14 Amazing Facts about Polycategories and their Fibrations

A quick compilation of facts about polycategories together with links to the appropriate litterature. Some are original work.

1. Polycategories have been introduced by Szabo in 1975 in Polycategories.
2. Polycategories are “categories with many-to-many morphisms”, i.e. polymaps have many inputs and many outputs.
3. Polycategories correspond to Gentzen classical sequent calculus.
4. Polycategories compose on one object.
5. Composition of Polycategories is the cut rule.
6. Representable polycategories, a.k.a. two-tensor polycategories, are linearly distributive categories, see Weakly distributive categories.
7. In a polycategory, $\otimes$, ⅋ and duality are characterised by universal properties.
8. Representable polycategories with duals are called $\ast$-representable.
9. Polycategories bifibred over $\ast$-representable one are $\ast$-representable, see this paper.
10. The terminal polycategory have one object $\ast$ and one polymap $\underline{(m,n)}$ for each arities.
11. A Polycategory is $\ast$-representable iff it is bifibred over the terminal one.
12. Universal polymaps are the cartesian polymaps, w.r.t. the functor into the terminal Polycategory.
13. Frobenius algebras are the generalised elements of polycategories.
14. There is a Grothendieck correspondence for polycategories.