Tag: category-theory

In this blog post, I want to explain a way in which I like to motivate Lie groupoids via the theorem on the existence and uniqueness of solutions to ordinary differential equations. The basic idea is that in trying to do away with the choices – the initial time and the initial conditions – we’re naturally led to consider a ‘universal fundamental solution’. This is a matrix function depending on two parameters, the initial and final times, and it must satisfy a homomorphism-type condition. In order to axiomatize this condition, we are automatically led to the definition of a Lie groupoid. Once this has been absorbed, we can then rephrase the existence and uniqueness theorem as a correspondence between ODEs and representations of the pair groupoid of the real line. This is a special case of Lie’s second theorem for Lie groupoids.

Continue reading “Lie groupoids and differential equations” →