Tannaka duality for enhanced triangulated categories. I: Reconstruction
Summary: We develop Tannaka duality theory for dg categories. To any dg functor from a dg category \(\mathcal A\) to finite-dimensional complexes, we associate a dg coalgebra \(C\) via a Hochschild homology construction. When the dg functor is faithful, this gives a quasi-equivalence between the derived dg categories of \(\mathcal A\)-modules and of \(C\)-comodules. When \(\mathcal A\) is Morita fibrant (i.e. an idempotent-complete pre-triangulated category), it is thus quasi-equivalent to the derived dg category of compact \(C\)-comodules. We give several applications for motivic Galois groups.
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