Gray (skew) multicategories: double and Gray-categorical cases

TL;DR

This paper introduces Gray (skew) multicategories, unifying double and Gray-categories, and explores their structures, products, and monoidal properties, advancing the understanding of higher categorical frameworks.

Contribution

It constructs Gray (skew) multicategories with various functor types, establishes their monoidal and skew-monoidal structures, and provides a unifying framework for higher categories.

Findings

Gray skew-multicategories are closed and representable.

Gray multicategories with strict functors are representable.

Categories of double and Gray-categories are skew monoidal or monoidal.

Abstract

We construct in a unifying way skew-multicategories and multicategories of double and Gray-categories that we call Gray (skew) multicategories. We study their different versions depending on the types of functors and higher transforms. We construct Gray type products by generators and relations and prove that Gray skew-multicategories are closed and representable on one side, and that the Gray multicaticategories taken with the strict type of functors are representable. We conclude that the categories of double and Gray-categories with strict functors underlying Gray (skew) multicategories are skew monoidal, respectively monoidal, depending on the type of the inner-hom and product considered. The described Gray (skew) multicategories we see as prototypes of general Gray (skew) multicategories, which correspond to (higher) categories of higher dimensional internal and enriched categories.