We observe that the existence of sequential and parallel composition supermaps in higher order physics can be formalised using enriched category theory. Encouraged by physically 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 are 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 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，高阶物理学中顺序与并行复合超映射的存在性可通过富范畴理论进行形式化。受物理相关实例（如酉超映射及高阶因果范畴中的层结构）启发，我们将高阶物理理论建模视为富幺半范畴的类比——正如物理理论通过幺半范畴建模。我们利用富幺半范畴框架，通过格罗滕迪克构造法构建了高阶物理理论间保结构映射的恰当定义。继而证明：高阶物理理论中便捷的柯里化特性，可被视为并行与顺序复合超映射存在的基本假设与链接附加特征相结合的产物。在第二个应用中，我们运用保结构映射定义证明：包含带完全忠实保结构映射的无限塔式富幺半范畴的范畴，必然导引出闭幺半结构。所提定义旨在为量子理论中新型因果结构的研究与比较提供广义框架，更广泛而言，旨在建立一种将静态与动态特征统一处理的物理理论范式。