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

2026-02-12

AI Summary

This work aims to unify the treatment of substitution for syntactic objects—such as λ-terms—that involve both free and bound variables across diverse mathematical frameworks, including presheaves, nominal sets, and renaming sets. To this end, the authors propose a general categorical approach: by leveraging actions of monoidal categories to derive closed monoidal structures, they systematically construct substitution mechanisms applicable to a wide range of semantic models. Key contributions include the first construction of a Fiore–Plotkin–Turi-style substitution tensor for nominal sets, the establishment of novel equivalences among nominal sets, renaming sets, and presheaf categories, and the successful recovery and generalization of existing substitution structures in presheaf models, yielding a series of new correspondences between substitution semantics and categorical frameworks.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Sustainability and carbon-aware systems for Web, mobile, and WoT

Abstract

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.

Problem

• substitution
• nominal sets
• presheaves
• monoidal structure
• bound variables

Innovation

• closed monoidal structure
• substitution tensor
• nominal sets
• presheaves
• categorical semantics