Relative and modified relative realizability

Regular functors and relative realisability categories

The relative realizability toposes that Awodey, Birkedal and Scott introduced in [1] 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 cat...

Models of intuitionistic set theory in subtoposes of nested realizability toposes

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. [16]) that RT(A,A#) appears as the exact/regular completion of its subcategory Asm(A,A#) of assemblies. In [5] the authors considered two complementary subtopo...

All realizability is relative

We introduce a category of basic combinatorial objects, encompassing PCAs and locales. Such a basic combinatorial object is to be thought of as a pre-realizability 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...

Relative and Modiied Relative Realizability

Workshop on Realizability Preliminary Version Local Realizability Toposes and a Modal Logic for Computability (extended Abstract)

This work is a step toward developing a logic for types and computation that includes both the usual spaces of mathematics and constructions and spaces from logic and domain theory. Using realizability, we investigate a connguration of three toposes, which we regard as describing a notion of relative computability. Attention is focussed on a certain local map of toposes, which we study rst axio...