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

Christopher L. Rogers, Chenchang Zhu

Published 2016-09-06Version 1

Lie $\infty$-groupoids are simplicial Banach manifolds that satisfy an analog of the Kan condition for simplicial sets. An explicit construction of Henriques produces certain Lie $\infty$-groupoids called "Lie $\infty$-groups" by integrating $L_\infty$-algebras. In order to study the compatibility between this integration procedure and the homotopy theory of $L_\infty$-algebras, we present a homotopy theory for Lie $\infty$-groupoids. Unlike Kan simplicial sets and the higher geometric groupoids of Behrend and Getzler, Lie $\infty$-groupoids do not form a category of fibrant objects (CFO), since the category of manifolds lacks pullbacks. Instead, we show that Lie $\infty$-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 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 homotopy equivalent $L_\infty$-algebras integrate to "Morita equivalent" Lie $\infty$-groups.