Homology, Homotopy and Applications

Volume 26 (2024)

Number 2

Homotopy characters as a homotopy limit

Pages: 1 – 20

DOI: https://dx.doi.org/10.4310/HHA.2024.v26.n2.a1

Authors

Sergey Arkhipov (Department of Mathematics, Aarhus University, Aarhus, Denmark)

Daria Poliakova (Centre for Quantum Mathematics, University of Southern Denmark, Odense, Denmark)

Abstract

For a Hopf DG‑algebra corresponding to a derived algebraic group, we compute the homotopy limit of the associated cosimplicial system of DG‑algebras given by the classifying space construction. The homotopy limit is taken in the model category of DG‑categories. The objects of the resulting DG‑category are Maurer–Cartan elements of $\operatorname{Cobar}(A)$, or 1‑dimensional $A_\infty$-comodules over $A$. These can be viewed as characters up to homotopy of the corresponding derived group. Their tensor product is interpreted in terms of Kadeishvili’s multibraces. We also study the coderived category of DG‑modules over this DG‑category.

Keywords

homotopy limit, representation up to homotopy, homotopy character, homotopy monoidal structure

2010 Mathematics Subject Classification

18A30, 55U35

Received 6 September 2020

Received revised 15 September 2022

Accepted 30 October 2022

Published 18 September 2024