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arXiv · 1808.06682

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

Camilo Arias Abad·Alexander Quintero Velez·Sebastian Velez Vasquez
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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.

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Camilo Arias Abad, Alexander Quintero Velez, Sebastian Velez Vasquez. 2018-08-20. An $\mathsf{A}_{\infty}$ version of the Poincar\'e lemma. https://doi.org/10.2140/pjm.2019.302.385

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