Meaning explanations at higher dimension

Angiuli, Carlo; Harper, Robert

doi:10.1016/j.indag.2017.07.010

Cited by 3 publications

(3 citation statements)

References 13 publications

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“…Here we consider a particular model to give meaning explanations for the judgments of type theory. For a certain notion of cubical sets, this has been done by Harper's group [6,5]. The benefit of the approach is that it guarantees that all constructions are computationally meaningful and make sense in the model.…”

Section: Computational Type Theoriesmentioning

confidence: 99%

Higher Structures in Homotopy Type Theory

Buchholtz

2019

Synthese Library

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The intended model of the homotopy type theories used in Univalent Foundations is the ∞-category of homotopy types, also known as ∞-groupoids. The problem of higher structures is that of constructing the homotopy types needed for mathematics, especially those that aren't sets. The current repertoire of constructions, including the usual type formers and higher inductive types, suffice for many but not all of these. We discuss the problematic cases, typically those involving an infinite hierarchy of coherence data such as semi-simplicial types, as well as the problem of developing the meta-theory of homotopy type theories in Univalent Foundations. We also discuss some proposed solutions.

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Section: Computational Type Theoriesmentioning

confidence: 99%

Higher Structures in Homotopy Type Theory

Buchholtz

2019

Synthese Library

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“…A crucial feature of dimension names is substitution: given any term M , any dimension name x, and any dimension term r, we have a dimension substitution operation M r/x which replaces all occurrences of x in M with r (note that r may be either a dimension name or ǫ). 2 Dimension substitution allows types and terms at arbitrary dimensions to be characterized as follows. The trivial case is, of course, that of types and terms at dimension zero.…”

Section: Dimension Termsmentioning

confidence: 99%

“…Our approach is based on the framework of Angiuli et al [4], which we shall refer to as 'computational cubical type theory'. Thus, although this paper is intended to be self-contained, the reader may find it helpful to refer to [4,3,2] for further clarification.…”

Section: Introductionmentioning

confidence: 99%

Cubical informal type theory: the higher groupoid structure

Bentzen¹

2018

Preprint

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Following a project of developing conventions and notations for informal type theory carried out in the homotopy type theory book for a framework built out of an augmentation of constructive type theory with axioms governing higher-dimensional constructions via Voevodsky's univalance axiom and higher-inductive types, this paper proposes a way of doing informal type theory with a cubical type theory as the underlying foundation instead. To that end, we adopt a cubical type theory recently proposed by Angiuli, Hou (Favonia) and Harper, a framework with a cumulative hierarchy of univalent Kan universes, full univalence and instances of higher-inductive types. In the present paper we confine ourselves to some elementary theorems concerning the higher groupoid structure of types.

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Groupoidal realizability for intensional type theory

Speight

2024

Math. Struct. Comp. Sci.

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We develop realizability models of intensional type theory, based on groupoids, wherein realizers themselves carry non-trivial (non-discrete) homotopical structure. In the spirit of realizability, this is intended to formalize a homotopical BHK interpretation, whereby evidence for an identification is a path. Specifically, we study partitioned groupoidal assemblies. Categories of such are parameterized by “realizer categories” (instead of the usual partial combinatory algebras) that come equipped with an interval qua internal cogroupoid. The interval furnishes a notion of homotopy as well as a fundamental groupoid construction. Objects in a base groupoid are realized by points in the fundamental groupoid of some object from the realizer category; isomorphisms in the base groupoid are realized by paths in said fundamental groupoid. The main result is that, under mild conditions on the realizer category, the ensuing category of partitioned groupoidal assemblies models intensional (1-truncated) type theory without function extensionality. Moreover, when the underlying realizer category is “untyped,” there exists an impredicative universe of 1-types (the modest fibrations). This is a groupoidal analog of the traditional situation.

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Meaning explanations at higher dimension

Cited by 3 publications

References 13 publications

Higher Structures in Homotopy Type Theory

Higher Structures in Homotopy Type Theory

Cubical informal type theory: the higher groupoid structure

Groupoidal realizability for intensional type theory

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