$2$-Categories from a Gray Perspective

TL;DR

This paper explores 2-category theory through Gray-categories and surface diagrams, providing new graphical tools to understand cones, limits, and lax functors in higher category theory.

Contribution

It introduces a graphical calculus for Gray-categories, extending surface diagrams with compositor sheets to analyze lax functors and their comparison structures.

Findings

Graphical calculus effectively represents 2-categories and Gray-categories.

Extended surface diagrams facilitate reasoning about cones and limits of 2-functors.

New interpretation of lax functors using compositor sheets enhances understanding.

Abstract

In this paper we present $2$-category theory from the perspective of Gray-categories using the graphical calculus of separated surface diagrams. As an extended example we consider cones and limits of $2$-functors. Then we use the canonical adjunction between $2$-computads and $2$-categories to interpret the comparison structure of lax functors and extend the surface diagram calculus with compositor sheets in order to represent and reason about them.