Structuralism, Invariance, and Univalence

Awodey, Steve

doi:10.1093/philmat/nkt030

Cited by 53 publications

(32 citation statements)

References 7 publications

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“…Awodey (2014) claims that, through its so-called 'Univalence Axiom', HoTT captures what is essentially right about the structuralist position. Shulman (forthcoming) agrees, describing it as a 'synthetic theory of structures', in the sense that nothing can be said about mathematical entities defined within it except structurally.…”

Section: Introductionmentioning

confidence: 99%

Expressing ‘the structure of’ in homotopy type theory

Corfield

2017

Synthese

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As a new foundational language for mathematics with its very different idea as to the status of logic, we should expect homotopy type theory to shed new light on some of the problems of philosophy which have been treated by logic. In this article, definite description, and in particular its employment within mathematics, is formulated within the type theory. Homotopy type theory has been proposed as an inherently structuralist foundational language for mathematics. Using the new formulation of definite descriptions, opportunities to express 'the structure of' within homotopy type theory are explored, and it is shown there is little or no need for this expression.

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Section: Introductionmentioning

confidence: 99%

Expressing ‘the structure of’ in homotopy type theory

Corfield

2017

Synthese

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“…Besides hasDimension landing in Prop, we can prove that n-Type is an (n + 1)-type, and we get the equivalence principle for the types of algebraic structures and for categories as mentioned in the Chapter by Ahrens-North. (See also [8].) We can also prove that the nth universe is not an n-type for any external natural number n [32].…”

Section: Some Constructions That Are Possiblementioning

confidence: 86%

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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“…In a way, this accords with the structuralist attempts to identify isomorphic structures, i.e. to take isomorphism as the identity condition for structures (Awodey 2014). And in fact such an identification, at least for algebraic structures, follows from the univalence axiom (Univalent Foundations Program 2013, pp.…”

Section: Some Basic Concepts Of Homotopy Theorymentioning

confidence: 92%

Identity and intensionality in Univalent Foundations and philosophy

Angere

2017

Synthese

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The Univalent Foundations project constitutes what is arguably the most serious challenge to set-theoretic foundations of mathematics since intuitionism. Like intuitionism, it differs both in its philosophical motivations and its mathematicallogical apparatus. In this paper we will focus on one such difference: Univalent Foundations' reliance on an intensional rather than extensional logic, through its use of intensional Martin-Löf type theory. To this, UF adds what may be regarded as certain extensionality principles, although it is not immediately clear how these principles are to be interpreted philosophically. In fact, this framework gives an interesting example of a kind of border case between intensional and extensional mathematics. Our main purpose will be the philosophical investigation of this system, and the relation of the concepts of intensionality it satisfies to more traditional philosophical or logical concepts such as those of Carnap and Quine.Keywords Univalent Foundations · Homotopy type theory · Intensionality · Identity The Univalent Foundations projectThe Univalent Foundations (UF) project is an approach to the foundations of mathematics currently under rapid development. It entails a major change both in philosophical motivation and form, ultimately taking its main inspiration from geometry rather than grammar or traditional logic. Michael Harris-himself working in the rather different field of number theory, and not being a fan of "Foundations" for math-B Staffan Angere

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Structuralism, Invariance, and Univalence

Cited by 53 publications

References 7 publications

Expressing ‘the structure of’ in homotopy type theory

Expressing ‘the structure of’ in homotopy type theory

Higher Structures in Homotopy Type Theory

Identity and intensionality in Univalent Foundations and philosophy

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