Generalised Process Theories

TL;DR

This paper introduces an operadic framework for process theories, extending traditional symmetric monoidal categories to encompass more general, higher-order, and enriched process structures, with applications in quantum foundations.

Contribution

It proposes an operad-based formalization of process theories, unifying and extending existing approaches beyond symmetric monoidal categories.

Findings

Operadic formalization captures a broader class of process theories.

Standard process theories are recovered as a special case.

Provides new insights into quantum foundations and compositional structures.

Abstract

Process theories provide a powerful framework for describing compositional structures across diverse fields, from quantum mechanics to computational linguistics. Traditionally, they have been formalized using symmetric monoidal categories (SMCs). However, various generalizations, including time-neutral, higher-order, and enriched process theories, do not naturally conform to this structure. In this work, we propose an alternative formalization using operad algebras, motivated by recent results connecting SMCs to operadic structures, which captures a broader class of process theories. By leveraging the string-diagrammatic language, we provide an accessible yet rigorous formulation that unifies and extends traditional process-theoretic approaches. Our operadic framework not only recovers standard process theories as a special case but also enables new insights into quantum foundations and compositional structures. This work paves the way for further investigations into the algebraic and operational properties of generalised process theories within an operadic setting.