Fundamentals of Lie categories

  • Žan Grad

    Instituto Superior Técnico, Lisbon, Portugal
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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  dictate the behavior of its invertible morphisms , we develop sufficient conditions for 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 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 of a Lie category to a Lie groupoid affects the ranks of morphisms and the algebroids of . 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.

Cite this article

Žan Grad, Fundamentals of Lie categories. J. Noncommut. Geom. (2024), published online first

DOI 10.4171/JNCG/563