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Motivated by rational homotopy theory, we study a representable presheaf ofgroups $\mathbf{\mathfrak{P}}$ on the homotopy category of cocommutativedifferential graded coalgebras, its Lie algebraic counterpart and its linearrepresentations. We prove a Tannaka type reconstruction theorem that$\mathbf{\mathfrak{P}}$ can be recovered from the dg-category of its linearrepresentations along with the forgetful dg-functor to the underlyingdg-category of chain complexes.

Representable presheaves of groups on the homotopy category of cocommutative dg-coalgebras and Tannakian reconstruction