Example of a (presentable $k$-linear $\infty$-)category which is dualizable but not compactly generated?

Is there an example of a presentable, stable, $k$-linear $\infty$-category which is dualizable but not compactly generated, where $k$ has characteristic zero, and which is $\text{QCoh}(X)$ (by which I mean the derived dg category of quasicoherent sheaves on $X$) for some prestack $X$?

Or, perhaps by removing some of the conditions above, e.g. being $\text{QCoh}(X)$, $k$ having characteristic zero, stability.

• What do you mean by $\mathrm{QCoh}(X)$? The derived ∞-category of $X$? The ∞-category of ind-pseudocoherent complexes? (I am more used to see that symbol denote the 1-category of quasicoherent sheaves) Mar 20, 2018 at 9:24
• I think this is open even if you drop the $\infty$, see this paper where they have a partial result in that direction. Mar 20, 2018 at 9:47
• @DenisNardin Yes, sorry, I mean the dg derived category of quasicoherent sheaves. Mar 21, 2018 at 0:59

If $X$ is a locally compact topological space, then $\mathrm{Shv}(X, \mathrm{Mod}_{k} )$ is a presentable $k$-linear stable $\infty$-category which is dualizable (in fact, self-dual), but is rarely compactly generated (for example, this fails for $X = \mathbf{R}$).
• If I may, a reference for this is section 21.1.7 in math.harvard.edu/~lurie/papers/SAG-rootfile.pdf. Also worth noting is that a stable $k$-linear $\infty$-category is dualizable if and only if it is a retract of a compactly generated one (Theorem D.7.0.7 in op. cit.). Mar 20, 2018 at 12:05