Univalence without function extensionality

2026-05-01

AI Summary

This study investigates whether categorical univalence—a weakened form of the univalence axiom—entails function extensionality. By adapting von Glehn’s polynomial model construction, the authors build a countermodel over a base type theory equipped with univalent universes, demonstrating the existence of a model satisfying categorical univalence while refuting function extensionality. This result establishes the logical independence of these two principles within Martin-Löf type theory, thereby disproving the conjecture that categorical univalence implies function extensionality. The findings contribute a refined understanding of the logical structure underlying univalent foundations.

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.

Problem

• univalence
• function extensionality
• categorical univalence
• homotopy type theory
• polynomial model

Innovation

• categorical univalence
• function extensionality
• polynomial model
• univalent universe
• homotopy type theory