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If $k$ is a field (probably of characteristic zero), the usual definition of a derived prestack is a functor $ X \colon {\operatorname{CDGA}}_{k}^{\le 0} \to \operatorname{Spaces} $ from (graded) commutative, connective DG-algebras into spaces. What happens to the ($\infty$-)category of derived prestacks if we replace $ \operatorname{Spaces} $ by $ \operatorname{Spectra} $? What kind of category do we get then? Do we get some kind of stabilization of the ($\infty$-)category of derived prestacks?

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  • $\begingroup$ If you let me consider X as landing in connective spectra (i.e., infinite loop spaces), then these are E_oo-group objects in derived prestacks. (In analogy to classical algebraic geometry: the functor of points of a scheme CAlg -> Set lands in Ab, then it is a commutative group scheme.) $\endgroup$
    – skd
    Mar 28 at 13:40
  • $\begingroup$ Yes, the stabilization of the $\infty$-category of functors from $\mathcal{C}$ to spaces is functors from $\mathcal{C}$ to spectra, for any $\infty$-category $\mathcal{C}$. $\endgroup$ Mar 28 at 17:15

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