Internal Universes in Models of Homotopy Type Theory (1801.07664v4)

Abstract: We begin by recalling the essentially global character of universes in various models of homotopy type theory, which prevents a straightforward axiomatization of their properties using the internal language of the presheaf toposes from which these model are constructed. We get around this problem by extending the internal language with a modal operator for expressing properties of global elements. In this setting we show how to construct a universe that classifies the Cohen-Coquand-Huber-M\"ortberg (CCHM) notion of fibration from their cubical sets model, starting from the assumption that the interval is tiny - a property that the interval in cubical sets does indeed have. This leads to an elementary axiomatization of that and related models of homotopy type theory within what we call crisp type theory.

• The paper constructs internal universes in models of Homotopy Type Theory, particularly cubical sets, by introducing a modal operator to handle the challenges posed by their global nature.
• It proposes a new dual-context "crisp" type theory to overcome limitations of internal universes, allowing modal constructs to separate global from local elements.
• The construction of internal universes of fibrant types relies on the "tiny interval" property of cubical sets and is formally verified using the Agda-flat theorem prover.

An Analysis of Internal Universes in Models of Homotopy Type Theory

The paper "Internal Universes in Models of Homotopy Type Theory" by Daniel R. Licata, Ian Orton, Andrew M. Pitts, and Bas Spitters advances the paper of type theory by addressing the construction of internal universes within the frameworks of Homotopy Type Theory (HoTT), especially using cubical sets. This research navigates through the complexities inherent in the axiomatization of universes in type theory models, particularly targeting the challenges that arise from the global nature of these universes.

Introduction and Motivation

The foundational motivation springs from Voevodsky's univalence axiom in HoTT, wherein invariance under isomorphism is a core tenet for constructions on structured types. The paper explores the semantics and computational interpretation of this axiom through various models, noting that existing models are yet to provide a fully satisfactory formal treatment of universes. This is mainly due to the universes' inherent global nature that resists straightforward internal axiomatization using internal languages of presheaf toposes.

Technical Overview

The paper presents a pathway to overcome the challenge by introducing a modal operator into the internal language, enabling the articulation of properties for global elements. By this approach, internal universes can classify the CCHM notion of fibration within cubical sets models, contingent upon the interval being 'tiny'—a property that indeed holds in those sets.

The primary contributions are structured as follows:

1. Definitions and Previous Work: The authors detail existing models of HoTT, specifically cubical sets, and discuss the lack of straightforward axiomatization for universes therein. The paper builds on work by making fibrant types explicit through external structures classified by a type, thus facilitating the construction of cubical set models of type theory using intensional Martin-Löf type theory (IMLTT).
2. Crisp Type Theory: To address the no-go theorem that restricts the existence of internal universes, the authors propose a new dual-context type theory. This allows for the expression of universes via type theory by treating these universes as modal objects that separate global elements from local ones. Crisp type theory enables the application of modal constructs to type-theoretical languages, allowing internal constructions to faithfully model the pathological examples that prevent internal universality.
3. Universe Construction and Tiny Interval Assumption: The type-theoretic construction of universes is achieved by leveraging the tiny interval property. This involves constructing a universe of fibrant types that classifies global fibrations using a right adjoint to the path constructor, extending the usefulness of adjunctions in type-theoretical settings.
4. Verification and Implementation: Theoretical results are verified using Agda-flat, a fork of Agda supporting the modal reasoning required by crisp type theory. This ensures formal soundness and checks the correctness of the theoretical constructs posited by the authors.

Implications and Future Work

The methodologies proposed have significant implications for both theoretical frameworks and practical implementations in HoTT. By facilitating universe hierarchies within crisp type theory, the research opens avenues for more intricate models like directed type theory and theories with hierarchies of universes.

Future research could explore extending these models to encompass various notions of fibrations and experiment with alternative topological and model-theoretic constructs. Understanding the richness of interactions between these structures could yield new insights into the computational content of type theories and their applications in synthetic approaches to mathematics.

In conclusion, the paper establishes a novel modal framework for understanding and constructing internal universes in models of type theory. It advances the field by addressing long-standing limitations and proposing a robust mechanism for greater flexibility and depth in modeling the universe within the axiomatically rich context of HoTT.