# nLab polycategory

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 single objects.

Note that a (multicolored) PROP can also be described in such a way; 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

${\circ }_{D}:\mathrm{Hom}\left(A,B;C,D,E\right)×\mathrm{Hom}\left(F,D,G;H\right)\to \mathrm{Hom}\left(F,A,B,G;C,H,E\right)$\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

${\circ }_{B,C}:\mathrm{Hom}\left(A;B,C\right)×\mathrm{Hom}\left(B,C;D\right)\to \mathrm{Hom}\left(A,D\right).$\circ_{B,C} \colon Hom(A; B,C) \times Hom(B,C; D) \to Hom(A,D).

Polycategories provide a natural categorical semantics for linear logic.

Created on May 7, 2012 22:42:18 by Mike Shulman (71.136.234.110)