Presheaves and nominal sets provide alternative abstract models of sets of syntactic objects with free and bound variables, such as lambda-terms. One distinguishing feature of the presheaf-based perspective is its elegant syntax-free characterization of substitution using a closed monoidal structure. In this paper, we introduce a corresponding closed monoidal structure on nominal sets, modeling substitution in the spirit of Fiore et al.'s substitution tensor for presheaves over finite sets. To this end, we present a general method to derive a closed monoidal structure on a category from a given action of a monoidal category on that category. We demonstrate that this method not only uniformly recovers known substitution tensors for various kinds of presheaf categories, but also yields notions of substitution tensor for nominal sets and their relatives, such as renaming sets. In doing so, we shed new light on different incarnations of nominal sets and (pre-)sheaf categories and establish a number of known and new correspondences between them.

翻译：预层与名义集为具有自由变量和约束变量的语法对象集合（如λ项）提供了替代的抽象模型。预层视角的一个显著特征在于，它通过封闭幺半群结构对替换操作给出了无语法的优雅刻画。本文针对名义集引入相应的封闭幺半群结构，以Fiore等人为有限集上预层定义的替换张量为精神，在名义集上模拟替换操作。为此，我们提出一种通用方法，从幺半群范畴对该范畴的给定作用出发，导出该范畴上的封闭幺半群结构。我们证明该方法不仅能统一地恢复各类预层范畴的已知替换张量，还能为名义集及其相关结构（如重命名集）导出替换张量的概念。通过这一工作，我们为名义集与（预）层范畴的不同实现提供了全新视角，并建立了它们之间若干已知及全新的对应关系。