Let $C$ be a [stable $\infty$-category][1].  Is there any categorical construction $C \mapsto T^* C$, where $T^* C$ is another stable $\infty$-category, that specializes to the following?

1.  When $C$ is the derived category of coherent sheaves on a variety $X$, $T^* C$ is the [derived category of coherent sheaves][2] on $T^*X$

2.  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][3].

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?


  [1]: https://ncatlab.org/nlab/show/stable+%28infinity%2C1%29-category
  [2]: https://ncatlab.org/nlab/show/triangulated+categories+of+sheaves
  [3]: https://www.math.uni-bielefeld.de/~wcrawley/dmvlecs.pdf