Category Archives: Univalence

Universe n is not an n-Type

Joint work with Christian Sattler Some time ago, at the UF Program in Princeton, I presented a proof that Universe n is not an n-type. We have now formalized that proof in Agda and want to present it here. One … Continue reading

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Homotopy Theory in Homotopy Type Theory: Introduction

Many of us working on homotopy type theory believe that it will be a better framework for doing math, and in particular computer-checked math, than set theory or classical higher-order logic or non-univalent type theory. One reason we believe this … Continue reading

Posted in Applications, Higher Inductive Types, Homotopy Theory, Uncategorized, Univalence | 3 Comments

Isomorphism implies equality

Thierry Coquand and I have proved that, for a large class of algebraic structures, isomorphism implies equality (assuming univalence). A class of algebraic structures Structures in this class consist of a type, some operations on this type, and propositional axioms … Continue reading

Posted in Applications, Code, Univalence | 13 Comments

Modeling Univalence in Inverse Diagrams

I have just posted the following preprint, which presents new set-theoretic models of univalence in categories of simplicial diagrams over inverse categories (or, more generally, diagrams over inverse categories starting from any existing model of univalence). The univalence axiom for … Continue reading

Posted in Models, Paper, Univalence | 5 Comments

Modeling Univalence in Subtoposes

In my recent post at the n-Category Café, I described a notion of “higher modality” in type theory, which semantically ought to represent a left-exact-reflective sub–category of an -topos — once we can prove that homotopy type theory has models … Continue reading

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Canonicity for 2-Dimensional Type Theory

A consequence of the univalence axiom is that isomorphic types are equivalent (propositionally equal), and therefore interchangable in any context (by the identity eliminaiton rule J). Type isomorphisms arise frequently in dependently typed programming, where types are often refined to … Continue reading

Posted in Foundations, Programming, Univalence | 8 Comments

Oberwolfach Report

Richard Garner has now completed and posted the report from the Oberwolfach meeting.

Posted in Foundations, Univalence | 1 Comment