An ${\mathsf{A}}_{\infty }$ version of the Poincaré lemma

Abstract

We prove a categorified version of the Poincaré lemma. The natural setting for our result is that of $\infty$-local systems. More precisely, we show that any smooth homotopy between maps $f$ and $g$ induces an ${\mathsf{A}}_{\infty }$-natural transformation between the corresponding pullback functors. This transformation is explicitly defined in terms of Chen's iterated integrals. In particular, we show that a homotopy equivalence induces a quasi-equivalence on the DG categories of $\infty$-local system.

arXiv comment24 pages, final version

References in corpus (7)

• Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence
• Loop Spaces and Connections
• Relative Calabi-Yau structures
• Iterated integrals of superconnections
• Morita cohomology
• String topology and the based loop space
• The colimit of an $\infty$-local system as a twisted tensor product

Cited by in corpus (2)

• Singular chains on Lie groups and the Cartan relations II
• Inverting covariant exterior derivative