An $\mathsf{A}_{\infty}$ version of the Poincar\'e lemma

TL;DR

This paper generalizes the classical Poincaré lemma to a categorified setting involving $$-local systems, establishing that homotopies induce $$-natural transformations via Chen's integrals, and showing homotopy equivalences lead to quasi-equivalences of DG categories.

Contribution

It introduces an $$-categorified Poincaré lemma, linking homotopies to $$-natural transformations in the context of $$-local systems, with explicit constructions.

Findings

Homotopies induce $$-natural transformations between pullback functors.

Homotopy equivalences induce quasi-equivalences of DG categories.

Explicit description of transformations via Chen's iterated integrals.

Abstract

We prove a categorified version of the Poincar\'e 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.