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

TL;DR

This paper develops a homotopy theory framework for Lie $ $-groupoids, enabling the study of their properties and the integration of $L_ $-algebras, with applications to their algebraic and geometric structures.

Contribution

It introduces an incomplete category of fibrant objects for Lie $ $-groupoids, establishing a homotopy theory compatible with their integration from $L_ $-algebras.

Findings

Lie $ $-groupoids form an incomplete category of fibrant objects

Integration functor preserves weak equivalences and fibrations

Acyclic fibrations are characterized as hypercovers

Abstract

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 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 arXiv:1809.05999, 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, 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 `quasi-split fibrations'. Such fibrations include acyclic fibrations as well as fibrations that arise in string-like extensions. In particular, integration sends $L_\infty$ quasi-isomorphisms to weak equivalences, quasi-split fibrations to Kan fibrations, and preserves acyclic fibrations, as well as pullbacks of acyclic/quasi-split fibrations.