96 research outputs found
Predicative toposes
We explain the motivation for looking for a predicative analogue of the
notion of a topos and propose two definitions. For both notions of a
predicative topos we will present the basic results, providing the groundwork
for future work in this area
The Herbrand Topos
We define a new topos, the Herbrand topos, inspired by the modified
realizability topos and our earlier work on Herbrand realizability. We also
introduce the category of Herbrand assemblies and characterise these as the
double-negation-separated objects in the Herbrand topos. In addition, we show
that the category of sets is included as the category of
double-negation-sheaves and prove that the inclusion functor preserves and
reflects validity of first-order formulas
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
More on Geometric Morphisms between Realizability Toposes
Geometric morphisms between realizability toposes are studied in terms of
morphisms between partial combinatory algebras (pcas). The morphisms inducing
geometric morphisms (the {\em 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.Comment: 20 page
The Modified Realizability Topos
The modified realizability topos is the semantic (and higher order) counterpart of a variant of Kreisel's modified realizability (1957). These years, this realizability has been in the limelight again because of its possibilities for modelling type theory (Streicher, Hyland-Ong-Ritter) and strong normalization. In this paper this topos is investigated from a general logical and topostheoretic point of view. It is shown that Mod (as we call the topos) is the closed complement of the effective topos inside another one; this turns out to have some logical consequences. Some important subcategories of Mod are described, and a general logical principle is derived, which holds in the larger topos and implies the well-known Independence of Premiss principle
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