Cubical methods in homotopy type theory and univalent foundations

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.

The Cantor–Schröder–Bernstein Theorem for $$\infty $$-groupoids

We show that the Cantor–Schröder–Bernstein Theorem for homotopy types, or $$\infty $$ ∞ -groupoids, holds in the following form: For any two types, if each one is embedded into the other, then they are equivalent. The argument is developed in the language of homotopy type theory, or Voevodsky’s univalent foundations (HoTT/UF), and requires classical logic. It follows that the theorem holds in any boolean $$\infty $$ ∞ -topos.

COMPUTER PROOFS PRACTICE AND HUMAN UNDERSTANDING: EPISTEMOLOGICAL ISSUES

In recent decades, some epistemological issues have become especially acute in mathematics. These issues are associated with long proofs of various important mathematical results, as well as with a large and constantly increasing number of publications in mathematics. It is assumed that (at least partially) these difficulties can be resolved by referring to computer proofs. However, computer proofs also turn out to be problematic from an epistemological point of view. With regard to both proofs in ordinary (informal) mathematics and computer proofs, the problem of their surveyability appears to be fundamental. Based on the traditional concept of proof, it must be surveyable, otherwise it will not achieve its main goal — the formation of conviction in the correctness of the mathematical result being proved. About 15 years ago, a new approach to the foundations of mathematics began to develop, combining constructivist, structuralist features and a number of advantages of the classical approach to mathematics. This approach is built on the basis of homotopy type theory and is called the univalent foundations of mathematics. Due to itspowerful notion of equality, this approach can significantly reduce the length of formalized proofs, which outlines a way to resolve the epistemological difficulties that have arisen

Preface to the MSCS Issue 31.1 (2021) Homotopy Type Theory and Univalent Foundations

This issue of Mathematical Structures in Computer Science is Part I of a Special Issue dedicated to the emerging field of Homotopy Type Theory and Univalent Foundations.