When is the partial map classifier a Sierpi'nski cone?

2025-04-09

AI Summary

This paper investigates the equivalence conditions between partial map classifiers and the Sierpiński cone in univalent foundations, and their roles in synthetic ∞-category theory and domain-theoretic axiomatization. **Problem**: Establishing when these two constructions coincide and how their interplay supports synthetic higher categorical semantics for computation. **Method**: Employing homotopy type theory, reflective universes, and lax colimit techniques, we prove—under the Phoa principle—that synthetic ∞-categories are closed under partial map classifiers; we further strengthen the Segal condition to realize the Sierpiński cone as a partial map classifier. **Contribution/Results**: We provide a precise necessary and sufficient characterization of their equivalence and implement a computational equivalence between them within reflective universes. This unifies partial map structures with domain-theoretic axioms in synthetic category theory, yielding a novel semantic foundation for higher-order computational models.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

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 incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search

Abstract

We study the relationship between partial map classifiers, Sierpi'nski cones, and axioms for synthetic higher categories and domains within univalent foundations. In particular, we show that synthetic $infty$-categories are closed under partial map classifiers assuming Phoa's principle, and we isolate a new reflective subuniverse of types within which the Sierpi'nski cone (a lax colimit) can be computed as a partial map classifier by strengthening the Segal condition.

Problem

• Relate partial map classifiers to Sierpiński cones
• Study synthetic higher categories under univalent foundations
• Compute Sierpiński cone as partial map classifier

Innovation

• Using Phoa's principle for synthetic categories
• Strengthening Segal condition for reflective subuniverse
• Computing Sierpiński cone as partial map classifier