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Homotopy type theory: unified foundations of mathematics and computation

Published: 28 January 2015 Publication History

Abstract

Homotopy type theory is a recently-developed unification of previously disparate frameworks, which can serve to advance the project of formalizing and mechanizing mathematics. One framework is based on a computational conception of the type of a construction, the other is based on a homotopical conception of the homotopy type of a space. The computational notion of type has its origins in Brouwer's program of intuitionism, and Church's λ-calculus, both of which sought to ground mathematics in computation (one would say "algorithm" these days). The homotopical notion comes from Grothendieck's late conception of homotopy types of spaces as represented by ∞-groupoids [Grothendieck 1983]. The computational perspective was developed most fully by Per Martin-Löf, leading in particular to his Intuitionistic Theory of Types [Martin-Löf and Sambin 1984], on which the formal system of homotopy type theory is based. The connection to homotopy theory was first hinted at in the groupoid interpretation of Hofmann and Streicher [Hofmann and Streicher 1994; 1995]. It was then made explicit by several researchers, roughly simultaneously. The connection was clinched by Voevodsky's introduction of the univalence axiom, which is motivated by the homotopical interpretation, and which relates type equality to homotopy equivalence [Kapulkin et al. 2012; Awodey et al. 2013].

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Published In

cover image ACM SIGLOG News
ACM SIGLOG News  Volume 2, Issue 1
January 2015
42 pages
EISSN:2372-3491
DOI:10.1145/2728816
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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 28 January 2015
Published in SIGLOG Volume 2, Issue 1

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View all
  • (2021)Conley's Fundamental Theorem for a Class of Hybrid SystemsSIAM Journal on Applied Dynamical Systems10.1137/20M133657620:2(784-825)Online publication date: 10-May-2021
  • (2018)Semantics columnACM SIGLOG News10.1145/3242953.32429615:3(52-53)Online publication date: 26-Jul-2018
  • (2016)Evolutionary Domains for Varying IndividualsProcedia Computer Science10.1016/j.procs.2016.07.44788(347-352)Online publication date: 2016

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