sufficient to omit the symmetric group actions; in addition one must place restrictions on the composition operations allowed so that no “wires cross”. (All the composition operations in a symmetric polycategory can then be recovered from planar composites together with symmetric group actions.)

One concrete consequence of this is that unlike for multicategories, it is not possible to freely generate a symmetric polycategory from a planar one by simply adding the symmetric group actions. That is, any planar multicategory gives rise to a symmetric multicategory by duplicating each $n$-ary morphism into $n!$ new morphisms with every possible re-ordering of the objects in the domain, but an attempt to do something similar for a planar polycategory fails because not all the putative composites of such morphisms can be expressed in terms of composition in the original planar polycategory. See for instance Example 1.3 of Koslowski.

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.

Related concepts

• dioperad
• PROP
• properad
• multicategory
• operad
• LNL Polycategory

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)

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

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

• Wee Liang Gan, Koszul duality for dioperads, arxiv

• Nicolas Blanco and Noam Zeilberger, Bifibrations of Polycategories and Classical Linear Logic, ENTCS 2020, doi