**Abstract**: In this talk we’ll investigate how to define vector fields and their flows in a tangent category, and how to prove a result about commutation of vector fields and flows in this setting. In the first half of the talk, we’ll introduce the notion of a “curve object” in a tangent category: an object which “uniquely solves ordinary differential equations in the tangent category”. In the second half of the talk, we’ll see how considering “vector fields and flows in the tangent categories of vector fields and flows” leads to a new proof of the commutation theorem for vector fields and flows (Proposition 18.5 in Lee’s “Introduction to Smooth Manifolds”).

**Bio**: Geoff is an Associate Professor at Mount Allison University in Sackville, NB. His general interest is category theory. He is currently investigating how to generalize as much of differential geometry as possible to the setting of tangent categories. Geoff received his PhD at Dalhousie University in 2009 under the supervision of Richard Wood, and subsequently did postdoctoral research at the University of Calgary (supervised by Robin Cockett) and the University of Ottawa (supervised by Rick Blute) before taking up his current position at Mount Allison.