A Category of Surface-Embedded Graphs

TL;DR

This paper introduces a categorical framework for rewriting surface-embedded graphs, enabling the representation of non-intersecting string diagrams with explicit boundary and surface topology considerations.

Contribution

It presents a novel double pushout rewriting formulation that records boundaries and incorporates surface topology via rotation systems.

Findings

Defines a new categorical formalism for surface-embedded graphs

Develops a boundary-aware double pushout rewriting method

Incorporates surface topology through rotation systems

Abstract

We introduce a categorical formalism for rewriting surface-embedded graphs. Such graphs can represent string diagrams in a non-symmetric setting where we guarantee that the wires do not intersect each other. The main technical novelty is a new formulation of double pushout rewriting on graphs which explicitly records the boundary of the rewrite. Using this boundary structure we can augment these graphs with a rotation system, allowing the surface topology to be incorporated.