In the case of ordinary schemes, all coproducts exist, so given any constant diagram $D_S:C\to \operatorname{Sch}$, the colimit over $D_S$ is isomorphic to the coproduct of $S$ over the connected components of $C$. In the homotopical setting, colimits over constant diagrams are much more interesting, representing tensors by homotopy types.

Do all colimits over constant diagrams exist in the ∞-category of derived schemes or spectral schemes? Where can I find a proof?

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    $\begingroup$ If you want interesting colimits, you should probably be taking them as derived stacks. $\endgroup$ Commented Oct 8, 2019 at 13:10
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    $\begingroup$ Yes, colimits of schemes will usually be stacks, except in special cases. Here is what I believe is supposed to be true: Let $DM$ be the $\infty$-category of spectral Deligne-Mumford stacks. Let $DM^{et}\subseteq DM$ be the non-full subcategory of al objects, but only etale morphisms. Then (i) $DM^{et}$ has all small colimits, and (ii) $DM^{et}\to DM$ preserves such colimits. Since identity maps are etale, this says that colimits of constant diagrams always exist. $\endgroup$ Commented Oct 8, 2019 at 15:33


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