The Discrete Objects in the Effective Topos

Hyland, J. M. E.; Robinson, Edmund; Rosolini, Giuseppe

doi:10.1112/plms/s3-60.1.1

Cited by 64 publications

(71 citation statements)

References 10 publications

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“…al. on small-complete small categories [10,12]. Now, the relevant realizability models all enjoy the property of being Π 0 2 -absolute, meaning that a Π 0 2 -sentence holds in the model if and only if it is true externally.…”

Section: Applicability Of Resultsmentioning

confidence: 99%

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Relational Parametricity for Computational Effects

Møgelberg

Simpson

2007

22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)

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Abstract. According to Strachey, a polymorphic program is parametric if it applies a uniform algorithm independently of the type instantiations at which it is applied. The notion of relational parametricity, introduced by Reynolds, is one possible mathematical formulation of this idea. Relational parametricity provides a powerful tool for establishing data abstraction properties, proving equivalences of datatypes, and establishing equalities of programs. Such properties have been well studied in a pure functional setting. Many programs, however, exhibit computational effects, and are not accounted for by the standard theory of relational parametricity. In this paper, we develop a foundational framework for extending the notion of relational parametricity to programming languages with effects.

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Section: Applicability Of Resultsmentioning

confidence: 99%

“…However, our perspective is slightly different. Rather than assuming a small category that is complete only in a restricted technical sense [12,34], our category C is assumed to be genuinely complete, but only weakly equivalent to a small category. This approach, which is taken from [35], offers several conveniences.…”

Section: Semantic Settingmentioning

confidence: 99%

Relational Parametricity for Computational Effects

Møgelberg

Simpson

2007

22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)

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“…It is known that the categories of extensional and complete extensional objects in any category of modest sets are (equivalent to) "small complete" categories (in the sense of [21,23]) within an ambient category of assemblies, cf. [30].…”

Section: Polymorphismmentioning

confidence: 99%

A Convenient Category of Domains

Battenfeld

Schröder

Simpson

2007

Electronic Notes in Theoretical Computer Science

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We show that the category of convenient vector spaces in the sense of Frölicher and Kriegl is a differential category. Differential categories were introduced by Blute, Cockett and Seely as the categorical models of the differential linear logic of Ehrhard and Regnier. Indeed we claim that this category fully captures the intuition of this logic. It was already evident in the monograph of Frölicher and Kriegl that the category of convenient vector spaces has remarkable structure. We here give much of that structure a logical interpretation. For example, this category supports a comonad for which the coKleisli category is the category of smooth maps on convenient vector spaces. We show this comonad models the exponential modality of linear logic. Furthermore, we show that the logical system suggests new structure. In particular , we demonstrate the existence of a codereliction map. Such a map allows for the differentiation of arbitrary maps by simple precomposition.

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“…In the effective topos, the full subcategory of discrete objects has been thoroughly studied (see, e.g. Hyland et al (1990)). I remind the reader that an object of the effective topos is called 'discrete' if it is a quotient of a subobject of the natural numbers object.…”

Section: Introductionmentioning

confidence: 99%