An $\mathsf A_\infty$ version of the Poincaré lemma

Camilo Arias Abad, Alexander Quintero Vélez and Sebastián Vélez Vásquez

Vol. 302 (2019), No. 2, 385–412

DOI: 10.2140/pjm.2019.302.385

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 quasiequivalence on the DG categories of $\infty$-local system.

Keywords

$\infty$-local system, DG category, DG functor, $\mathsf A_\infty$-natural transformation, iterated integral

Mathematical Subject Classification 2010

Primary: 55P65

Secondary: 51H25

Milestones

Received: 29 August 2018

Revised: 25 January 2019

Accepted: 16 March 2019

Published: 27 November 2019