Descent for sheaves on compact Hausdorff spaces

Descent for sheaves on compact Hausdorff spaces Haine

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Descent for sheaves on compact Hausdorff spaces 2022 10.48550/arxiv.2210.00186 haine2022-descent-for-sheaves-on-compact-hau haine2022-descent-for-sheaves-on-compact-hau.txt, the project's staged copy 3534313f36bd72acc23fc9f87041c301e9338527 open-access higher-category-theory

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0 spans located by searching the stored text for the quote's exact words, allowing any whitespace between them existence Quotation withheld because this record is not cleared for redistribution. descent for sheaves on compact Hausdorff spaces Quotation withheld because this record is not cleared for redistribution. mathematical-proof mathematical-proof Quotation withheld because this record is not cleared for redistribution. mechanisms Descent fails because every compact Hausdorff space is covered by a profinite set, and sheaves on profinite sets are already Postnikov complete while sheaves on a general compact Hausdorff space are not, so the target of the limit is forced to satisfy a completeness the source lacks. Quotation withheld because this record is not cleared for redistribution. The descent and full-faithfulness results fail before Postnikov completion; the paper devotes sections to explaining why. Quotation withheld because this record is not cleared for redistribution. Whether every-object-is-the-limit-of-its-Postnikov-tower is strictly weaker than Postnikov completeness is left open. Quotation withheld because this record is not cleared for redistribution. The protruncated shape, one of the derived consequences, does not preserve pullbacks. Quotation withheld because this record is not cleared for redistribution. Three competing sheaf theories agree on all truncated objects, so which one a result used is invisible until unbounded objects are involved. what this limits: a result stated for sheaves on a space: sheaves, hypersheaves and Postnikov complete sheaves all have the same truncated objects, and the differences between the three theories appear only on unbounded objects - so two such results can be compared safely in the truncated case and not otherwise Quotation withheld because this record is not cleared for redistribution. Two incompatible set-theoretic foundations are in use in this literature; the difference is declared inessential and then not mentioned again. what this limits: a comparison between results in this paper and results of Clausen and Scholze: the two use different set-theoretic foundations, one with universes and one avoiding them, which the author states is a minor difference not affecting any argument in an essential way Quotation withheld because this record is not cleared for redistribution. The agreement between three cohomology theories is claimed for spaces satisfying a stated structural condition, not in general. what this limits: the claim that condensed and sheaf cohomology agree: the three-way agreement is asserted for a topological space admitting a locally finite CW structure, so it is a conditional identification rather than a general one Quotation withheld because this record is not cleared for redistribution. hypersheaves hypersheaves Quotation withheld because this record is not cleared for redistribution. Postnikov complete sheaves Postnikov complete sheaves Quotation withheld because this record is not cleared for redistribution. condensed cohomology condensed cohomology Quotation withheld because this record is not cleared for redistribution. compact Hausdorff spaces compact Hausdorff spaces Quotation withheld because this record is not cleared for redistribution. hypercompleteness hypercompleteness Quotation withheld because this record is not cleared for redistribution. sheaf theory - THREE OF THEM, DISTINGUISHED ONLY OFF THE TRUNCATED PART: sheaves, hypersheaves and Postnikov complete sheaves all extend the classical theory and share their truncated objects, so the choice between them is undetectable in the bounded case Quotation withheld because this record is not cleared for redistribution. the subject's name - CONDENSED/PYKNOTIC, BOTH CARRIED: the field is written with a slash between two names for it, in a paper by one of the pyknotic authors thanking the condensed ones Quotation withheld because this record is not cleared for redistribution. the set theory - A DIFFERENCE DECLARED AND THEN SET ASIDE: universes here, no universes there, and an explicit decision not to mention it again Quotation withheld because this record is not cleared for redistribution. extends → Dyckhoff [7, Theorem 3.11; 8] and Clausen–Scholze [23, Theorem 3.2] comparison of sheaf and condensed cohomology Quotation withheld because this record is not cleared for redistribution. builds-on → Clausen–Scholze's condensed/pyknotic mathematics, Barwick–Haine, and Lurie's work on ultracategories Quotation withheld because this record is not cleared for redistribution. Postnikov complete sheaves —distinct-from→ hypersheaves Postnikov complete sheaves hypersheaves strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. For a compactly assembled coefficient ∞-category, the assignment of the ∞-category of Postnikov complete sheaves to a locally compact Hausdorff space satisfies descent along proper surjections. Quotation withheld because this record is not cleared for redistribution. The same result gives proper descent for left complete derived ∞-categories and shows that Postnikov complete sheaves form a sheaf on compact Hausdorff spaces for the topology of finite jointly surjective families. Quotation withheld because this record is not cleared for redistribution. Neither ordinary sheaves nor hypersheaves define a sheaf on the site of compact Hausdorff spaces. Quotation withheld because this record is not cleared for redistribution. Postnikov complete sheaves on a locally compact Hausdorff space embed fully faithfully into condensed objects over that space. Quotation withheld because this record is not cleared for redistribution. Consequently the sheaf cohomology and the condensed cohomology of a locally compact Hausdorff space coincide. Quotation withheld because this record is not cleared for redistribution. For a space carrying a locally finite CW structure, condensed, singular and sheaf cohomology all coincide. Quotation withheld because this record is not cleared for redistribution. Sheaves on a profinite set are Postnikov complete, whereas sheaves on the Hilbert cube are not, so Postnikov completeness is not automatic for compact Hausdorff spaces. Quotation withheld because this record is not cleared for redistribution. The protruncated shape functor fails to preserve pullbacks of compact Hausdorff spaces. Quotation withheld because this record is not cleared for redistribution. Postnikov completion Postnikov completion Quotation withheld because this record is not cleared for redistribution. Postnikov complete Postnikov complete Quotation withheld because this record is not cleared for redistribution. proper descent proper descent Quotation withheld because this record is not cleared for redistribution. compactly assembled ∞-category compactly assembled ∞-category Quotation withheld because this record is not cleared for redistribution. hypercomplete objects hypercomplete objects Quotation withheld because this record is not cleared for redistribution. van Kampen colimit van Kampen colimit Quotation withheld because this record is not cleared for redistribution. Postnikov complete sheaves Postnikov complete sheaves Quotation withheld because this record is not cleared for redistribution.

The full text of this article is not redistributable, so it is not included here. The evidence sentences are withheld with it; the tag layer and its grounding status remain. The article itself is at https://doi.org/10.48550/arxiv.2210.00186.