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On the homotopy theory for Lie $\infty$–groupoids, with an application to integrating $L_\infty$–algebras

Christopher L Rogers and Chenchang Zhu

Algebraic & Geometric Topology 20 (2020) 1127–1219
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.

Keywords
simplicial manifold, Lie $\infty$–groupoid, $L_\infty$–algebra, category of fibrant objects, hypercover
Mathematical Subject Classification 2010
Primary: 17B55, 18G30, 22A22, 55U35
References
Publication
Received: 5 March 2017
Revised: 15 January 2019
Accepted: 1 May 2019
Published: 27 May 2020
Authors
Christopher L Rogers
Department of Mathematics and Statistics
University of Nevada, Reno
Reno, NV
United States
Chenchang Zhu
Mathematisches Institut and Courant Research Centre “Higher Order Structures”
Georg-August-Universität Göttingen
Göttingen
Germany