Localization in Homotopy Type Theory

Christensen, J. Daniel; Opie, Morgan; Rijke, Egbert; Scoccola, Luis

doi:10.21136/hs.2020.01

Cited by 12 publications

(21 citation statements)

References 9 publications

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“…Otherwise, the property of belonging to different connected components would be a property by means of which we could discern them. 12 But if indiscernible terms necessarily belong to the same connected component, they can be connected by a path, i.e. they are propositionally equal.…”

Section: The Homotopic Semanticsmentioning

confidence: 99%

“…4 Below diagram summarizes this sequence of successive extensions of MLTT. Mathematical introductions to UF can be found in [7,12,18] and more conceptual-oriented discussions in [19][20][21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

“…these expressions are not definitionally equal). By contrast, propositional equality is more extensional since (for instance) it can be shown that m + n = N n + m ( [12], Prop.5.6.4, p.48).…”

mentioning

confidence: 99%

“…A critical assessment of the interpretation of propositional equalities in terms of the notion of indiscernibility can be found in[10] 12. This presupposes that the elements in the set of connected components can be discerned from each other.…”

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confidence: 99%

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Abstraction, equality and univalence

Catren

2023

Phil. Trans. R. Soc. A.

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We shall propose a conceptual-oriented discussion of the so-called Univalent Foundations Program, that is, of Martin–Löf type theory enriched with a homotopic interpretation, together with the univalence axiom proposed by Voevodsky. We shall argue that the type-theoretic notion of propositional equality encodes the notion of indiscernibility , we shall address the homotopic interpretation of Martin–Löf type theory, and we shall analyse whether Leibniz’s principle of the identity of indiscernibles holds or not in Univalent Foundations. We shall finally argue that univalence can be understood as a particular implementation of a constructive notion of abstraction that resolves Fregean abstraction. This article is part of the theme issue ‘Identity, individuality and indistinguishability in physics and mathematics’.

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Section: The Homotopic Semanticsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Abstraction, equality and univalence

Catren

2023

Phil. Trans. R. Soc. A.

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“…Now, the notion of mathematical equality (and, a fortiori , the notion of identity) has undergone a far-reaching process of reconceptualization that started with the development of category theory , continued with the enhancement of the latter to higher category theory, and has recently entered into a new phase with the development of homotopy type theory in the early new millennium [26,27] (see also [28] and references therein). Category theory made clear that the notion of strict equality is indeed too strict.…”

Section: Revisiting Mathematical Equalitymentioning

confidence: 99%

Identity, individuality and indistinguishability in physics and mathematics

Catren

2023

Phil. Trans. R. Soc. A.

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In this brief survey, we discuss some of the scientific and philosophical problems and debates that underlie the notions of identity , individuality and indistinguishability in physics and mathematics. We critically analyse the different positions for or against the existence of indistinguishable objects in different scientific theories, notably quantum mechanics and gauge theories in physics and homotopy type theory in mathematics. We argue that the different forms of indistinguishability that occur in many areas of physics and mathematics—far from being a problem to be eradicated—exhibit a rich formal structure that plays a key role in the corresponding theories that needs to be properly understood. This article is part of the theme issue ‘Identity, individuality and indistinguishability in physics and mathematics’.

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Algebras of Complemented Subsets

Petrakis¹,

Misselbeck-Wessel²

2022

Revolutions and Revelations in Computability

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We introduce cs-topologies, or topologies of complemented subsets, as a new approach to constructive topology that preserves the duality between open and closed subsets of classical topology. Complemented subsets were used successfully by Bishop in his constructive formulation of the Daniell approach to measure and integration. Here we use complemented subsets in topology, in order to describe simulataneously an open set, the first-component of an open complemented subset, and its complement as a closed set, the second component of an open complemented subset. We analyse the canonical cs-topology induced by a metric, and we introduce the notion of a mudulus of openness for a cs-open subset of a metric space. Pointwise and uniform continuity of functions between metric spaces are formulated with respect to the way these functions inverse open complemented subsets together with their moduli of openness. The addition of moduli of openness in the concept of an open subset, given a base for the cs-topology, makes possible to define the notion of uniform continuity of functions between csb-spaces, that is cs-spaces with a given base. In this way, the notions of pointwise and uniform continuity of functions between metric spaces are directly generalised to the notions of pointwise and uniform continuity between csb-topological spaces. Constructions and facts from the classical theory of topologies of open subsets are translated to our constructive theory of topologies of open complemented subsets.

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Localization in Homotopy Type Theory

Cited by 12 publications

References 9 publications

Abstraction, equality and univalence

Abstraction, equality and univalence

Identity, individuality and indistinguishability in physics and mathematics

Algebras of Complemented Subsets

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