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Let $Sch$ be the 1-category of schemes. Is there a cosimplicial scheme $D^\bullet$ and a sequence of schemes $S_1, S_2, ...$ such that the geometric realization of the simplicial set $Hom_{Sch}(D^\bullet,S_n)$ is homotopy equivalent to the topological $n$-sphere $S^n$?

Note: One can ask the same question with $Sch$ replaced by any category, e.g. $Top$, where there is such data: take $D^\bullet$ to be the topological simplices $\Delta^\bullet_\text{top}$ and take $S_n=S^n$. This is the situation I want to "algebraicize."

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  • $\begingroup$ The category of compact Hausdorff spaces embeds in schemes. Just take $D^n$ to be the spectrum of the ring of continuous real valued functions on the simplex. . . . But is this what you really want? Why do you want this? . . . It might be possible to produce an example related to varieties over fields using valuation rings, which have finite spectra, similar to how you can mimic simplices in finite topological spaces, whose posets of opens are the posets of in the category of simplices. $\endgroup$
    – Ben Wieland
    Commented Jan 29, 2022 at 22:23

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