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.

翻译：我们观察到，在变换的高阶理论中，顺序与并行组合超映射的存在性可通过富范畴论形式化。受酉超映射及高阶因果范畴（HOCCs）内层级等相关实例的启发，我们采用富幺半范畴对高阶物理理论进行建模，类比于使用幺半范畴对物理理论的建模。我们利用富幺半设定，通过格罗滕迪克构造，构建了高阶物理理论间结构保持映射的合适定义。随后证明，高阶物理理论中便捷的柯里化特征，可视为原始假设（即并行与顺序组合超映射的存在性）与额外链接特征相结合的结果。进而，我们运用所定义的结构保持映射，证明了包含无限塔状富幺半范畴的范畴，若其间存在全且忠实的结构保持映射，则必然导致闭幺半结构的产生。所提定义旨在逐步构建一个广泛框架，以用于研究和比较量子理论中的新型因果结构，并在更广义层面，为物理理论提供一种将静态与动态特征统一处理的范式。