Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Natural Commuting of Vanishing Cycles and the Verdier Dual   We prove that the shifted vanishing cycles and nearby cycles commute with
Verdier dualizing up to a {\bf natural} isomorphism, even when the coefficients
are not in a field.

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The categories of this paper are: hep-th
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The categories of this paper are: funct-an
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The categories of this paper are: acc-phys
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The categories of this paper are: q-fin
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph
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The categories of this paper are: cs
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The categories of this paper are: physics
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The categories of this paper are: q-alg
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The categories of this paper are: eess
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The categories of this paper are: quant-ph
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The categories of this paper are: supr-con
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-lat
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The categories of this paper are: plasm-ph