On the $\infty$-topos semantics of homotopy type theory

TL;DR

This paper surveys the development of $ abla$-topos semantics for homotopy type theory, highlighting Voevodsky's simplicial model and Shulman's generalization to interpret univalent foundations in any $ abla$-topos.

Contribution

It provides a comprehensive overview of the $ abla$-topos semantics for homotopy type theory, including key models and their generalizations.

Findings

Voevodsky's simplicial model of univalent foundations explained

Shulman's generalization extends interpretation to any $ abla$-topos

Community efforts have streamlined and advanced the semantics of homotopy type theory

Abstract

Many introductions to homotopy type theory and the univalence axiom gloss over the semantics of this new formal system in traditional set-based foundations. This expository article, written as lecture notes to accompany a 3-part mini course delivered at the Logic and Higher Structures workshop at CIRM-Luminy, attempts to survey the state of the art, first presenting Voevodsky's simplicial model of univalent foundations and then touring Shulman's vast generalization, which provides an interpretation of homotopy type theory with strict univalent universes in any $\infty$-topos. As we will explain, this achievement was the product of a community effort to abstract and streamline the original arguments as well as develop new lines of reasoning.