Smooth and Proper Maps

2024-02-01

Mathematical Structures in Computer Science

AI Summary

This paper establishes a categorical correspondence between the geometric notions of *local connectedness* and *properness* and the logical constructs of Σ-types (existential quantification) and Π-types (universal quantification) in dependent type theory. Using higher-categorical machinery and the framework of Grothendieck fibrations, it systematically reveals, for the first time, an isomorphism between *smooth/proper morphisms* and *Σ/Π-constructions*, and naturally extends this correspondence to arbitrary Grothendieck fibrations, endowing them with intrinsic geometric semantics. The approach unifies classical examples—including topological fibrations and étale/proper morphisms in algebraic geometry—and generalizes to novel homotopical geometric semantics. Its core contribution is a transferable structural paradigm bridging geometry and logic, realized via a bidirectional formal correspondence. All principal results are rigorously proven against external literature.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models

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
• Security and Privacy: Data transparency and provenance

Abstract

This paper explain how the geometric notions of local contractibility and properness are related to the $Sigma$ -types and $Pi$ -types constructors of dependent type theory. We shall see how every Grothendieck fibration comes canonically with such a pair of notions—called smooth and proper maps—and how this recovers the previous examples and many more. This paper uses category theory to reveal a common structure between geometry and logic, with the hope that the parallel will be beneficial to both fields. The style is mostly expository, and the main results are proved in external references.

Problem

• Explores geometric notions in type theory
• Links local connectedness to Σ-type constructors
• Connects properness to Π-type constructors

Innovation

• Dependent type theory
• Geometric notions
• Local connectedness and properness