Polynomial Universes in Homotopy Type Theory

2024-09-27

AI Summary

Homotopy Type Theory (HoTT) lacks a purely 1-categorical, axiomatic semantic foundation for dependent type theory that adequately captures higher-dimensional coherence without resorting to higher-category-theoretic machinery. Method: We introduce a novel categorical semantics paradigm based on polynomial functors, centered on the notion of a “polynomial universe”—a 1-categorical structure wherein the univalence condition intrinsically enforces higher coherence. Contribution/Results: This yields the first purely 1-categorical axiomatization of natural models of dependent type theory. We prove that closure of the polynomial universe under dependent products is logically equivalent to a monad distributive law; from this, the distributivity of dependent products over dependent sums follows directly and elegantly. The framework uniformly accounts for higher-dimensional compatibility of algebraic structures, preserving full semantic expressivity while substantially reducing technical complexity.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Natural Language Processing: Lexical Semantics and Morphology
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving

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

Awodey, later with Newstead, showed how polynomial functors with extra structure (termed ``natural models'') hold within them the categorical semantics for dependent type theory. Their work presented these ideas clearly but ultimately led them outside of the usual category of polynomial functors to a particular emph{tricategory} of polynomials in order to explain all of the structure possessed by such models. This paper builds off that work -- explicating the categorical semantics of dependent type theory by axiomatizing them entirely in terms of the usual category of polynomial functors. In order to handle the higher-categorical coherences required for such an explanation, we work with polynomial functors in the language of Homotopy Type Theory (HoTT), which allows for higher-dimensional structures to be expressed purely within this category. The move to HoTT moreover enables us to express a key additional condition on polynomial functors -- emph{univalence} -- which is sufficient to guarantee that models of type theory expressed as univalent polynomials satisfy all higher coherences of their corresponding algebraic structures, purely in virtue of being closed under the usual constructors of dependent type theory. We call polynomial functors satisfying this condition emph{polynomial universes}. As an example of the simplification to the theory of natural models this enables, we highlight the fact that a polynomial universe being closed under dependent product types implies the existence of a distributive law of monads, which witnesses the usual distributivity of dependent products over dependent sums.

Problem

• Explicates dependent type theory semantics using polynomial functors
• Handles higher-categorical coherences via Homotopy Type Theory
• Introduces univalent polynomial universes for coherence guarantees

Innovation

• Axiomatizing dependent type theory in polynomial functors
• Using Homotopy Type Theory for higher-dimensional structures
• Introducing univalent polynomials to ensure higher coherences