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Consider the site $\mathbf{CHaus}$ of compact Hausdorff spaces together with the finitely jointly surjective families of maps as coverings. Restriction induces an equivalence of categories $$ \mathbf{Sh}_{\mathbf{An}}(\mathbf{CHaus})^{\mathrm{hyp}} \simeq \mathbf{Sh}_{\mathbf{An}}(\mathbf{ExtrDisc}) = \mathbf{Cond(An)} $$ also known as the condensed anima.

What is an example of a sheaf $F: \mathbf{CHaus}^{\mathrm{op}} \to \mathbf{An}$ which is not a hypersheaf?

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    $\begingroup$ You're probably aware, but in the (almost) equivalent language of pyknotic and pseupyknotic spaces, this is also claimed in Barwick and Haine, Pyknotic objects I, Warning 2.2.7. No example is given there, presumably because it's pretty hard to write down anything at all that is not constructed from representables. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Sep 21, 2023 at 22:17
  • $\begingroup$ @R.vanDobbendeBruyn Thank you for your comment! I'm aware indeed, maybe it would already be a good start to provide a proof of the claim that $\mathbf{Sh}_{\mathbf{An}}(\mathbf{CHaus})$ is not hypercomplete which I already cannot see. $\endgroup$
    – Qi Zhu
    Commented Sep 24, 2023 at 15:42
  • $\begingroup$ Hmm, I thought that was the same statement, as a hypersheaf is the same thing as a hypercomplete object [HTT, Corollary 6.5.3.13], and an $\infty$-topos is hypercomplete if and only if all of its objects are [HTT, text before Lemma 6.5.2.9 and text before Remark 6.5.2.11]. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Sep 24, 2023 at 18:33
  • $\begingroup$ @R.vanDobbendeBruyn It's funny, my master's thesis supervisor said the same thing! I was rather thinking of the following though: In the beginning question I asked for an explicit example of such a sheaf but possibly there is an abstract argument (without explicitly constructing an example) that shows that $\mathbf{Sh}_{\mathbf{An}}(\mathbf{CHaus})$ is not hypercomplete (maybe by showing that it doesn't have some property that hypercomplete topoi would have?). $\endgroup$
    – Qi Zhu
    Commented Sep 25, 2023 at 9:08

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