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 -local systems. More precisely, we show that any smooth homotopy between maps f and g induces an A-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 -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
Authors
Camilo Arias Abad
Escuela de matemáticas
Universidad Nacional de Colombia Sede Medellín
Medellín
Colombia
Alexander Quintero Vélez
Escuela de matemáticas
Universidad Nacional de Colombia Sede Medellín
Medellín
Colombia
Sebastián Vélez Vásquez
Escuela de matemáticas
Universidad Nacional de Colombia Sede Medellín
Medellín
Colombia