一句话：将 props 的张量积推广为带点有向空间的 smash 乘积，统一 monoidal theories 并生成高维 coherence 数据。
Hackney 与 Robertson 曾将 operad 的 Boardman–Vogt 积推广为 prop 的张量积，使交换幺半群、双代数等理论可分解为张量积。本文为该构造给出拓扑解释并进行大幅推广：将张量积看作“带点有向空间”的 smash 乘积的低维投影。作者用 diagrammatic sets 作为有向空间的组合模型，并以 Gray 积替代笛卡尔积，从而把该运算提升到更高维理论。文章建立了 diagrammatic sets、pros、probs、props 与 Gray-categories 之间的伴随网，说明这些结构如何相互对应。关键结果是，smash 乘积可直接作用于高维理论的 presentation，并系统性地产生高维 coherence 数据。这项工作为 monoidal theories 的分解与高阶代数语义提供了统一框架。
The tensor product of props was defined by Hackney and Robertson as an extension of the Boardman–Vogt product of operads to more general monoidal theories. Theories that factor as tensor products include the theory of commutative monoids and the theory of bialgebras. We give a topological interpretation (and vast generalisation) of this construction as a low-dimensional projection of a "smash product of pointed directed spaces". Here directed spaces are embodied by combinatorial structures called diagrammatic sets, while Gray products replace cartesian products. The correspondence is mediated by a web of adjunctions relating diagrammatic sets, pros, probs, props, and Gray-categories. The smash product applies to presentations of higher-dimensional theories and systematically produces higher-dimensional coherence data.