A Unified Treatment of Substitution for Presheaves, Nominal Sets, Renaming Sets, and so on

Presheaves and nominal sets provide alternative abstract models of sets of syntactic objects with free and bound variables, such as λ-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, 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 the process, we shed new light on different incarnations of nominal sets and (pre-)sheaf categories and establish a number of correspondences between them.