# The Sierpinski Object in the Scott Realizability Topos

## TL;DR

This paper investigates the properties of the Sierpinski object within the Scott realizability topos, revealing its structure, the nature of order-discrete objects, and constructing a homotopy model where these objects have only constant paths.

## Contribution

It introduces the concept of order-discrete objects in the Scott realizability topos, characterizes the Sierpinski object as a dominance, and constructs a homotopy model based on these findings.

## Key findings

- Order-discrete objects form a reflective subcategory.
- The Sierpinski object is not closed under unions.
- A homotopy model with constant paths is constructed.

## Abstract

We study the Sierpinski object $\Sigma$ 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 $\Sigma ^N$, where $N$ is the natural numbers object. We define order-discrete objects by orthogonality to $\Sigma$. 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 $\Sigma$ as a dominance; we explicitly construct the lift functor and characterize $\Sigma$-subobjects. Contrary to our expectations the dominance $\Sigma$ 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.