DG-category and simplicial bar complex

TL;DR

This paper establishes a homotopy equivalence between certain DG categories related to a differential graded algebra and comodules over its simplicial bar complex, with applications to classifying structures in algebraic geometry.

Contribution

It proves a homotopy equivalence involving DG categories and comodules over the bar complex, and constructs coalgebras for classifying nilpotent variations of mixed Tate Hodge structures.

Findings

Homotopy equivalence between DG categories and comodules over bar complex.

Equivalence of homotopy categories of A-connections and comodules on homology.

Construction of coalgebras for classifying mixed Tate Hodge structures.

Abstract

In this paper, we prove that the DG category of DG complex of DG category of a differential graded algebra A is homotopy equivalent to that of comodules over the simplicial bar complex of A. Under the assuption of connectedness of A, we show the homotopy category of A-connection is equivalent to comodules on the homology of bar complex. As an application, we construct coalgebras classifying nilpotent variation of mixed Tate Hodge structures on algebraic varieties.