Skip to main content
2 of 2
Added links and tags
David White
  • 30.3k
  • 9
  • 154
  • 250

Is there a cotangent bundle of a stable $\infty$-category?

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?

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

  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.

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?

David Treumann
  • 4.9k
  • 26
  • 36