The Sierpinski Object in the Scott Realizability Topos

Abstract

We study the Sierpinski object $Σ$ in the realizability topos based on Scott's graph model of the $λ$-calculus. Our starting observation is that the object of realizers in this topos is the exponential $Σ^N$, where $N$ is the natural numbers object. We define order-discrete objects by orthogonality to $Σ$. We show that the order-discrete objects form a reflective subcategory of the topos, and that many fundamental objects in higher-type arithmetic are order-discrete. Building on work by Lietz, we give some new results regarding the internal logic of the topos. Then we consider $Σ$ as a dominance; we explicitly construct the lift functor and characterize $Σ$-subobjects. Contrary to our expectations the dominance $Σ$ is not closed under unions. In the last section we build a model for homotopy theory, where the order-discrete objects are exactly those objects which only have constant paths.