On the $\infty$-topos 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 -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.
Keywords
Cite
@article{arxiv.2212.06937,
title = {On the $\infty$-topos semantics of homotopy type theory},
author = {Emily Riehl},
journal= {arXiv preprint arXiv:2212.06937},
year = {2024}
}
Comments
These lecture notes were written to accompany a mini-course delivered at CIRM - Luminy from 21-25 February 2022. Video is available at https://www.carmin.tv/en/collections/logic-and-higher-structures-logique-et-structures-superieures; v2 incorporates feedback from the referee; v3 is the final journal version with updated section numbers to conform to journal style