Univalence without function extensionality

Abstract: It is a well-known theorem of homotopy type theory, originally due to Voevodsky, that function extensionality holds inside any univalent universe. We consider a weaker variant of the univalence axiom, asserting that the wild category formed by the universe is univalent, which we call categorical univalence. We show that categorical univalence does not imply function extensionality by an analysis of Von Glehn's polynomial model construction, which produces models of Martin-Löf type theory that always refute function extensionality. We find in particular that when the base model has a univalent universe, its polynomial model has a universe that is categorically univalent but lacks function extensionality.

• The paper establishes that categorical univalence does not imply function extensionality using sophisticated polynomial model constructions.
• It introduces familial categorical univalence to extend the analysis to families of types and clearly separates it from function extensionality.
• The findings refine the foundations of Homotopy Type Theory by delineating a precise dependency structure among various univalence principles.

Univalence without Function Extensionality: An Expert Summary

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Introduction and Motivation

The relationship between the univalence axiom (UA) and function extensionality (FE) is a fundamental aspect of Homotopy Type Theory (HoTT). Voevodsky's original formulation of univalence asserted that for intensional Martin-Löf Type Theory (ITT), univalence implies function extensionality: two dependent functions are equal if they are pointwise equal (homotopic). However, it has been unclear whether this implication is purely structural or an artifact of UA's formulation. Specifically, the question is open as to whether weaker versions of univalence, such as categorical univalence (CUA), suffice to derive FE, or whether they are strictly independent.

The paper "Univalence without function extensionality" (2605.00812) rigorously investigates this distinction, introducing categorical univalence as a weakening of UA and demonstrating, via polynomial model constructions, that categorical univalence does not imply function extensionality. The work not only resolves open questions dating back to Dorais (2013) but also systematizes the interaction between univalence principles and extensionality axioms using advanced categorical models.

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Structural Decomposition of Univalence

The univalence axiom for a universe $\mathcal{U}$, denoted $\mathsf{UA}_{\mathcal{U}}$, asserts that the canonical function $$\mathsf{idToEquiv}: (A =_{\mathcal{U}} B) \to (A \simeq B)$$ is an equivalence for all $A, B : \mathcal{U}$, where $A \simeq B$ denotes homotopy equivalence. Function extensionality (FE), on the other hand, asserts that pointwise homotopic (dependent) functions are equal.

Dorais's proposal is a structural weakening: categorical univalence ($\mathsf{CUA}_{\mathcal{U}}$) replaces homotopy equivalence in UA with categorical equivalence, i.e., invertible functions up to strictly equal two-sided inverses rather than homotopies, corresponding to isomorphisms in the ambient wild category of types and functions.

The authors show that in the presence of FE, the logical triangle

$$A =_{\mathcal{U}} B \to^{\mathsf{idToCEq}} A \cong B \to^{\mathsf{CEqToEq}} A \simeq B$$

consists of equivalences, but without FE, the passage from categorical equivalence to homotopy equivalence is nontrivial.

They formalize "familial categorical univalence" ($\mathsf{FamCUA}_{\mathcal{U}}$), extending CUA to all $\mathcal{U}^I$ for $I: \mathcal{U}$ (i.e., to all families of types), and analyze its status relative to FE.

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Von Glehn's Polynomial Model: Construction and Consequences

To separate CUA (and its familial variant) from FE, the authors deeply analyze Von Glehn's polynomial model construction $\mathsf{UA}_{\mathcal{U}}$0, which, given a suitable base model $\mathsf{UA}_{\mathcal{U}}$1 of ITT with specific coproduct properties, builds a new model $\mathsf{UA}_{\mathcal{U}}$2.

Key properties of this construction include:

• It always refutes function extensionality, regardless of the base model, leveraging the structure of "topgun types" that can separate homotopic but judgmentally distinct functions.
• The construction preserves familial categorical univalence when the base model possesses it.

