#### Vol. 302, No. 2, 2019

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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
##### 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
Primary: 55P65
Secondary: 51H25