# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1903.09206