Graphs for Juncture

TL;DR

This paper introduces a graph-theoretic approach to 2-categories using a new operation called juncture, which generalizes composition and relates to proof theory and operads, offering an alternative foundation for bicategories.

Contribution

It develops a detailed planarity framework for juncture and graphs of 2-cells, proposing an alternative foundation for bicategories and weak n-categories.

Findings

Juncture generalizes vertical and horizontal composition of 2-cells.

A detailed planarity characterization of graphs of 2-cells is provided.

Connections between juncture, proof theory, and operads are established.

Abstract

An alternative foundation for 2-categories is explored by studying graph-theoretically a partial operation on 2-cells named juncture, which can replace vertical and horizontal composition. Juncture is a generalized vertical composition of 2-cells that need not involve the whole target and the whole source; it may involve them only partly, provided the result is again a 2-cell. Since commuting diagrams of arrows of ordinary categories may be conceived as invertible 2-cells, this study concerns ordinary category theory too. The operation of juncture has a connection with proof theory, where it corresponds to a kind of cut rule on sequents, and it is related also to an operation on which the notion of operad can be based. The main achievement of the work is a detailed description of the specific planarity involved in juncture and graphs of 2-cells, comparable to the usual combinatorial characterizations of planarity in graph theory. This work points out to an alternative foundation for bicategories, i.e. weak 2-categories, and more generally weak n-categories.