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Cubical methods in homotopy type theory and univalent foundations

Published online by Cambridge University Press:  10 December 2021

Anders Mörtberg Open the ORCID record for Anders Mörtberg [Opens in a new window]
Anders Mörtberg*
Affiliation:
Department of Mathematics, Stockholm University, Stockholm, Sweden
*
*Corresponding author. Email: anders.mortberg@math.su.se
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Abstract

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Cubical methods have played an important role in the development of Homotopy Type Theory and Univalent Foundations (HoTT/UF) in recent years. The original motivation behind these developments was to give constructive meaning to Voevodsky’s univalence axiom, but they have since then led to a range of new results. Among the achievements of these methods is the design of new type theories and proof assistants with native support for notions from HoTT/UF, syntactic and semantic consistency results for HoTT/UF, as well as a variety of independence results and establishing that the univalence axiom does not increase the proof theoretic strength of type theory. This paper is based on lecture notes that were written for the 2019 Homotopy Type Theory Summer School at Carnegie Mellon University. The goal of these lectures was to give an introduction to cubical methods and provide sufficient background in order to make the current research in this very active area of HoTT/UF more accessible to newcomers. The focus of these notes is hence on both the syntactic and semantic aspects of these methods, in particular on cubical type theory and the various cubical set categories that give meaning to these theories.

Keywords

Homotopy type theoryunivalent foundationscubical type theorycubical set models
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Special Issue 10: Homotopy Type Theory 2019 , November 2021 , pp. 1147 - 1184
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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