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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?

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    $\begingroup$ Not sure if this is relevant or not, but the categorical Hochschild homology (CHH) of a monoidal category seems to be a reasonable candidate for part 1 (no idea about part 2). By CHH I mean the category $\mathcal C \otimes{\mathcal C \times \mathcal C^{op}} \mathcal C$ associated to a monoidal category $(\mathcal C, \otimes)$. In the case $\mathcal C = QC(X)$, for X a scheme $CHH(\mathcal C) = QC(LX)$, where $LX = X\times_{X\times X} X$ is the derived loopspace (by a result of Ben-Zvi-Francis-Nadler). The derived loopspace is the same as the odd tangent complex by HKR. $\endgroup$
    – Sam Gunningham
    Commented Nov 22, 2014 at 1:00
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    $\begingroup$ You may complain that this is more like the tangent bundle than the cotangent, but I think some kind of Koszul duality should relate sheaves on this odd tangent bundle with sheaves on the cotangent bundle (at least up to shifts by 2). $\endgroup$
    – Sam Gunningham
    Commented Nov 22, 2014 at 1:02
  • $\begingroup$ Thanks Sam. What if I insist that the construction of $T^* C$ should not depend on any monoidal structure $\otimes$? For example it doesn't seem like such a structure is available in part 2. $\endgroup$
    – David Treumann
    Commented Nov 24, 2014 at 3:30

1 Answer 1

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In the years since this question was asked, some of the theory the OP wanted was developed. First, in Lurie's Higher Algebra, Definition 7.3.2.14 defines the absolute cotangent complex functor $L: C\to T_C$ associated to a presentable $\infty$-category $C$. The next section deals with relative cotangent complexes. This theory is generalized in Lurie's Spectral Algebraic Geometry, Definition 17.1.1.3 and Variant 17.1.1.4 (for the relative version).

Y. Harpaz, J. Nuiten, M. Prasma extended this theory from $\infty$-categories to model categories, in the following three papers:

  • The abstract cotangent complex and Quillen cohomology of enriched categories, arXiv:1612.02608.
  • Tangent categories of algebras over operads, arXiv:1612.02607.
  • The tangent bundle of a model category, arXiv:1802.08031.

Furthermore, Porta and Sala's 2022 paper Two-dimensional categorified Hall algebras works out the connection between cotangent $\infty$-categories and coherent sheaves (see especially 2.3 and 3.2) and includes a discussion of cotangent bundles of quivers on page 10. There, they relate the cotangent stack of a quiver with cohomological Hall algebras, and recover the description in (2) of the OP. There they cite:

See also 2.3 of the Porta and Sala paper Simpson’s shapes of schemes and stacks, from 2020. My expertise is really more on the model category side, but hopefully these references help (and get this old question off the "unanswered queue").

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