Homotopy theoretic models of identity types

Awodey, Steve; Warren, Michael A.

doi:10.1017/s0305004108001783

Cited by 194 publications

(303 citation statements)

References 17 publications

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“…The common definition of connectedness "there is no partition of [0,1] into two open inhabited sets" is classically equivalent to our definition but has a constructively weaker conclusion, namely that something does not exist. Constructive mathematics encourages a positive attitude: instead of saying that A is not empty, say that A has an element; instead of saying that x is not zero, say that x is negative or positive; instead of saying that a finite set is one that is not infinite, say that it is one that is in bijection with {0, .…”

Section: Proposition 52 If a And B Are Inhabited Open Subsets Ofmentioning

confidence: 96%

“…The situation with computer scientists is worse, as some of them actually help spread constructive mathematics with slogans such as "propositions are types" [36]. The recently discovered homotopy-theoretic interpretation of Martin-Löf type theory [1,34], a most extreme form of constructivism, has made some homotopy theorists and category theorists into allies of constructive mathematics. They even profess a new foundation of mathematics [30,35] in which logic and sets are just two levels of an infinite hierarchy of homotopy types.…”

Section: Depressionmentioning

confidence: 99%

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Five stages of accepting constructive mathematics

Bauer

2016

Bull. Amer. Math. Soc.

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Abstract. On the odd day, a mathematician might wonder what constructive mathematics is all about. They may have heard arguments in favor of constructivism but are not at all convinced by them, and in any case they may care little about philosophy. A typical introductory text about constructivism spends a great deal of time explaining the principles and contains only trivial mathematics, while advanced constructive texts are impenetrable, like all unfamiliar mathematics. How then can a mathematician find out what constructive mathematics feels like? What new and relevant ideas does constructive mathematics have to offer, if any? I shall attempt to answer these questions.From a psychological point of view, learning constructive mathematics is agonizing, for it requires one to first unlearn certain deeply ingrained intuitions and habits acquired during classical mathematical training. In her book On Death and Dying [17] psychologist Elisabeth Kübler-Ross identified five stages through which people reach acceptance of life's traumatizing events: denial, anger, bargaining, depression, and acceptance. We shall follow her path.

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Section: Proposition 52 If a And B Are Inhabited Open Subsets Ofmentioning

confidence: 96%

Section: Depressionmentioning

confidence: 99%

Five stages of accepting constructive mathematics

Bauer

2016

Bull. Amer. Math. Soc.

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“…Evaluation of such a function with an argument y of type A is given by substituting the expression naming y for each instance of x in Φ (with renaming to avoid collisions, as usual), thus producing an expression of type B. 8 For any types A and B there is a function type A → B whose tokens are functions defined as above. Given a token of A → B and a token of A we can combine them to produce a token of B.…”

Section: Logic and The Rules For Manipulating Typesmentioning

confidence: 99%

“…It is important to note that the elimination rule for identity types that we are calling (following the HoTT Book) 'path induction' is part of the original formulation of Martin-Löf type theory [2]. Martin-Löf's motivation for this principle of course owes nothing to the homotopy interpretation which was introduced several decades later by Awodey and Warren [8]. There is a wealth of literature that discusses other aspects of the justification of Martin-Löf's work.…”

Section: Introductionmentioning

confidence: 99%

Identity in Homotopy Type Theory, Part I: The Justification of Path Induction

Ladyman

Presnell

2015

Philosophia Mathematica

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Homotopy type theory (HoTT) is a new branch of mathematics that connects algebraic topology with logic and computer science, and which has been proposed as a new language and conceptual framework for mathematical practice. Much of the power of HoTT lies in the correspondence between the formal type theory and ideas from homotopy theory, in particular the interpretation of types, tokens, and equalities as (respectively) spaces, points, and paths. Fundamental to the use of identity and equality in HoTT is the powerful proof technique of path induction. In the 'HoTT Book' [1] this principle is justified through the homotopy interpretation of type theory, by treating identifications as paths and the induction step as a homotopy between paths. This is incompatible with HoTT being an autonomous foundation for mathematics, since any such foundation must be able to justify its principles without recourse to existing areas of mathematics. In this paper it is shown that path induction can be motivated from pre-mathematical considerations, and in particular without recourse to homotopy theory. This makes HoTT a candidate for being an autonomous foundation for mathematics.

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“…Homotopy type theory is an extension of MLTT based on a correspondence with homotopy theory and higher category theory [3,10,12,13,23,[33][34][35]. In homotopy theory, one studies topological spaces by way of their points, paths (between points), homotopies (paths or continuous deformations between paths), homotopies between homotopies (paths between paths between paths), and so on.…”

Section: Introductionmentioning

confidence: 99%

Homotopical patch theory

et al. 2014

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Homotopy type theory is an extension of Martin-Löf type theory, based on a correspondence with homotopy theory and higher category theory. The propositional equality type becomes proofrelevant, and acts like paths in a space. Higher inductive types are a new class of datatypes which are specified by constructors not only for points but also for paths. In this paper, we show how patch theory in the style of the Darcs version control system can be developed in homotopy type theory. We reformulate patch theory using the tools of homotopy type theory, and clearly separate formal theories of patches from their interpretation in terms of basic revision control mechanisms. A patch theory is presented by a higher inductive type. Models of a patch theory are functions from that type, which, because function are functors, automatically preserve the structure of patches. Several standard tools of homotopy theory come into play, demonstrating the use of these methods in a practical programming context.

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Homotopy theoretic models of identity types

Abstract: This paper presents a novel connection between homotopical algebra and mathematical logic. It is shown that a form of intensional type theory is valid in any Quillen model category, generalizing the Hofmann-Streicher groupoid model of Martin-Loef type theory.Comment: 11 page

Cited by 194 publications

References 17 publications

Five stages of accepting constructive mathematics

Five stages of accepting constructive mathematics

Identity in Homotopy Type Theory, Part I: The Justification of Path Induction

Homotopical patch theory

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