A finite approach to representable multicategories and related structures
Gabriele Lobbia
TL;DR
The paper develops a finite presentation for multicategorical structures by introducing short multicategories, truncating multimaps to arity at most four, and proving that representable and closed variants of these short structures are equivalent to their full (classical or skew) counterparts. Central to the approach are map classifiers and universal properties that recover tensor products, internal homs, and units from finite data, enabling a robust translation between short multicategories and (skew) monoidal categories. The work extends these correspondences to closed, braided, and biclosed settings, providing a coherent framework that covers both classical and skew theories and yields practical finite descriptions for complex multicategorical phenomena. The results facilitate construction and manipulation of (skew) multicategorical structures in practice and connect braided/ symmetric variants through a unified equivalence network.
Abstract
It is known that monoidal categories have a finite definition, whereas multicategories have an infinite (albeit finitary) definition. Since monoidal categories correspond to representable multicategories, it goes without saying that representable multicategories should also admit a finite description. With this in mind, we give a new finite definition of a structure called a short multicategory, which only has multimaps of dimension at most four, and show that under certain representability conditions short multicategories correspond to various flavours of representable multicategories. This is done in both the classical and skew settings.