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2020 On the homotopy theory for Lie $\infty$–groupoids, with an application to integrating $L_\infty$–algebras
Christopher L Rogers, Chenchang Zhu
Algebr. Geom. Topol. 20(3): 1127-1219 (2020). DOI: 10.2140/agt.2020.20.1127

Abstract

Lie –groupoids are simplicial Banach manifolds that satisfy an analog of the Kan condition for simplicial sets. An explicit construction of Henriques produces certain Lie –groupoids called “Lie –groups” by integrating finite type Lie n–algebras. In order to study the compatibility between this integration procedure and the homotopy theory of Lie n–algebras introduced in the companion paper (1371–1429), we present a homotopy theory for Lie –groupoids. Unlike Kan simplicial sets and the higher geometric groupoids of Behrend and Getzler, Lie –groupoids do not form a category of fibrant objects (CFO), since the category of manifolds lacks pullbacks. Instead, we show that Lie –groupoids form an “incomplete category of fibrant objects” in which the weak equivalences correspond to “stalkwise” weak equivalences of simplicial sheaves. This homotopical structure enjoys many of the same properties as a CFO, such as having, in the presence of functorial path objects, a convenient realization of its simplicial localization. We further prove that the acyclic fibrations are precisely the hypercovers, which implies that many of Behrend and Getzler’s results also hold in this more general context. As an application, we show that Henriques’ integration functor is an exact functor with respect to a class of distinguished fibrations, which we call “quasisplit fibrations”. Such fibrations include acyclic fibrations as well as fibrations that arise in string-like extensions. In particular, integration sends L–quasi-isomorphisms to weak equivalences and quasisplit fibrations to Kan fibrations, and preserves acyclic fibrations, as well as pullbacks of acyclic/quasisplit fibrations.

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Christopher L Rogers. Chenchang Zhu. "On the homotopy theory for Lie $\infty$–groupoids, with an application to integrating $L_\infty$–algebras." Algebr. Geom. Topol. 20 (3) 1127 - 1219, 2020. https://doi.org/10.2140/agt.2020.20.1127

Information

Received: 5 March 2017; Revised: 15 January 2019; Accepted: 1 May 2019; Published: 2020
First available in Project Euclid: 5 June 2020

zbMATH: 07207572
MathSciNet: MR4105550
Digital Object Identifier: 10.2140/agt.2020.20.1127

Subjects:
Primary: 17B55, 18G30, 22A22, 55U35

Rights: Copyright © 2020 Mathematical Sciences Publishers

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