Derived analogues of the Tannakian reconstruction theorems of Lurie and Bhatt-Halpern-Leistner are proved over animated rings, with QCoh enhanced to a Θ-category carrying the symmetric algebra monad.
Theta-Categories and Tannakian duality
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abstract
We introduce a notion of $\Theta$-categories, which is a refinement of the notion of symmetric monoidal $\infty$-categories. We use this notion to prove a Tannakian duality statement, relating $\Theta$-categories with fpqc-stacks by means of a certain stack of fiber functors in the context of $\Theta$-categories. This provides, over a base ring of arbitrary characteristic, a strong link between Tannakian $\Theta$-categories and the schematic homotopy types.
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Tannakian reconstruction in derived algebraic geometry
Derived analogues of the Tannakian reconstruction theorems of Lurie and Bhatt-Halpern-Leistner are proved over animated rings, with QCoh enhanced to a Θ-category carrying the symmetric algebra monad.