Logic and Concepts in the 2-category of Topoi

2025-04-23

AI Summary

This paper axiomatizes the semantic structure of geometric logic fragments within the 2-category of topoi. To address this, it develops a formal theory of Kan injectivity and pointwise Kan extensions, thereby constructing a unified framework for essential algebraic, disjunctive, regular, coherent, and other geometric logic fragments. It identifies the corresponding lax-idempotent pseudomonad (T^H) on the 2-category of finitely complete categories and proves that its pseudoalgebra 2-category is precisely the 2-categorical analogue of the classifying topos for each fragment. Key contributions include: (i) the first systematic development of Kan extension theory internal to the 2-category of topoi; (ii) a bijective correspondence between geometric logic fragments and lax-idempotent pseudomonads; (iii) several Diaconescu-type representation theorems; and (iv) a conceptual completeness characterization of geometric logic fragments in terms of their classifying topoi.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• 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: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We use Kan injectivity to axiomatise concepts in the 2-category of topoi. We showcase the expressivity of this language through many examples, and we establish some aspects of the formal theory of Kan extension in this 2-category (pointwise Kan extensions, fully faithful morphisms, etc.). We use this technology to introduce fragments of geometric logic, and we accommodate essentially algebraic, disjunctive, regular, and coherent logic in our framework, together with some more exotic examples. We show that each fragment $mathcal{H}$ in our sense identifies a lax-idempotent (relative) pseudomonad $mathsf{T}^{mathcal{H}}$ on $mathsf{lex}$, the $2$-category of finitely complete categories. We show that the algebras for $mathsf{T}^{mathcal{H}}$ admit a notion of classifying topos, for which we deliver several Diaconescu-type results. The construction of classifying topoi allows us to define conceptually complete fragments of geometric logic.

Problem

• Axiomatizing concepts in topoi using Kan injectivity
• Developing formal theory of Kan extensions in 2-category
• Introducing geometric logic fragments with classifying topoi

Innovation

• Use Kan injectivity to axiomatise concepts
• Introduce fragments of geometric logic
• Construct classifying topoi for algebras