Abstract
AbstractCubical 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.
Highlights
This paper is based on lecture notes for a course given at the 2019 Homotopy Type Theory Summer School at Carnegie Mellon University in Pittsburgh.1 The course covered the basics of cubical type theory with its semantics in cubical sets, and this paper closely follows the structure of the course
Homotopy Type Theory and Univalent Foundations, as formulated by Voevodsky (2010, 2011, 2014, 2015) and in the HoTT book (Univalent Foundations Program, 2013), are axiomatic extensions of type theory initially developed by the Swedish logician Per Martin-Löf
There are multiple different type theories of this kind, see, for example, Martin-Löf (1975, 1982, 1984, 1998), and we will here use the term “Martin-Löf type theory” (MLTT) for type theories specified in the MLTT tradition, that is, type theories specified by the four hypothetical judgments: A
Summary
This paper is based on lecture notes for a course given at the 2019 Homotopy Type Theory Summer School at Carnegie Mellon University in Pittsburgh. The course covered the basics of cubical type theory with its semantics in cubical sets, and this paper closely follows the structure of the course. The univalence axiom of Voevodsky (2014) is at the center for HoTT/UF and provides a type theoretic rendering of the idea that all constructions should be invariant up to some form of equivalence This is a very common informal practice in mathematics; for instance, a mathematician typically makes no distinction between the quotient ring R/(0) and R itself even though they formally are different objects. The paper first introduces both the type theoretical and semantical setting in which the rest of the paper is formulated (Section 2); it continues with a general introduction to cubical type theories and their models (Section 3), followed by a discussion of the cubical transport operation and why it is not sufficient to get a constructive model of HoTT/UF (Section 4), this naturally leads to the more general cubical Kan composition operations that lets us correct the sides of transported elements (Section 5), and in order to be able to prove the univalence axiom and give it computational content Glue types are introduced (Section 6). In order to get a deeper understanding of lifting problems and the content of Section 5.4, see Riehl (2014, Chapter 11), but beware that this requires much more categorical background than the rest of the material and are not necessary unless one wants to study the homotopy theoretical aspects of the field
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