# Derived prestacks regarded as functors into spectra

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?

• 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.)
– skd
Mar 28 at 13:40
• 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 $\mathcal{C}$. Mar 28 at 17:15