Realizability categories

Abstract

The thesis is a comprehensive analysis of realizability toposes, which is divided into three central chapters. The first chapter considers a class of realizability models which are a generalization of realizability triposes, and derives a collection of properties which characterizes these structures up to equivalence. The second chapter explains the construction of realizability categories from the generalized realizability triposes and focusses on the functors between these categories. The last chapter uses the theory of the first two chapters to solve a couple of problems in the theory of realizability toposes. It contains a characterization of the effective topos, a demonstration of a complete internal subcategory and a construction of classical realizability models from relative realizability models