Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Sierpinski Object in the Scott Realizability Topos   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.

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The categories of this paper are: math-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: plasm-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: physics
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The categories of this paper are: gr-qc
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The categories of this paper are: q-alg
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The categories of this paper are: q-fin
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The categories of this paper are: econ
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The categories of this paper are: hep-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: nlin
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ex
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The categories of this paper are: dg-ga