Abstract Peter Johnstone

Title: Realizability toposes: from stamp-collecting to geometry

Abstract: Realizability toposes originated in Martin Hyland's work in the late 1970s on constructing a topos model for Kleene's recursive realizability interpretation of intuitionistic logic. In joint work with Andy Pitts and myself, he showed that such toposes arise from any Sch"onfinkel algebra (also called a partial combinatory algebra). Many examples of Schonfinkel algebras are known, giving rise to many different realizability toposes; but for a long time our knowledge of them was stuck at the "stamp-collecting" stage: we had lots of beautiful examples, but no general theory of how they fitted together as a category. However, thanks to work of several people (principally John Longley, Jaap van Oosten, Pieter Hofstra and myself) over the last twenty years, we now have a complete description of the 2-category of realizability toposes and their geometric morphisms. The object of my talk is to explain how this came about.

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