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Working over a base field $k$ of characteristic $0$, say $K$ is a field (over $k$) and $X$ is a ("nice" if necessary) dg scheme in the sense of Toen-Vezzosi and others, and say $X^0$ is the reduced locus in $X$, in particular ignoring everything in nonzero cohomological degree. The category of dg schemes has a (more or less) simplicial model structure (alternatively: it is an $\infty$-category), so we have a simplicial Hom set $Hom_\Delta(\operatorname{Spec}(K), X).$

Question. Is the homotopy type of $Hom_\Delta(\operatorname{Spec}(K), X)$ discrete and equivalent to $X^0(K),$ or are there examples with nontrivial topology?

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    $\begingroup$ Yes it’s discrete. For any derived scheme X, and any discrete ring R, the space of R-points X(R) is discrete. $\endgroup$
    – crystalline
    Commented Sep 15, 2019 at 3:43

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