What's the Hochschild homology of the category of constructible sheaves?

Let $X$ be a manifold. Does the Hochschild homology/cohomology of the category of constructible sheaves on $X$ have a more familiar name?

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Aren't there some subtleties in the definition of a HH_* for an $\infty$-category which is not compactly generated? How do you define it? –  Daniel Pomerleano Apr 24 '14 at 12:49
It's defined for any dualizable $\infty$-category (presentable say), not necessarily compactly generated, as the composition of evaluation and coevaluation maps.. then again it's harder to show you're dualizable without compact generation... –  David Ben-Zvi Apr 24 '14 at 16:47