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
–algebras.
In order to study the compatibility between this integration procedure and the homotopy theory
of Lie
–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
–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
