Categories for Types

Crole, Roy L.

doi:10.1017/cbo9781139172707

Cited by 93 publications

(102 citation statements)

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“…By now there are many textbook treatments of these ideas and their ramifications, ranging from introductions that do not assume prior knowledge of category theory [12,29], to more advanced texts that do [7,16,20,26]. Lawvere has also described how to do classical physics in a topos [22,23].…”

Section: Discussionmentioning

confidence: 99%

Quantum Quandaries: A Category-Theoretic Perspective

Baez¹

2006

The Structural Foundations of Quantum Gravity

109

135

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General relativity may seem very different from quantum theory, but work on quantum gravity has revealed a deep analogy between the two. General relativity makes heavy use of the category nCob, whose objects are (n − 1)-dimensional manifolds representing 'space' and whose morphisms are n-dimensional cobordisms representing 'spacetime'. Quantum theory makes heavy use of the category Hilb, whose objects are Hilbert spaces used to describe 'states', and whose morphisms are bounded linear operators used to describe 'processes'. Moreover, the categories nCob and Hilb resemble each other far more than either resembles Set, the category whose objects are sets and whose morphisms are functions. In particular, both Hilb and nCob but not Set are * -categories with a noncartesian monoidal structure. We show how this accounts for many of the famously puzzling features of quantum theory: the failure of local realism, the impossibility of duplicating quantum information, and so on. We argue that these features only seem puzzling when we try to treat Hilb as analogous to Set rather than nCob, so that quantum theory will make more sense when regarded as part of a theory of spacetime.

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Section: Discussionmentioning

confidence: 99%

Quantum Quandaries: A Category-Theoretic Perspective

Baez¹

2006

The Structural Foundations of Quantum Gravity

109

135

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“…We hope the material has been written in a way that will make it accessible to readers familiar with standard denotational semantics and types, e.g. [10,17,40].…”

Section: Structure Of the Tutorialmentioning

confidence: 99%

Nominal Game Semantics

Murawski

Tzevelekos

2016

FNT in Programming Languages

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“…Crole presents in his book (Crole, 1994) a categorical interpretation of higher-order polymorphic types, which could presumably be instantiated to the concrete syntactic relations used here.…”

Section: Related Workmentioning

confidence: 99%

Parametricity, type equality, and higher-order polymorphism

Vytiniotis¹,

Weirich²

2010

J. Funct. Prog.

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Propositions that express type equality are a frequent ingredient of modern functional programming|they can encode generic functions, dynamic types, and GADTs. Via the Curry-Howard correspondence, these propositions are ordinary types inhabited by proof terms, computed using runtime type representations. In this paper we show that two examples of type equality propositions actually do re ect type equality; they are only inhabited when their arguments are equal and their proofs are unique (up to equivalence.) We show this result in the context of a strongly normalizing language with higher-order polymorphism and primitive recursion over runtime type representations by proving Reynolds's abstraction theorem. We then use this theorem to derive \free" theorems about equality types. STEPHANIE WEIRICH University of Pennsylvania AbstractPropositions that express type equality are a frequent ingredient of modern functional programming-they can encode generic functions, dynamic types, and GADTs. Via the Curry-Howard correspondence, these propositions are ordinary types inhabited by proof terms, computed using runtime type representations. In this paper we show that two examples of type equality propositions actually do reflect type equality; they are only inhabited when their arguments are equal and their proofs are unique (up to equivalence.) We show this result in the context of a strongly normalizing language with higher-order polymorphism and primitive recursion over runtime type representations by proving Reynolds's abstraction theorem. We then use this theorem to derive "free" theorems about equality types.

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Categories for Types

Cited by 93 publications

References 0 publications

Quantum Quandaries: A Category-Theoretic Perspective

Quantum Quandaries: A Category-Theoretic Perspective

Nominal Game Semantics

Parametricity, type equality, and higher-order polymorphism

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