Let $C$ be a stable $\infty$-category. Is there any categorical construction $C \mapsto T^* C$, where $T^* C$ is another stable $\infty$-category, that specializes to the following?
When $C$ is the derived category of coherent sheaves on a variety $X$, $T^* C$ is the derived category of coherent sheaves on $T^*X$
When $C$ is the derived category of representations of a quiver, $T^* C$ is the derived category of representations of the preprojective algebra attached to the quiver.
I guess a good negative answer would be an example of varieties $X$ and $Y$ with $D(X) = D(Y)$ but $D(T^* X) \neq D(T^* Y)$. Is there a pair of varieties like that?
