Principal actions of stacky Lie groupoids

Abstract

Stacky Lie groupoids are generalizations of Lie groupoids in which the "space of arrows" of the groupoid is a differentiable stack. In this paper, we consider actions of stacky Lie groupoids on differentiable stacks and their associated quotients. We provide a characterization of principal actions of stacky Lie groupoids, i.e., actions whose quotients are again differentiable stacks in such a way that the projection onto the quotient is a principal bundle. As an application, we extend the notion of Morita equivalence of Lie groupoids to the realm of stacky Lie groupoids, providing examples that naturally arise from non-integrable Lie algebroids.

Weak representations, representations up to homotopy, and VB-groupoids

“…Weak representations are, in particular, weak actions of groupoids; the definition of a weak action is a repurposed version of Burszytyn, Noseda, and Zhu's definition of the action of a stacky Lie groupoid on a stack (see Definition 3.15, [4]). Definition 4.1.…”

“…Proof. See Lemma 3.21, [4]. Now, we may define weak representations of a groupoid G. As is the case with groupoid actions, weak representations are groupoid actions on smooth linear spaces.…”

“…Remark 5.2. As noted above, weak groupoid actions are inspired by the definition of a stacky Lie groupoid actions of Bursztyn, Noseda, and Zhu [4]. Similarly, the construction of an action groupoid associated to a weak groupoid action is also inspired by their work.…”

“…To define equivariant maps, we again refer to the work of Bruszytyn, Noseda, and Zhu (see Definition 3.20, [4]): Definition 4.3. Let f : H → G 0 and f ′ : H ′ → G 0 be two Lie groupoid maps to G 0 and let (A, α, ε) and (A ′ , α ′ , ε ′ ) be weak actions of G on H and H ′ , respectively.…”

String Principal Bundles and Courant Algebroids

“…14 (action Courant algebroids). A quadratic Lie algebra (k, [−, −] k , (−, −) k ) gives rise to a string Lie 2-algebra [42, Sect.…”