Fundamentals of Lie categories

Abstract

We introduce the basic notions and present examples and results on Lie categories – categories internal to the category of smooth manifolds. Demonstrating how the units of a Lie category $\mathcal{C}$ dictate the behavior of its invertible morphisms $\mathcal{G}(\mathcal{C})$, we develop sufficient conditions for $\mathcal{G}(\mathcal{C})$ to form a Lie groupoid. We show that the construction of Lie algebroids from the theory of Lie groupoids carries through, and ask when the Lie algebroid of $\mathcal{G}(\mathcal{C})$ is recovered. We reveal that the lack of the invertibility assumption on morphisms leads to a natural generalization of rank from linear algebra, develop its general properties, and show how the existence of an extension $\mathcal{C}\hookrightarrow \mathcal{G}$ of a Lie category to a Lie groupoid affects the ranks of morphisms and the algebroids of $\mathcal{C}$. Furthermore, certain completeness results for invariant vector fields on Lie monoids and Lie categories with well-behaved boundaries are obtained. Interpreting the developed framework in the context of physical processes, we yield a rigorous approach to the theory of statistical thermodynamics by observing that entropy change, associated to a physical process, is a functor.