 
Summary: A General Notion of Realizability
L. Birkedal \Lambda
School of Computer Science
Carnegie Mellon University
Pittsburgh, USA
Abstract
We present a general notion of realizability encompass
ing both standard Kleene style realizability over partial
combinatory algebras and Kleene style realizability over
more general structures, including all partial cartesian
closed categories. We show how the general notion of re
alizability can be used to get models of dependent predicate
logic, thus obtaining as a corollary (the known result) that
the category Equ of equilogical spaces models dependent
predicate logic. Moreover, we characterize when the gen
eral notion of realizability gives rise to a topos.
1. Introduction
There has recently been a lot of interest in understanding
how models based on realizability over partial combinatory
algebras (PCAs) may be generalized to models based on re
