preprints available

[jvoosten@math.uu.nl]

The following preprints are available from the WWW:

Jaap van Oosten
Realizability: A Historical Essay
http://www.math.uu.nl/publications/preprints/1131.ps.gz

Abstract: historical survey on Realizability. Focuses on
notions of realizability used in the study of metamathematics of
arithmetical theories, and topos-theoretic developments.
Bibliography contains 96 items.24 pages

Lars Birkedal and Jaap van Oosten
Relative and Modified Relative Realizability
http://www.math.uu.nl/publications/preprints/1146.ps.gz

Abstract: We approach `relative realizability' from an abstract point
of view, studying internal partial combinatory algebras in an
arbitrary topos E. Let RT(E,A) denote the standard realizability
topos over E w.r.t. A.
We define the notion of `elementary subobject'
in a topos; if, for two internal pca's A and B in E, there is
an embedding which maps A as elementary subobject into B, there
is a local geometric morphism from RT(E,B) to RT(E,A).
Next we study the situation where an internal topology j is given;
we have a tripos over E using only the j-closed subobjects of A,
giving a topos RTj(E,A). RTj(E,A) is a subtopos of RT(E,A) and we
have a pullback diagram of toposes:

  Sh_j(E)--->RTj(E,A)
    |           |
    |           |
    V           V
    E   ---->  RT(E,A)

If A--> B is an embedding with A elementary subobject of B, the local
geometric morphism restricts to a local geometric morphism:
RTj(E,B)-->RTj(E,A)
Moreover if A --> B is a j-dense embedding, there is a logical functor
(a filter-quotient situation): RTj(E,A) --> RTj(E,B).
If j is an open topology, the inclusion RTj(E,A)-->RT(E,A) is open
too, and it makes sense to consider its closed complement; we call this
the modified relative realizability topos Mj(E,A) w.r.t. A and j.
We have an automatic pullback diagram

  Sh_k(E)---->Mj(E,A)
    |           |
    |           |
    V           V
    E  ---->  RT(E,A)

where k denotes the closed complement of j.
In a section `Examples' we show that our treatment generalizes former
definitions of relative realizability (in Awodey, Birkedal, Scott) and
modified realizability.
16 pages.