Fundamentals of Lie categories

TL;DR

This paper develops the theory of Lie categories, generalizing Lie groupoids, exploring their properties, and connecting them to physical processes and thermodynamics through a functorial approach.

Contribution

It introduces Lie categories, generalizes the concept of invertibility, and links the structure to Lie groupoids, algebroids, and thermodynamics.

Findings

Conditions for Lie categories to form Lie groupoids.

Generalization of rank concept from linear algebra.

Framework for entropy change as a functor in thermodynamics.

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 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.