Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science 2018
DOI: 10.1145/3209108.3209197
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On Higher Inductive Types in Cubical Type Theory

Abstract: Cubical type theory provides a constructive justification to certain aspects of homotopy type theory such as Voevodsky's univalence axiom. This makes many extensionality principles, like function and propositional extensionality, directly provable in the theory. This paper describes a constructive semantics, expressed in a presheaf topos with suitable structure inspired by cubical sets, of some higher inductive types. It also extends cubical type theory by a syntax for the higher inductive types of spheres, to… Show more

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Cited by 62 publications
(98 citation statements)
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“…The other two theories, on the other hand, were each presented with examples of what we call CITs. Coquand et al have expanded on their initial offering with further examples (such as pushouts and two approaches to the torus), sketched a schema, and proven consistency of these with a semantics in cubical sets [Coquand et al 2018]. Our definitions of the values (including the use of fhcom), operational semantics, and rules for a non-indexed HIT specialize (roughly speaking) to the definitions given for these examples.…”
Section: Eliminationmentioning
confidence: 99%
“…The other two theories, on the other hand, were each presented with examples of what we call CITs. Coquand et al have expanded on their initial offering with further examples (such as pushouts and two approaches to the torus), sketched a schema, and proven consistency of these with a semantics in cubical sets [Coquand et al 2018]. Our definitions of the values (including the use of fhcom), operational semantics, and rules for a non-indexed HIT specialize (roughly speaking) to the definitions given for these examples.…”
Section: Eliminationmentioning
confidence: 99%
“…Semantics for different subclasses of HITs are given by [8,24,25,26,27]. Cubical type theories were shown to support some HITs in a computational way [28,29].…”
Section: Related Workmentioning
confidence: 99%
“…In the paper we define higher inductive types as initial algebras for a certain signature (Section 1.3). In the formalisation, we use higher inductive types as implemented in cubical Agda, which are based on [CHM18]. Although it is natural to assume that the Agda higher inductive type should be initial in the sense of Section 1.3, proving this fact is actually one of the main difficulties in the formalisation (Proposition 2.6).…”
Section: Formalization In Cubical Agdamentioning
confidence: 99%
“…For the formalisation we use cubical Agda [AMV19] which is based on the cubical type theory of [CCHM18]. The development of HIITs in Agda is based on [CHM18].…”
Section: Introductionmentioning
confidence: 99%