The Sierpinski Object in the Scott Realizability Topos

We study the Sierpinski object $\mathrm{\Sigma }$ in the realizability topos based on Scott’s graph model of the $\lambda$-calculus. Our starting observation is that the object of realizers in this topos is the exponential ${\mathrm{\Sigma }}^N$, where $N$ is the natural numbers object. We define order-discrete objects by orthogonality to $\mathrm{\Sigma }$. 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 $\mathrm{\Sigma }$ as a dominance; we explicitly construct the lift functor and characterize $\mathrm{\Sigma }$-subobjects. Contrary to our expectations the dominance $\mathrm{\Sigma }$ 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.