Fundamentals of Lie categories
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866909491475251200 |
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| author | Grad, Žan |
| author_facet | Grad, Žan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_05233 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fundamentals of Lie categories Grad, Žan Differential Geometry Mathematical Physics Category Theory 22A99, 22A22, 53Z05, 58H05 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. |
| title | Fundamentals of Lie categories |
| topic | Differential Geometry Mathematical Physics Category Theory 22A99, 22A22, 53Z05, 58H05 |
| url | https://arxiv.org/abs/2302.05233 |