Vector Fields and Flows on Differentiable Stacks

Abstract

This paper introduces the notions of vector field and flow on a general differentiable stack.Our main theorem states that the flow of a vector field on a compact proper differentiable stack exists and is unique up to a uniquely determined 2-cell.This extends the usual result on the existence and uniqueness of flows on a manifold as well as the author's existing results for orbifolds.It sets the scene for a discussion of Morse Theory on a general proper stack and also paves the way for the categorification of other key aspects of differential geometry such as the tangent bundle and the Lie algebra of vector fields.