Relative and modified relative realizability

Birkedal, Lars; Oosten, Jaap van

doi:10.1016/s0168-0072(01)00122-1

Cited by 19 publications

(31 citation statements)

References 11 publications

(3 reference statements)

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“…a pca A together with a subpca A # gives rise to a pca internal to Set 2 op from which one may construct the "nested realizability topos" RT(A, A # ) as described in [5,16]. 1 Within RT(A, A # ) there is a unique nontrivial subterminal object u giving rise to the open subtopos induced by the closure operator u → (−) and the complementary subtopos induced by the closure operator u ∨ (−) as described in [5]. Next we will give more elementary descriptions of RT(A, A # ) and the above mentioned subtoposes.…”

Section: Nested Realizability Toposes and Some Of Their Subtoposesmentioning

confidence: 99%

“…In RT(A, A # ) there is a nontrivial subterminal u = (A, ∅) giving rise to two complementary subtoposes induced by the closure operators o u (p) = u → p and c u (p) = u ∨ p as in [5]. We denote the open subtopos induced by o u by RT r (A, A # ) and the complementary subtopos induced by c u by RT m (A, A # ).…”

Section: Some Subtoposes Of Rt(a a # )mentioning

confidence: 99%

“…Given a partial combinatory algebra (pca) A (see e.g. [16]) together with a subpca A # of A we will construct the nested realizability topos RT(A, A # ) as described in [5] (without giving it a proper name there). It is well known (from e.g.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Models of intuitionistic set theory in subtoposes of nested realizability toposes

Maschio

Streicher

2015

Annals of Pure and Applied Logic

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Section: Nested Realizability Toposes and Some Of Their Subtoposesmentioning

confidence: 99%

Section: Some Subtoposes Of Rt(a a # )mentioning

confidence: 99%

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Models of intuitionistic set theory in subtoposes of nested realizability toposes

Maschio

Streicher

2015

Annals of Pure and Applied Logic

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“…This amounts to the observation that deterministic higher order cartesian C-structures naturally embrace the concept of relative realizability (see e.g. [3]). Informally, the idea behind relative realizability is to consider (for instance) a TPCA A equipped with a sub-TPCA A of elements we regard as 'computable'.…”

Section: Higher Order C-structuresmentioning

confidence: 99%

Computability structures, simulations and realizability

Longley¹

2013

Math. Struct. Comp. Sci.

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We generalize the standard construction of realizability models (specifically, of categories of assemblies) to a wide class of computability structures, broad enough to embrace models of computation such as labelled transition systems and process algebras. We consider a general notion of simulation between such computability structures, and show how these simulations correspond precisely to certain functors between the realizability models. Furthermore, we show that our class of computability structures has good closure properties -in particular, it is 'cartesian closed' in a slightly relaxed sense. Finally, we investigate some important subclasses of computability structures and of simulations between them. We suggest that our 2-category of computability structures and simulations may offer a useful framework for investigating questions of computational power, abstraction and simulability for a wide range of models.

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“…Also, Birkedal and van Oosten (2002) described relative realisability as realisability over an internal partial combinatory algebra in another topos.…”

Section: Realisability Toposesmentioning

confidence: 99%

Regular functors and relative realisability categories

Stekelenburg

2013

Math. Struct. Comp. Sci.

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The relative realisability toposes introduced by Awodey, Birkedal and Scott in Awodey et al. (2002) satisfy a universal property involving regular functors to other categories. We use this universal property to define what relative realisability categories are when they are based on categories other than the topos of sets. This paper explains the property and gives a construction for relative realisability categories that works for arbitrary base Heyting categories. The universal property also provides some new geometric morphisms to relative realisability toposes.

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Relative and modified relative realizability

Cited by 19 publications

References 11 publications

Models of intuitionistic set theory in subtoposes of nested realizability toposes

Models of intuitionistic set theory in subtoposes of nested realizability toposes

Computability structures, simulations and realizability

Regular functors and relative realisability categories

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