[Paper Review] Separating Path and Identity Types in Presheaf Models of Univalent Type Theory

Andrew Swan|arXiv (Cornell University)|Aug 2, 2018

Logic, programming, and type systems20 references4 citations

TL;DR

This paper demonstrates that in presheaf models of univalent type theory—particularly over the first and second Kleene algebras—path types cannot serve as identity types when propositional truncation is present. Using Brouwerian counterexamples and realizability arguments, it proves that the reflexivity map cannot be shown constructively to be a trivial cofibration, thereby ruling out a direct identification of path and identity types in such models.

ABSTRACT

We give a collection of results regarding path types, identity types and univalent universes in certain models of type theory based on presheaves. The main result is that path types cannot be used directly as identity types in any Orton-Pitts style model of univalent type theory with propositional truncation in presheaf assemblies over the first and second Kleene algebras. We also give a Brouwerian counterexample showing that there is no constructive proof that there is an Orton-Pitts model of type theory in presheaves when the universe is based on a standard construction due to Hofmann and Streicher, and path types are identity types. A similar proof shows that path types are not identity types in internal presheaves in realizability toposes as long as a certain universe can be extended to a univalent one. We show that one of our key lemmas has a purely syntactic variant in intensional type theory and use it to make some minor but curious observations on the behaviour of cofibrations in syntactic categories.

Motivation & Objective

• To investigate whether path types can be used as identity types in Orton-Pitts style models of univalent type theory based on presheaves.
• To identify conditions under which the reflexivity map $ r^X: X \to X^\mathbb{I} $ fails to be a trivial cofibration constructively.
• To show that in realizability toposes and presheaf assemblies, path types cannot serve as identity types when univalent universes are present.
• To explore the limitations of using path types as identity types in constructive type theories with univalence and propositional truncation.
• To clarify the tension between univalence, canonicity, and the definitional identification of path and identity types in type-theoretic models.

Proposed method

• Constructs a Brouwerian counterexample to show that the inclusion of constant paths into all paths is not pointwise decidable in certain realizability models.
• Uses the assumption that path types are identity types to derive that certain maps must be cofibrations, leveraging the structure of fibrations and lifting properties.
• Applies univalent universes to show that cofibrations are pointwise stable under double negation, leading to non-constructive consequences.
• Combines these techniques to derive results that are constructively invalid in realizability models, such as the failure of decidable image for trivial cofibrations.
• Employs a syntactic variant of a key lemma in intensional type theory to analyze cofibration behavior in syntactic categories.
• Analyzes models based on presheaves over the first and second Kleene algebras, focusing on cubical assemblies and realizability toposes.

Experimental results

Research questions

• RQ1 Can path types be identified with identity types in Orton-Pitts style models of univalent type theory with propositional truncation in presheaf assemblies over the first and second Kleene algebras?
• RQ2 Is there a constructive proof that the reflexivity map $ r^X: X \to X^\mathbb{I} $ is a trivial cofibration when path types are used as identity types?
• RQ3 What are the limitations of using path types as identity types in realizability toposes when a univalent universe is present?
• RQ4 To what extent can the regularity condition (e.g., $ \sigma $-rule) be used to reconcile path types with identity types while preserving univalence?
• RQ5 Can a type theory satisfy univalence, good computational properties (like canonicity and decidable type checking), and definitional isomorphism between path and identity types simultaneously?

Key findings

• In presheaf assemblies over the first and second Kleene algebras, path types cannot be used as identity types when propositional truncation is present.
• A Brouwerian counterexample shows that there is no constructive proof that the reflexivity map $ r^X $ is a trivial cofibration in such models.
• The assumption that path types are identity types forces certain maps to be cofibrations, which leads to non-constructive consequences in realizability models.
• In internal presheaves in realizability toposes, path types are not identity types if a univalent universe can be extended to a univalent one.
• A key lemma from the paper has a purely syntactic variant in intensional type theory, which reveals subtle behavior of cofibrations in syntactic categories.
• The results suggest that it is impossible to simultaneously satisfy univalence, good computational properties (like canonicity and decidable type checking), and definitional identification of path and identity types in a single type theory.