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The smashing spectrum of sheaves

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arxiv 2406.03969 v1 pith:EW6ENZPW submitted 2024-06-06 math.CT math.ACmath.AT

Ko Aoki This is my paper · ORCID
classification math.CTmath.ACmath.AT
keywords inftysmashingcategoryresultsheavesspectrumadjunctionarbitrary
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abstract

For an arbitrary $\infty$-topos, we classify the smashing localizations in the $\infty$-category of sheaves valued in derived vector spaces: Any of them is the restriction functor to a (unique) closed subtopos. Our proof is based on the existence of a Boolean cover. This result in particular gives us the first example of a nonzero presentably symmetric monoidal stable $\infty$-category whose smashing spectrum has no points. Combining this with the sheaves-spectrum adjunction, we obtain a Tannaka-type categorical reconstruction result for locales.

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  1. Very Schwartz coidempotents and continuous spectrum

    math.CT 2025-05 conditional novelty 7.0 of 10

    The sheaf functor Shv(-;Sp) is fully faithful on stably compact spaces, with right adjoint given by the new continuous spectrum functor Smcon.

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