[Paper Review] THE SIMPLICIAL MODEL OF UNIVALENT FOUNDATIONS

Krzysztof Kapulkin, Peter LeFanu Lumsdaine+1 more|arXiv (Cornell University)|Jan 1, 2014

Homotopy and Cohomology in Algebraic Topology29 references76 citations

TL;DR

This paper constructs a model of Univalent Foundations in the category of simplicial sets using a novel universe-based coherence technique. It establishes a weakly universal Kan fibration, proves the Univalence Axiom holds in this model, and shows that Univalent Foundations are consistent relative to ZFC with two inaccessible cardinals.

ABSTRACT

In this paper, we construct and investigate a model of the Univalent Foundations of Mathematics in the category of simplicial sets. To this end, we rst give a new technique for constructing models of dependent type theory, using universes to obtain coherence. We then construct a (weakly) universal Kan bration, and use it to exhibit a model in simplicial sets. Lastly, we introduce the Univalence Axiom, in several equivalent formulations, and show that it holds in our model. As a corollary, we conclude that Univalent Foundations are at least as consistent as ZFC with two inaccessible cardinals.

Motivation & Objective

• To develop a new technique for constructing models of dependent type theory using universes to ensure coherence.
• To construct a weakly universal Kan fibration in the category of simplicial sets.
• To demonstrate that the Univalence Axiom holds in this simplicial model.
• To establish the consistency of Univalent Foundations relative to ZFC with two inaccessible cardinals.

Proposed method

• Introduce a universe-based method for constructing models of dependent type theory, ensuring coherence across type constructors.
• Define and construct a weakly universal Kan fibration in the category of simplicial sets.
• Use the weakly universal Kan fibration to interpret type universes and dependent types in simplicial sets.
• Formulate the Univalence Axiom in multiple equivalent ways within the simplicial model.
• Verify that all formulations of the Univalence Axiom are satisfied in the constructed model.
• Establish that the consistency strength of Univalent Foundations does not exceed that of ZFC with two inaccessible cardinals.

Experimental results

Research questions

• RQ1 Can a coherent model of dependent type theory be constructed using universes in simplicial sets?
• RQ2 Does a weakly universal Kan fibration exist in the category of simplicial sets?
• RQ3 Are the various formulations of the Univalence Axiom equivalent in this simplicial model?
• RQ4 Does the Univalence Axiom hold in the constructed simplicial model of type theory?
• RQ5 What is the consistency strength of Univalent Foundations relative to standard set theory?

Key findings

• A new universe-based technique enables coherent modeling of dependent type theory in simplicial sets.
• A weakly universal Kan fibration is successfully constructed in the category of simplicial sets.
• The Univalence Axiom holds in the simplicial model, confirming its validity in this setting.
• Multiple formulations of the Univalence Axiom are shown to be equivalent within the model.
• The consistency of Univalent Foundations is established relative to ZFC with two inaccessible cardinals.