A $\textbf{Gray}$-categorical pasting theorem

TL;DR

This paper provides a formal foundation for pasting diagrams in Gray-categories, establishing that composites are uniquely determined up to a contractible groupoid of choices, thus justifying their use in 3D category theory.

Contribution

It introduces a rigorous approach to pasting in Gray-categories and proves the uniqueness of composites up to a contractible groupoid, filling a gap in the literature.

Findings

Formalized pasting in Gray-categories.

Proved uniqueness of composites up to a contractible groupoid.

Justified the use of 2D pasting diagrams in 3D category theory.

Abstract

The notion of $\textbf{Gray}$-category, a semi-strict $3$-category in which the middle four interchange is weakened to an isomorphism, is central in the study of three-dimensional category theory. In this context it is common practice to use $2$-dimensional pasting diagrams to express composites of $2$-cells, however there is no thorough treatment in the literature justifying this procedure. We fill this gap by providing a formal approach to pasting in $\textbf{Gray}$-categories and by proving that such composites are uniquely defined up to a contractible groupoid of choices.