A Mathematical Framework for Transformations of Physical Processes

We observe that the existence of sequential and parallel composition supermaps in higher order physics can be formalized using enriched category theory. Encouraged by physically relevant examples such as unitary supermaps and layers within higher order causal categories (HOCCs), we treat the modeling of higher order physics with enriched monoidal categories in analogy with the process theoretic framework in which physical theories are modeled 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". In a second application, we show more generally that categories containing infinite towers of enriched monoidal categories with full and faithful structure-preserving maps between them inevitably lead to closed monoidal structure. The aim of the proposed definitions is to give a broad framework for the study and comparison of novel causal structures in quantum theory, and, more broadly, provide a paradigm of physical theory where static and dynamical features are treated in a unified way.