Formally, they show that if the base model $\mathsf{UA}_{\mathcal{U}}$3, then $\mathsf{UA}_{\mathcal{U}}$4, but $\mathsf{UA}_{\mathcal{U}}$5. This is achieved by a precise analysis of the wild category structures, morphisms, and isomorphism types in the polynomial model.

By instantiating $\mathsf{UA}_{\mathcal{U}}$6 with a univalent (cubical) model satisfying FE and univalence, they construct a model supporting familial categorical univalence while failing to satisfy function extensionality, thereby establishing the independence.

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Implications, Variants, and Hierarchy of Univalence Principles

The study highlights several non-equivalent weakenings of univalence:

• $\mathsf{UA}_{\mathcal{U}}$7: the universe is a univalent wild category (up to strict invertibility).
• $\mathsf{UA}_{\mathcal{U}}$8: univalence for all wild categories of families.
• $\mathsf{UA}_{\mathcal{U}}$9: strengthening $$\mathsf{idToEquiv}: (A =_{\mathcal{U}} B) \to (A \simeq B)$$0 by demanding $$\mathsf{idToEquiv}: (A =_{\mathcal{U}} B) \to (A \simeq B)$$1 be a categorical equivalence, which is strictly stronger, but still does not imply FE.
• $$\mathsf{idToEquiv}: (A =_{\mathcal{U}} B) \to (A \simeq B)$$2: approximate univalence, demanding only a section for $$\mathsf{idToEquiv}: (A =_{\mathcal{U}} B) \to (A \simeq B)$$3 up to homotopy.

The constructed models provide strict separations between these properties:

• $$\mathsf{idToEquiv}: (A =_{\mathcal{U}} B) \to (A \simeq B)$$4 does not imply $$\mathsf{idToEquiv}: (A =_{\mathcal{U}} B) \to (A \simeq B)$$5
• $$\mathsf{idToEquiv}: (A =_{\mathcal{U}} B) \to (A \simeq B)$$6 does not imply $$\mathsf{idToEquiv}: (A =_{\mathcal{U}} B) \to (A \simeq B)$$7
• $$\mathsf{idToEquiv}: (A =_{\mathcal{U}} B) \to (A \simeq B)$$8 is strictly stronger than $$\mathsf{idToEquiv}: (A =_{\mathcal{U}} B) \to (A \simeq B)$$9 but does not reach $A, B : \mathcal{U}$0
• The status of $A, B : \mathcal{U}$1 relative to FE remains open.

Moreover, the work identifies that the polynomial and parametric exceptional model constructions systematically destroy function extensionality while preserving certain categorical forms of univalence. The authors demonstrate careful preservation and failure properties via explicit manipulation of the underlying contexts, types, and categorical actions.

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Theoretical and Practical Consequences

The logical independence of function extensionality from categorical univalence demonstrated in this work has several foundational and practical implications for type theory:

• The traditional identification of UA with FE is contingent on the "homotopical" nature of UA; minor variations or strictifications (such as in CUA) do not generally reconstitute FE.
• Foundations of HoTT and type theory need more refined notions of univalence when FE is not available, and the architecture of polynomial and related syntactic models provides canonical counterexamples and insight into the subtleties of type equivalence.
• For formal verification or theorem proving in type theories lacking FE by design (e.g. certain realizability or computational models), adopting categorical univalence principles yields markedly weaker reasoning about equality of functions, and the interaction with universes becomes delicate.

The results suggest richer landscapes in the space of univalence principles and point towards a more granular taxonomy of categorical versus homotopical equivalence in type theory.

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Conclusion

"Univalence without function extensionality" rigorously delineates the dependency structure between various formulations of univalence and function extensionality. By leveraging the Von Glehn polynomial model, the paper constructs ITT models where categorical and even familial categorical univalence holds, while function extensionality fails. Consequently, the standard implication $A, B : \mathcal{U}$2 is not reversible for a wide array of natural weakenings of univalence. This formal separation has substantial consequences for both the semantics and the foundations of type theory, and opens a landscape for further investigations into variants of univalence and their roles both in theory and formalized mathematics.