تقرير
Fundamentals of Lie categories
| العنوان: | Fundamentals of Lie categories |
|---|---|
| المؤلفون: | Grad, Žan |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics Mathematical Physics |
| مصطلحات موضوعية: | Mathematics - Differential Geometry, Mathematical Physics, Mathematics - Category Theory, 22A99, 22A22, 53Z05, 58H05 |
| الوصف: | 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. Comment: 38 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2302.05233 |
| رقم الأكسشن: | edsarx.2302.05233 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |