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

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