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 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 novel 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 novel correspondences between them.

💡 Research Summary

The paper presents a unified categorical framework for modelling substitution across two major abstract settings for syntax with variables: presheaf categories and nominal sets (including the related renaming sets). The authors observe that in the presheaf approach, substitution is captured by a closed monoidal structure known as the substitution tensor, originally introduced by Fiore, Plotkin, and Turi. This tensor is defined using a “power” operation ⊳ that, for a context A and a presheaf X, yields the A‑fold product X_A, i.e., the set of A‑indexed families of elements of X. The substitution tensor X ⋄ Y is then the coequaliser of the obvious action of morphisms on terms and on substitutions, formally (X_A × (A⊳Y))/∼. The unit of this monoidal structure is the presheaf of variables, and the internal hom is given by (Y −⋄ Z)_A = Nat(Y_A, Z), the clone of operations from Y to Z.

The core contribution of the paper is a general method that extracts such a substitution tensor from any left action ⊳ of a monoidal category A on a category C. The authors prove two main theorems. First (Theorem 4.8) they show that the coequaliser construction yields a monoidal product ⋄ on C whenever the action satisfies mild coherence conditions. Second (Theorem 4.10) they introduce a property called “contextuality” – essentially the requirement that every object of C can be expressed as a suitable colimit of objects obtained by the action – and prove that under this condition the monoidal product is right‑closed, i.e., there exists an internal hom making (C, ⋄, I) a closed monoidal category.

Applying this abstract machinery, the authors uniformly recover the four known substitution tensors for presheaf categories over the finite‑set based context categories F (cartesian), I (weakening), S (contraction), and B (exchange). In each case the appropriate context category supplies the monoidal action, and contextuality holds, yielding the familiar tensors in a single conceptual framework.

The novel part of the work is the extension of the same construction to nominal sets. Nominal sets are sets equipped with an action of the finite permutation group on a countable set of names; the action ⊳ is interpreted as forming the set of elements supported by a finite set A of names. The authors verify that the contextuality condition holds for nominal sets (essentially because any nominal element has a finite support) and thus obtain a substitution tensor on Nom, together with an explicit description of the internal hom. This tensor is considerably simpler than the presheaf counterpart, because the underlying action already encodes α‑equivalence and capture‑avoiding renaming.

A further contribution is the treatment of renaming sets, which generalize nominal sets by allowing arbitrary (not necessarily injective) renamings. Here the acting monoidal category consists of finite functions between name sets, and the same construction yields a closed monoidal substitution tensor on Ren. The paper provides concrete element‑wise formulas for both the tensor and the internal hom, showing that they behave exactly as expected for simultaneous substitution.

In the final technical section the authors establish new categorical equivalences linking the presheaf and nominal worlds. They prove that presheaves over B are equivalent to nominal sets with support‑preserving maps, and presheaves over S are equivalent to renaming sets where renamings respect minimal supports. These equivalences are constructed via Yoneda embeddings, left Kan extensions, and Day convolution, and they fill gaps in the literature concerning the precise relationship between the two styles of syntax modelling.

Overall, the paper delivers a powerful, uniform recipe for building substitution tensors from monoidal actions, demonstrates its applicability to both presheaf and nominal settings, and uncovers previously unknown correspondences between them. This advances the theory of abstract syntax, providing a common language for reasoning about substitution in a wide variety of logical and computational systems, including linear, affine, and resource‑sensitive calculi. Future work may explore extensions to infinite contexts, dependent types, or higher‑order abstract syntax, leveraging the same action‑based approach.