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Towards a double operadic theory of systems

Sophie Libkind, David Jaz Myers

TL;DR

The paper proposes a double operadic theory of systems (DOTS) that unifies the composition of systems and the maps between them via symmetric monoidal loose right modules over symmetric monoidal double categories of interfaces and interactions. It develops a versatile toolkit of doctrines to construct diverse system theories (e.g., spans, cospans, lenses, tangencies) and provides concrete exemplars like open Petri nets and Moore machines, illustrating how interactions act on systems and how system maps realize refinements and simulations. A central contribution is the systematic use of pseudo-functorial constructions to build modules of systems and the demonstration that many diagrammatic languages arise as free processes in these doctrines. The framework supports both the operadic perspective (via wiring diagrams) and the process-theoretic perspective (via double categories of interfaces/interactions), enabling a flexible, compositional analysis of complex systems with a clear pathway to future developments such as Yoneda theory, assume-guarantee reasoning, and time-varying behavior. Overall, DOTS provides a robust categorical foundation for modular design and analysis of heterogeneous systems, with broad applicability across diagrammatic formalisms and dynamical systems modeling.

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.

Towards a double operadic theory of systems

TL;DR

The paper proposes a double operadic theory of systems (DOTS) that unifies the composition of systems and the maps between them via symmetric monoidal loose right modules over symmetric monoidal double categories of interfaces and interactions. It develops a versatile toolkit of doctrines to construct diverse system theories (e.g., spans, cospans, lenses, tangencies) and provides concrete exemplars like open Petri nets and Moore machines, illustrating how interactions act on systems and how system maps realize refinements and simulations. A central contribution is the systematic use of pseudo-functorial constructions to build modules of systems and the demonstration that many diagrammatic languages arise as free processes in these doctrines. The framework supports both the operadic perspective (via wiring diagrams) and the process-theoretic perspective (via double categories of interfaces/interactions), enabling a flexible, compositional analysis of complex systems with a clear pathway to future developments such as Yoneda theory, assume-guarantee reasoning, and time-varying behavior. Overall, DOTS provides a robust categorical foundation for modular design and analysis of heterogeneous systems, with broad applicability across diagrammatic formalisms and dynamical systems modeling.

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.

Paper Structure

This paper contains 51 sections, 34 theorems, 73 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Categorical systems theories
  3. Basic ideas of categorical systems theory
  4. Operadic systems theories and process theories
  5. Double operadic systems theory
  6. Contributions of this paper
  7. Future work
  8. Preliminaries
  9. 2-Categories and $\mathcal{F}$-sketches
  10. Double categorical preliminaries
  11. Adequate triples, spans, and lenses
  12. Adequate triples and spans
  13. Lex and rex categories
  14. Lex categories
  15. Rex categories
  16. ...and 36 more sections

Key Result

Lemma 2.6

The djm-00IR is djm-00AN.

Figures (4)

  • Figure 1: Ontology of double categorical systems theory
  • Figure 2: Systems theory as labelled double category
  • Figure 3: Ontology of double categorical systems theory
  • Figure 4: Doctrine of systems theories

Theorems & Definitions (150)

  • Remark 1.2: Symmetric monoidal loose right modules and algebras of double operads
  • Definition 2.1: Cartesian 2-category
  • Remark 2.2: Terminology for $\mathcal{F}$-categories
  • Definition 2.3: $\mathcal{S}\mathsf {M}$ and $\mathcal{C}\mathsf {art}$
  • Definition 2.4: The 2-category of double categories
  • Lemma 2.6: $\mathcal{D}\mathsf {bl}$ is cartesian
  • Lemma 2.7: Tight isomorphisms are commuter cells
  • proof
  • Definition 2.8: Commuter cells
  • Definition 2.9: Adequate triple
  • ...and 140 more