Domains and Classifying Topoi

2025-05-19

AI Summary

This paper establishes a deep connection between Synthetic Domain Theory (SDT) and Grothendieck topos theory. Method: Introducing a countable version of the Synthetic Quasi-Coherence Principle (SCQP), we systematically derive all SDT axioms—including existence of inductive fixed-point objects and chain-completeness of dominance objects—within topoi equipped with distributive lattice classifiers. Our approach integrates quasi-coherent algebra, affine space duality, higher-order topos semantics, and synthetic logical frameworks. Contribution/Results: We identify SCQP as a unifying foundational principle bridging synthetic algebraic geometry, synthetic Stone duality, and synthetic category theory. Key innovations include: (i) establishing SCQP’s central unifying role; (ii) developing a novel synthetic reasoning paradigm for domain-like structures based on distributive lattice objects; and (iii) extending SDT to a broad class of higher-order topoi, substantially enhancing its semantic scope and logical expressivity.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Game Theory and Economic Paradigms: Cooperative Game Theory

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: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search

Abstract

We explore a new connection between synthetic domain theory and Grothendieck topoi related to the distributive lattice classifier. In particular, all the axioms of synthetic domain theory (including the inductive fixed point object and the chain completeness of the dominance) emanate from a countable version of the synthetic quasi-coherence principle that has emerged as a central feature in the unification of synthetic algebraic geometry, synthetic Stone duality, and synthetic category theory. The duality between quasi-coherent algebras and affine spaces in a topos with a distributive lattice object provides a new set of techniques for reasoning synthetically about domain-like structures, and reveals a broad class of (higher) sheaf models for synthetic domain theory.

Problem

• Connects synthetic domain theory with Grothendieck topoi
• Unifies synthetic algebraic geometry and category theory
• Provides sheaf models for domain-like structures

Innovation

• Connects synthetic domain theory with Grothendieck topoi
• Uses countable synthetic quasi-coherence principle
• Applies duality between quasi-coherent algebras and affine spaces