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

TL;DR

This paper develops a Tannakian reconstruction framework for a presheaf of groups on the homotopy category of cocommutative dg-coalgebras, linking it to its linear representations and dg-category structures.

Contribution

It introduces a Tannaka type theorem that reconstructs the presheaf of groups from its dg-category of linear representations, advancing rational homotopy theory.

Findings

Proves a Tannaka reconstruction theorem for presheaves of groups.

Establishes a correspondence between the presheaf and its linear representations.

Links the presheaf to dg-categories and chain complexes.

Abstract

Motivated by rational homotopy theory, we study a representable presheaf of groups $\mathbf{\mathfrak{P}}$ 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 $\mathbf{\mathfrak{P}}$ 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.