# Tannaka duality for enhanced triangulated categories I: reconstruction

## TL;DR

This paper develops a Tannaka duality framework for dg categories, establishing a correspondence between dg categories and dg coalgebras, with applications to motivic Galois groups.

## Contribution

It introduces a novel Tannaka duality theory for dg categories using Hochschild homology, linking dg categories to dg coalgebras and their comodules.

## Key findings

- Faithful dg functors induce quasi-equivalences between derived categories.
- Morita fibrant dg categories are quasi-equivalent to categories of compact comodules.
- Applications include insights into motivic Galois groups.

## Abstract

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.