Realizability toposes and ordered PCA's

Abstract

Partial combinatory algebras (pca's, for short), are well-known to form the basic ingredient for the construction of various realizability toposes, of which the Eective Topos is undoubtedly the most famous. There is more than one way to present the realizability topos associated to a pca; one may take the exact completion of the category of partitioned assemblies (see [7]), or one can use tripos theory. Triposes built from pca's are, together with those from locales, the most important and most extensively studied instances of triposes, but from a structural point of view, there are important dierences between the two; whereas locales are organized in a well-behaved category, which is a re ective subcategory of the category of toposes, it is not immediately clear what an appropriate category for pca's may look like. Moreover, there are various nice properties in the localic case, such as the fact that there is a one-to-one correspondence between maps of locales and geometric morphisms between the corresponding sheaf toposes, and also the fact that this correspondence preserves epi-mono factorizations; such anintimate connection is absent for pca's.