Towards a double operadic theory of systems

TL;DR

This paper develops a unified categorical framework for systems theory, modeling open systems and their interactions using advanced structures like symmetric monoidal double categories and modules, with concrete examples such as Petri nets and Moore machines.

Contribution

It introduces a novel, comprehensive categorical framework for systems, including doctrines for various system types, extending and generalizing existing categorical systems theory work.

Findings

Module of open Petri nets over wiring diagrams

Module of deterministic Moore machines over lenses

Introduction of doctrines for port-plugging and variable sharing systems

Abstract

We present a unified framework for categorical systems theory which packages a collection of open systems, their interactions, and their maps into a symmetric monoidal loose right module of systems over a symmetric monoidal double category of interfaces and interactions. As examples, we give detailed descriptions of (1) the module of open Petri nets over undirected wiring diagrams and (2) the module of deterministic Moore machines over lenses. We define several pseudo-functorial constructions of modules of systems in the form of doctrines of systems theories. In particular, we introduce doctrines for port-plugging systems, variable sharing systems, and generalized Moore machines, each of which generalizes existing work in categorical systems theory. Finally, we observe how diagrammatic interaction patterns are free processes in particular doctrines.