ScalingStacks

Proof. If $X$ is perfect, so that $\qc(X)=\operatorname{Ind}\operatorname{Perf}(X)$, we claim that compact and dualizable objects agree, and hence compact objects generate, so that (1) implies (2).

To see the claim, it suffices to show that the full $\infty$-subcategory of dualizable objects of $\qc(X)$ is idempotent complete (since it is also stable). Since $\qc(X)$ is idempotent complete, this is equivalent to showing that dualizable objects are closed under retracts. However, for a retract $N$ of a dualizable object $M$ one can explicitly write down the unit and trace maps for $N\otimes{\mathcal{H}om}(N,\mathcal{O}_{X})$ and confirm the necessary conditions. We leave this to the reader.

Conversely, if $\qc(X)$ is compactly generated, then $\qc(X)$ is the Ind-category of its compact objects, and hence by assumption also the Ind-category of its dualizable objects, so that (2) implies (1). ∎