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- W3089422605 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." @default.
- W3089422605 created "2020-10-08" @default.
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- W3089422605 date "2019-01-01" @default.
- W3089422605 modified "2023-09-27" @default.
- W3089422605 title "The Sierpinski Object in the Scott Realizability Topos." @default.
- W3089422605 hasPublicationYear "2019" @default.
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