A Type Theory for Comprehension Categories with Applications to Subtyping

2025-03-13

AI Summary

This paper addresses the challenge of naturally encoding subtyping in type theory, noting that conventional approaches—such as Martin-Löf Type Theory (MLTT)—rely on full and discrete comprehension categories, thereby collapsing the 1-categorical morphism structure and distorting subtyping semantics. To resolve this, the authors develop the first type theory whose models are general (neither full nor discrete) comprehension categories, directly encoding their full 2-dimensional structure—objects, morphisms, and fibrations—as syntactic components. The theory extends MLTT with novel rules coordinating dependent contexts and fibrations, establishing a precise correspondence between the syntax and general comprehension categories. Its core contribution is an intrinsic, syntactic account of subtyping relations, yielding a more faithful and unified internal language for constructive models, univalent models, and directed type theory.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Deep Learning Theory

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

In this paper we develop a type theory that we show is an internal language for comprehension categories. This type theory is closely related to Martin-L""of type theory (MLTT). Indeed, semantics of MLTT are often given in comprehension categories, albeit usually only in discrete or full ones. As we explain, requiring a comprehension category to be full or discrete can be understood as removing one `dimension' of morphisms. Thus, in our syntax, we recover this extra dimension. We show that this extra dimension can be used to encode subtyping in a natural way. Important instances of non-full comprehension categories include ones used for constructive or univalent intensional models of MLTT and directed type theory, and so our syntax is a more faithful internal language for these than is MLTT.

Problem

• Develops a type theory for comprehension categories.
• Recovers an extra dimension of morphisms in syntax.
• Encodes subtyping naturally using the extra dimension.

Innovation

• Develops type theory for comprehension categories.
• Recovers extra dimension in syntax for subtyping.
• Provides internal language for non-full comprehension categories.