A Mathematical Framework for Transformations of Physical Processes
Matt Wilson, Giulio Chiribella
TL;DR
This work formulates a universal mathematical framework for higher-order physical theories by treating them as enriched symmetric monoidal categories with enrichment in a base category $\mathbf{V}$. It shows that sequential and parallel composition of higher-order processes arise naturally from monoidal enrichment and linking between levels, and defines structure-preserving pm-functors to compare theories across layers via the Grothendieck construction. A central result is that linked, faithful enrichment yields closed monoidal structure, and this extends to towers of higher-order theories, where a merger apex becomes closed monoidal. The framework provides a principled approach to analyze and compare novel causal structures in quantum theory and beyond, aiming to unify static and dynamical aspects of physical theories. It opens directions toward universal properties, causality integration, and broader generalizations such as multicategorical or promonoidal settings.
Abstract
We observe that the existence of sequential and parallel composition supermaps in higher order theories of transformations can be formalised using enriched category theory. Encouraged by relevant examples such as unitary supermaps and layers within higher order causal categories (HOCCs), we treat the modelling of higher order physical theories with enriched monoidal categories in analogy with the modelling of physical theories with monoidal categories. We use the enriched monoidal setting to construct a suitable definition of structure preserving map between higher order physical theories via the Grothendieck construction. We then show that the convenient feature of currying in higher order physical theories can be seen as a consequence of combining the primitive assumption of the existence of parallel and sequential composition supermaps with an additional feature of linking. We then use our definition of structure preserving map to show that categories containing infinite towers of enriched monoidal categories with full and faithful structure preserving maps between them inevitably lead to closed monoidal structures. The aim of the proposed definitions is to step towards providing a broad framework for the study and comparison of novel causal structures in quantum theory, and, more broadly, a paradigm of physical theory where static and dynamical features are treated in a unified way.