The reconstruction part of the Tannaka-Krein duality states that a compact group X can be recovered from the category of its finite dimensional unitary representations. Grothendieck suggested and Saavedra showed in his thesis that an analogous reconstruction holds true for affine group schemes. This result lead to the notion of Tannakian categories, which was further studied by Deligne.
In this talk, we study representable presheaves of groups on the category of cocommutative differential graded coalgebras, motivated by rational homotopy theory. Such presheaves can be considered as a dual notion to differential graded affine group schemes. We introduce an analogous reconstruction result for these presheaves. More precisely, we reconstruct such presheaf from the category of its (not necessarily finite dimensional) representations. As a consequence, we give an alternative reconstruction for (differential graded) affine group schemes.
This is a joint work with Jae-Suk Park.