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All realizability is relative
 Mathematical Proceedings of the Cambridge Philosophical Society 141
, 2006
"... We introduce a category of basic combinatorial objects, encompassing PCAs and locales. Such a basic combinatorial object is to be thought of as a prerealizability notion. To each such object we can associate an indexed preorder, generalizing the construction of triposes for various notions of reali ..."
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We introduce a category of basic combinatorial objects, encompassing PCAs and locales. Such a basic combinatorial object is to be thought of as a prerealizability notion. To each such object we can associate an indexed preorder, generalizing the construction of triposes for various notions of realizability. There are two main results: first, the characterization of triposes which arise in this way, in terms of ordered PCAs equipped with a filter. This will include “Effective Toposlike ” triposes, but also the triposes for relative, modified and extensional realizability and the dialectica tripos. Localic triposes can be identified as those arising from ordered PCAs with a trivial filter. Second, we give a classification of geometric morphisms between such triposes in terms of maps of the un
Iterated realizability as a comma construction
 Math. Proc. Cambridge Philos. Soc
, 2008
"... We show that the 2category of partial combinatory algebras, as well as various related categories, admit a certain type of lax comma objects. This not only reveals some of the properties of such categories, but it also gives an interpretation of iterated realizability, in the following sense. Let φ ..."
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We show that the 2category of partial combinatory algebras, as well as various related categories, admit a certain type of lax comma objects. This not only reveals some of the properties of such categories, but it also gives an interpretation of iterated realizability, in the following sense. Let φ: A → B be a morphism of PCAs, giving a comma object A nφ B. In the realizability topos RT(B) over B, the object (A,φ) is an internal PCA, so we can construct the realizability topos over (A,φ). This topos is equivalent to the realizability topos over the commaPCA AnφB. This result is both an analysis and a generalization of a special case studied by Pitts in the context of the effective monad. 1
Krivine’s Classical Realizability from a Categorical Perspective
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2010
"... ... introduced his notion of Classical Realizability for classical second order logic and ZermeloFraenkel set theory. Moreover, in more recent work (Krivine 2008) he has considered forcing constructions on top of it with the ultimate aim of providing a realizability interpretation for the axiom of ..."
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... introduced his notion of Classical Realizability for classical second order logic and ZermeloFraenkel set theory. Moreover, in more recent work (Krivine 2008) he has considered forcing constructions on top of it with the ultimate aim of providing a realizability interpretation for the axiom of choice. The aim of this paper is to show how Krivine’s classical realizability can be understood as an instance of the categorical approach to realizability as started by Hyland in (Hyland 1982) and described in detail in (van Oosten 2008). Moreover, we will give an intuitive explanation of the iteration of realizability as can be found in (Krivine 2008).
THE GLEASON COVER OF A REALIZABILITY TOPOS
"... Abstract. Recently Benno van den Berg [1] introduced a new class of realizability toposes which he christened Herbrand toposes. These toposes have strikingly different properties from ordinary realizability toposes, notably the (related) properties that the ‘constant object ’ functor from the topos ..."
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Abstract. Recently Benno van den Berg [1] introduced a new class of realizability toposes which he christened Herbrand toposes. These toposes have strikingly different properties from ordinary realizability toposes, notably the (related) properties that the ‘constant object ’ functor from the topos of sets preserves finite coproducts, and that De Morgan’s law is satisfied. In this paper we show that these properties are no accident: for any Schönfinkel algebra Λ, the Herbrand realizability topos over Λ may be obtained as the Gleason cover (in the sense of [8]) of the ordinary realizability topos over Λ. As a corollary, we obtain the functoriality of the Herbrand realizability construction on the
More on Geometric Morphisms between
, 2014
"... Geometric morphisms between realizability toposes are studied in terms of morphisms between partial combinatory algebras (pcas). The morphisms inducing geometric morphisms (the computationally dense ones) are seen to be the ones whose ‘lifts ’ to a kind of completion have right adjoints. We charac ..."
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Geometric morphisms between realizability toposes are studied in terms of morphisms between partial combinatory algebras (pcas). The morphisms inducing geometric morphisms (the computationally dense ones) are seen to be the ones whose ‘lifts ’ to a kind of completion have right adjoints. We characterize topos inclusions corresponding to a general form of relative computability. We characterize pcas whose realizability topos admits a geometric morphism to the effective topos.
Effective Operations of Type 2 in Pcas
, 2014
"... We exhibit a way of “forcing a functional to be an effective operation” for arbitrary partial combinatory algebras (pcas). This gives a method of defining new pcas from old ones for some fixed functional, where the new partial functions can be viewed as computable relative to that functional. It is ..."
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We exhibit a way of “forcing a functional to be an effective operation” for arbitrary partial combinatory algebras (pcas). This gives a method of defining new pcas from old ones for some fixed functional, where the new partial functions can be viewed as computable relative to that functional. It is shown that this generalizes a notion of computation relative to a functional as defined by Kleene for the natural numbers. The construction can be used to study subtoposes of the Effective Topos. We will do this for a particular functional that forces every arithmetical set to be decidable. In this paper we also prove the convenient result that the two definitions of a pca that are common in the literature are essentially the same. 1
MORE ON GEOMETRIC MORPHISMS BETWEEN REALIZABILITY TOPOSES
"... Abstract. Geometric morphisms between realizability toposes are studied in terms of morphisms between partial combinatory algebras (pcas). The morphisms inducing geometric morphisms (the computationally dense ones) are seen to be the ones whose ‘lifts ’ to a kind of completion have right adjoints. W ..."
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Abstract. Geometric morphisms between realizability toposes are studied in terms of morphisms between partial combinatory algebras (pcas). The morphisms inducing geometric morphisms (the computationally dense ones) are seen to be the ones whose ‘lifts ’ to a kind of completion have right adjoints. We characterize topos inclusions corresponding to a general form of relative computability. We characterize pcas whose realizability topos admits a geometric morphism to the effective topos.