Realization of relational presheaves

2025-12-09

AI Summary

This paper addresses the challenge of unifying geometric structures with syntactic and semantic formalisms in modeling concurrent systems. We propose an implementation theory based on relational presheaves—generalizations of classical presheaves defined via lax functors from index categories into the category of sets and binary relations—enabling natural representation of unbounded and multi-bordered concurrent behaviors. Our primary contribution is the first construction of an implementation functor for relational presheaves, rigorously proven to yield a Cartesian theory model in the opposite category. Crucially, this functor assigns a precise syntactic counterpart to the semantic “inflation” operation, thereby achieving structural unification between geometric modeling and symbolic reasoning. The approach integrates category-theoretic methods, cocomplete category techniques, and higher-order logical inference, establishing a novel paradigm for semantic analysis of concurrent systems.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

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: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

Relational presheaves generalize traditional presheaves by going to the category of sets and relations (as opposed to sets and functions) and by allowing functors which are lax. This added generality is useful because it intuitively allows one to encode situations where we have representables without boundaries or with multiple boundaries at once. In particular, the relational generalization of precubical sets has natural application to modeling concurrency. In this article, we study categories of relational presheaves, and construct realization functors for those. We begin by observing that they form the category of set-based models of a cartesian theory, which implies in particular that they are locally finitely presentable categories. By using general results from categorical logic, we then show that the realization of such presheaves in a cocomplete category is a model of the theory in the opposite category, which allows characterizing situations in which we have a realization functor. Finally, we explain that our work has applications in the semantics of concurrency theory. The realization namely allows one to compare syntactic constructions on relational presheaves and geometric ones. Thanks to it, we are able to provide a syntactic counterpart of the blowup operation, which was recently introduced by Haucourt on directed geometric semantics, as way of turning a directed space into a manifold.

Problem

• Generalizing presheaves to relational presheaves for modeling concurrency
• Constructing realization functors for relational presheaves in cocomplete categories
• Applying relational presheaves to provide syntactic counterparts for geometric semantics

Innovation

• Relational presheaves generalize presheaves using sets and relations
• Realization functors constructed for relational presheaves categories
• Syntactic counterpart of blowup operation for concurrency semantics