# Compact generation of quasicoherent sheaves on mapping stack

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 $$G$$?

• 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. – dhy Nov 28 '18 at 6:40