A skew approach to enrichment for Gray-categories

TL;DR

This paper introduces skew monoidal closed structures on Gray-categories to model various higher-dimensional transformations, addressing limitations of existing monoidal structures.

Contribution

It presents new skew monoidal closed structures on Gray-categories that capture higher lax and pseudo-transformations, advancing the modeling of weak higher-dimensional transformations.

Findings

Several skew monoidal closed structures are constructed.

One structure captures higher lax transformations.

Another models higher pseudo-transformations.

Abstract

It is well known that the category of Gray-categories does not admit a monoidal biclosed structure that models weak higher-dimensional transformations. In this paper, the first of a series on the topic, we describe several skew monoidal closed structures on the category of Gray-categories, one of which captures higher lax transformations, and another which models higher pseudo-transformations.