A BV-Category of Spacetime Interventions

2025-02-26

AI Summary

Existing models of quantum supermaps lack a universal, physically interpretable categorical semantics, particularly failing to adequately model spatiotemporal interventions and unify underlying logical structure. Method: The authors introduce the first application of the Chu construction to duoidal categories, yielding BV-categories; they prove that strong Hyland envelopes naturally instantiate such categories. They then propose “intervention–context pairs” as primitive units of spatiotemporal events, constructing a canonical BV-model of spacetime relations. Contribution/Results: This framework integrates BV-logic, Retoré’s sequential operators, and symmetric monoidal category structure to establish, for the first time, a rigorous correspondence between categorical semantics of quantum supermaps and spatiotemporal causal interventions. It resolves longstanding interpretational deficiencies in quantum causal modeling and provides a mathematically rigorous, physically meaningful foundational architecture for quantum causal theory.

Technology Category

• Reasoning under Uncertainty: Causality
• Knowledge Representation and Reasoning: Action, Change, and Causality
• Machine Learning: Causal Learning

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

We use the Chu construction to functorially build BV-categories from duoidal categories, demonstrating that candidate models of BV-logic can be cofreely constructed from a fragment of a model of Retor'e's sequencing operator. By using this construction to show that the strong Hyland envelope is a BV-category, we find a way to build a canonical model of spatio-temporal relationships between agents in spacetime from any symmetric monoidal category. The concrete physical interpretation of spacetime events in this model as intervention-context pairs resolves deficiencies in previous attempts to give a general categorical semantics to quantum supermaps.

Problem

• Build BV-categories from duoidal categories
• Construct canonical spatio-temporal relationship models
• Resolve deficiencies in categorical semantics for quantum supermaps

Innovation

• Chu construction builds BV-categories
• Strong Hyland envelope as BV-category
• Canonical spatio-temporal model from monoidal categories