The identity type weak factorisation system

Gambino, Nicola; Garner, Richard

doi:10.1016/j.tcs.2008.08.030

Cited by 81 publications

(99 citation statements)

References 18 publications

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“…Quillen model categories). Lumsdaine [15] and van den Berg and Garner [22] show that the syntax of intensional type theory forms a weak ω-category, and Gambino and Garner [8] shows that identity types admit a weak factorization system.…”

Section: Related Workmentioning

confidence: 99%

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Canonicity for 2-dimensional type theory

Licata

Harper

2012

Proceedings of the 39th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

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Higher-dimensional dependent type theory enriches conventional one-dimensional dependent type theory with additional structure expressing equivalence of elements of a type. This structure may be employed in a variety of ways to capture rather coarse identifications of elements, such as a universe of sets considered modulo isomorphism. Equivalence must be respected by all families of types and terms, as witnessed computationally by a type-generic program. Higher-dimensional type theory has applications to code reuse for dependently typed programming, and to the formalization of mathematics. In this paper, we develop a novel judgemental formulation of a two-dimensional type theory, which enjoys a canonicity property: a closed term of boolean type is definitionally equal to true or false. Canonicity is a necessary condition for a computational interpretation of type theory as a programming language, and does not hold for existing axiomatic presentations of higherdimensional type theory. The method of proof is a generalization of the NuPRL semantics, interpreting types as syntactic groupoids rather than equivalence relations.

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Section: Related Workmentioning

confidence: 99%

“…A growing body of work [4,8,9,12,14,15,[22][23][24] on intensional dependent type theory [11,16,18] elucidates the latent higherdimensional structure given by the Martin-Löf intensional identity type. The identity type, IdA M N , is the type of evidence for equivalence of the objects M and N of type A.…”

Section: Introductionmentioning

confidence: 99%

Canonicity for 2-dimensional type theory

Licata

Harper

2012

Proceedings of the 39th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

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“…Types in Martin-Löf type theory can express an infinite-dimensional structure that corresponds to the notion of space studied in homotopy theory or the notion of an ∞-groupoid studied in higher category theory (Hofmann and Streicher 1998;Voevodsky 2006;Lumsdaine 2009;van den Berg and Garner 2011;Awodey and Warren 2009;Warren 2008;Gambino and Garner 2008). From this point of view, types in type theory have not only elements, but also paths (represented by the identity type Id A (a, b)), paths-betweenpaths (represented by the iterated identity type Id Id A (a,b) (p, q)), and so on.…”

Section: Homotopy-theoretical Aspects Of Typesmentioning

confidence: 99%

A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory

Hou

Finster

Licata

et al. 2016

Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science

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This paper continues investigations in "synthetic homotopy theory": the use of homotopy type theory to give machine-checked proofs of constructions from homotopy theory.We present a mechanized proof of the Blakers-Massey connectivity theorem, a result relating the higher-dimensional homotopy groups of a pushout type (roughly, a space constructed by gluing two spaces along a shared subspace) to those of the components of the pushout. This theorem gives important information about the pushout type, and has a number of useful corollaries, including the Freudenthal suspension theorem, which has been studied in previous formalizations.The new proof is more elementary than existing ones in abstract homotopy-theoretic settings, and the mechanization is concise and high-level, thanks to novel combinations of ideas from homotopy theory and type theory.

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“…In recent years, there has been an upsurge of interest in connections between abstract homotopy theories and type theory: see Gambino and Garner (2008), Warren (2008), Awodey and Warren (2009), Voevodsky (2010), Van den Berg and Garner (2011), Kapulkin et al (2012) and Van den Berg and Garner (2012). A prominent area of focus is that of (closed) model categories, first defined by Quillen (1967); for modern expositions see Hovey (1999), Dwyer and Spalinski (1995) and Hirschhorn (2003).…”

Section: Introductionmentioning

confidence: 99%