arXiv subjects

Jaap van Oosten

Publications and source records attributed to Jaap van Oosten.

9 recordsLinked to original sources

arXiv2020

The Sierpinski Object in the Scott Realizability Topos

We study the Sierpinski object $Σ$ in the realizability topos based on Scott's graph model of the $λ$-calculus. Our starting observation is that the object of realizers in this topos is the exponential $Σ^N$, where $N$ is the natural numbers object. We define order-discrete objects by orthogonality to $Σ$. We show that the order-discrete objects form a reflective subcategory of the topos, and that many fundamental objects in higher-type arithmetic are order-discrete. Building on work by Lietz, we give some new results regarding the internal logic of the topos. Then we consider $Σ$ as a dominance; we explicitly construct the lift functor and characterize $Σ$-subobjects. Contrary to our expectations the dominance $Σ$ is not closed under unions. In the last section we build a model for homotopy theory, where the order-discrete objects are exactly those objects which only have constant paths.

cs.LO↗

arXiv2016

Extensions of Scott's Graph Model and Kleene's Second Algebra

We use a way to extend partial combinatory algebras (pcas) by forcing them to represent certain functions. In the case of Scott's Graph model, equality is computable relative to the complement function. However, the converse is not true. This creates a hierarchy of pcas which relates to similar structures of extensions on other pcas. We study one such structure on Kleene's second model and one on a pca equivalent but not isomorphic to it. For the recursively enumerable sub pca of the Graph model, results differ as we can compute the (partial) complement function using the equality.

math.LO↗

arXiv2016

Classical and Relative Realizability

We show that every abstract Krivine structure in the sense of Streicher can be obtained, up to equivalence of the resulting tripos, from a filtered opca (A,A') and a subobject of 1 in the relative realizability topos RT(A',A); the topos is always a Booleanization of a closed subtopos of RT(A',A). We exhibit a range of non-localic Boolean subtoposes of the Kleene-Vesley topos.

math.CT↗

arXiv2014

Effective Operations of Type 2 in Pcas

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.

math.LO↗

arXiv2014

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.

math.LO↗

arXiv2013

Realizability with a Local Operator of A.M. Pitts

We study a notion of realizability with a local operator J which was first considered by A.M. Pitts in his thesis. Using the Suslin-Kleene theorem, we show that the representable functions for this realizability are exactly the hyperarithmetical functions. We show that there is a realizability interpretation of nonstandard arithmetic, which, despite its classical character, lives in a very nonclassical universe, where the Uniformity Principle holds and Konig's Lemma fails. We conjecture that the local operator gives a useful indexing of the hyperarithmetical functions.

math.LO↗

arXiv2012

Basic Subtoposes of the Effective Topos

We employ a new tool (sights) to investigate local operators in the Effective Topos. A number of new such local operators is analyzed using this machinery. Moreover, we investigate a local operator defined in the thesis of A. Pitts, and establish that its corresponding subtopos satisfies true arithmetic.

math.LO↗

arXiv2009

Partial Combinatory Algebras of Functions

We employ the notions of `sequential function' and `interrogation' (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley's preorder-enriched category of partial combinatory algebras and decidable applicative structures. We also investigate total combinatory algebras of partial functions. One of the results is, that every realizability topos is a quotient of a realizability topos on a total combinatory algebra.

math.LO↗

arXiv2005

A general form of relative recursion

For every partial combinatory algebra (pca) $A$ and every partial endofunction on $A$, a pca $A[f]$ is constructed such that in $A[f]$, the function $f$ is representable by an element; a universal property of the construction is formulated in terms of Longley's 2-category of pcas and decidable applicative morphisms.

math.LO↗