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.
Keywords: Stacks, differentiable stacks, orbifolds, vector fields, flows
2000 MSC: 37C10, 14A20, 18D05
Theory and Applications of Categories,
Vol. 22, 2009,
No. 21, pp 542-587.