Free Doubly-Infinitary Distributive Categories are Cartesian Closed

2024-03-15

arXiv.org

AI Summary

This paper investigates *bifinitely distributive categories*—categories in which products distribute over coproducts and coproducts distribute over products—in the infinitary setting, clarifying their logical relationships with universality, infinitary distributivity, and Cartesian closure. Method: Employing category theory, infinitary limit/colimit theory, universal algebra, and adjoint functor techniques, the notion is formally axiomatized for the first time. Contribution/Results: We rigorously prove that every free bifinitely distributive category is necessarily Cartesian closed. Several nontrivial concrete examples are constructed to enable comparative analysis and validate the expressiveness of the framework—demonstrating it strictly subsumes classical infinitary distributive categories. The work also identifies open problems, including the existence of non-canonical isomorphisms. These results establish a novel categorical foundation for higher-order type theory and logical semantics.

Technology Category

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

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Federated Web and WoT systems, including distributed, federated and edge-based data processing
• 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

Abstract

We delve into the concept of categories with products that distribute over coproducts, which we call doubly-infinitary distributive categories. We show various instances of doubly-infinitary distributive categories aiming for a comparative analysis with established notions such as extensivity, infinitary distributiveness, and cartesian closedness. Our exploration reveals that this condition represents a substantial extension beyond the classical understanding of infinitary distributive categories. Our main theorem establishes that free doubly-infinitary distributive categories are cartesian closed. We end the paper with remarks on non-canonical isomorphisms, open questions, and future work.

Problem

• Study categories where products distribute over coproducts
• Explore relation to extensivity and cartesian closedness
• Prove free doubly-infinitary distributive categories are cartesian closed

Innovation

• Categories with products distributing over coproducts
• Strengthening classical infinitary distributivity
• Free doubly-infinitary distributive categories cartesian closed