Models of Intuitionistic Set Theory in Subtoposes of Nested Realizability Toposes

With every pca $\mathcal{A}$ and subpca $\mathcal{A}_\#$ we associate the nested realizability topos $\mathsf{RT}(\mathcal{A},\mathcal{A}_\#)$ within which we identify a class of small maps $\mathcal{S}$ giving rise to a model of intuitionistic set theory within $\mathsf{RT}(\mathcal{A},\mathcal{A}_\#)$. For every subtopos $\mathcal{E}$ of such a nested realizability topos we construct an induced class $\mathcal{S_E}$ of small maps in $\mathcal{E}$ giving rise to a model of intuitionistic set theory within $\mathcal{E}$. This covers relative realizability toposes, modified relative realizability toposes, the modified realizability topos and van den Berg's recent Herbrand topos

Regular Functors and Relative Realizability Categories

Relative realizability toposes satisfy a universal property that involves regular functors to other categories. We use this universal property to define what relative realizability categories are, when based on other categories than of the topos of sets. This paper explains the property and gives a construction for relative realizability categories that works for arbitrary base Heyting categories. The universal property shows us some new geometric morphisms to relative realizability toposes too

Realizability Models Separating Various Fan Theorems

Abstract. We develop a realizability model in which the realizers are the reals not just Turing computable in a fixed real but rather the reals in a countable ideal of Turing degrees. This is then applied to prove several separation results involving variants of the Fan Theorem

Computability structures, simulations and realizability

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.

Relative and Modified Relative Realizability

this paper) and it was described by means of tripos theory right from the beginnings of that theory, see, e.g., [17, Section 1.5, item (ii)]. Recently there has been a renewed interest in Relative Realizability, both in Thomas Streicher's "Topos for Computable Analysis" [18] and in [2, 1, 4]. The idea is, that instead of doing realizability with one partial combinatory algebra A one uses an inclusion of partial combinatory algebras A ] ` A (such that there are combinators k; s 2 A ] which also serve as combinators for A), the principal point being that "(A ] -) computable" functions may also act on data (in A) that need not be computabl