Fundamentals of Lie categories
Žan Grad
TL;DR
The paper develops a differential-geometric framework for Lie categories, categories internal to smooth manifolds, and investigates how invertible morphisms (the core) interact with boundary structures. It generalizes Lie groupoid concepts via left/right invariant vector fields to define Lie algebroids $A^L(\\mathcal{C})$ and $A^R(\\mathcal{C})$, and analyzes how extendability to Lie groupoids aligns these algebroids and ranks under suitable conditions. A refined notion of morphism rank extends classical linear rank, with openness and semi-continuity results guiding when regular morphisms form subcategories and how extension affects algebroid structure. Completeness results for invariant vector fields and a generalized exponential map on Lie monoids with normal boundaries are established, and these tools are applied to a physics-inspired construction in statistical thermodynamics, where entropy is modeled as a functor and the Gibbs equilibrium emerges from a maximum-entropy principle within the Lie-category framework.
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.