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I am asking for references and discussions of statements of the form

Every bounded hypercover can be refined by an ordinary cover

By "bounded" I mean "finite height". E.g., are there conditions for a site making this statement true?

My impression is that the statement is true at least for paracompact topological spaces with the Grothendieck topology of open sets, and that, for example, Lemma 7.2.3.5 in

Lurie, Jacob, Higher topos theory, Annals of Mathematics Studies 170. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14049-0/pbk; 978-0-691-14048-3/hbk). xv, 925 p. (2009). ZBL1175.18001.

can be used to prove this (though some kind of induction would be needed). Has this been proved anywhere explicitly?

My question is related to the question how descent w.r.t. covers differs from descent w.r.t. hypercovers, which has been discussed lately in Necessity of hypercovers for sheaf condition for simplicial sheaves, in When is the localization of all hypercovers equivalent to that of Čech covers?, or in descent implies hyperdescent. In particular, I am aware of the Appendix A in

Dugger, Daniel; Hollander, Sharon; Isaksen, Daniel C., Hypercovers and simplicial presheaves, Math. Proc. Camb. Philos. Soc. 136, No. 1, 9-51 (2004). ZBL1045.55007.

There, Theorem A.10 states that descent w.r.t. to bounded hypercovers is equivalent to descent w.r.t. covers, without any conditions on the site. But I am interested in the more special question about refinement of bounded hypercovers, which, I guess, would then imply the statement about descent.

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  • $\begingroup$ Just for ease of checking the actual statement of Lemma 7.2.3.5, here is Higher Topos Theory. $\endgroup$
    – David Roberts ♦
    Commented Aug 19, 2021 at 6:44
  • $\begingroup$ Presumably if one takes the numerable site (i.e. all topological spaces, with numerable open covers instead of all open covers), then a result analogous to the one for paracompact (Hausdorff?) spaces holds. Not particularly helpful, but at least a mildly different example. $\endgroup$
    – David Roberts ♦
    Commented Sep 15, 2021 at 7:15
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    $\begingroup$ I thought the best constructive reference was Prop 3.6.63 in arxiv.org/pdf/1310.7930.pdf#page287. Maybe there is some subtlety I’m missing but the proof made sense to me. $\endgroup$
    – cheyne
    Commented Jun 28, 2023 at 15:43
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    $\begingroup$ @cheyne: yes, this is exactly what I was thinking. Thank you! $\endgroup$
    – Konrad Waldorf
    Commented Jun 29, 2023 at 19:32

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