Let $k$ be a field of characteristic $0$, and let $\mathcal{C}= \mathbf{Vect}_k^{\leq 0}$ be the $\infty$-category of vector spaces concentrated in degrees $\leq 0$. Consider the category $\mathbf{Pr}(\mathcal{C}):= \operatorname{Fun}(\mathbf{CAlg}(\mathcal{C}), \mathcal{S})$ of prestacks over $k$, where $\mathcal{S}$ is the $\infty$-category of spaces or $\infty$-groupoids.

Suppose we have a grouplike prestack $G \in \mathbf{Pr}(\mathcal{C})$. That is, a functor $G: \mathbf{CAlg}(\mathcal{C}) \to \mathbf{Sp}^{\text{cn}}$, where $\mathbf{Sp}^{\text{cn}}$ is the $\infty$-category of connective spectra, thought of as a functor to spaces by composing with the forgetful functor $\mathbf{Sp}^{\text{cn}} \to \mathcal{S}$. We can then form the iterated classifying spaces $B^nG$.

Suppose we have a nice enough stack $X \in \mathbf{Pr}(\mathcal{C})$ (e.g. a perfect stack). When will the category $\mathbf{QCoh}(\text{Map}(X,B^nG))$ of quasicoherent sheaves on the mapping stack be compactly generated? Is the assumption that $X$ be perfect enough? Do we have to make any assumptions on $G$?

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    $\begingroup$ I don't see any reason for it to be compactly generated, even in a case as simple as $X$ a smooth projective curve, $n=1$, and $G=SL_2.$ There $\operatorname{Map}(X,BG)\equiv\operatorname{Bun}_G(X)$ is not quasi-compact, which you can maybe leverage to show that its category of quasicoherent sheaves is not compactly generated, maybe using techniques along the lines of section $12$ of arxiv.org/abs/1112.2402. $\endgroup$ – dhy Nov 28 '18 at 6:40

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