When Only Topology Matters

Abstract

Graphical languages are symmetric monoidal categories presented by generators and equations. The string diagrams notation allows to transform numerous axioms into low dimension topological rules we are comfortable with as three dimensional space citizens. This aspect is often referred to by the Only Topology Matters paradigm (OTM). However OTM remains quite informal and its exact meaning in terms of rewriting rules is ambiguous. In this paper we define three precise aspects of the OTM paradigm, namely flexsymmetry, flexcyclicity and flexibility of Frobenius algebras. We investigate how this new framework can simplify the presentation of known graphical languages based on Frobenius algebras.