Realizability Toposes from Specifications

We investigate a framework of Krivine realizability with I/O effects, and present a method of associating realizability models to specifications on the I/O behavior of processes, by using adequate interpretations of the central concepts of `pole' and `proof-like term'. This method does in particular allow to associate realizability models to computable functions. Following recent work of Streicher and others we show how these models give rise to triposes and toposes

A 2-Categorical Analysis of the Tripos-to-Topos Construction

We characterize the tripos-to-topos construction of Hyland, Johnstone and Pitts as a biadjunction in a bicategory enriched category of equipment-like structures. These abstract concepts are necessary to handle the presence of oplax constructs --- the construction is only oplax functorial on certain classes of cartesian functors between triposes. A by-product of our analysis is the decomposition of the tripos-to-topos construction into two steps, the intermediate step being a weakened version of quasitoposes

Triposes as a Generalization of Localic Geometric Morphisms

We discuss how triposes may be understood as generalizations of localic geometric morphisms.Comment: We have updated some references and included a reference to a comment in Hyland, Johnstone and Pitts's original paper where they discuss whether triposes over Set inducing the same localic topos over Set are unique up to equivalenc

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 toposes and ordered PCA's

Partial combinatory algebras (pca's, for short), are well-known to form the basic ingredient for the construction of various realizability toposes, of which the Eective Topos is undoubtedly the most famous. There is more than one way to present the realizability topos associated to a pca; one may take the exact completion of the category of partitioned assemblies (see [7]), or one can use tripos theory. Triposes built from pca's are, together with those from locales, the most important and most extensively studied instances of triposes, but from a structural point of view, there are important dierences between the two; whereas locales are organized in a well-behaved category, which is a re ective subcategory of the category of toposes, it is not immediately clear what an appropriate category for pca's may look like. Moreover, there are various nice properties in the localic case, such as the fact that there is a one-to-one correspondence between maps of locales and geometric morphisms between the corresponding sheaf toposes, and also the fact that this correspondence preserves epi-mono factorizations; such anintimate connection is absent for pca's