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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeFeb 15th 2023
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## Definition An [[internal category]] object in the category of [[smooth manifolds]] in which the source and target maps are [[submersions]]. Sometimes, the smooth manifold of morphisms is allowed to have a boundary, in which case the restrictions of the source and target maps to the boundary are required to be [[submersions]] themselves. ## Related concepts * [[Lie groupoid]], [[internal category]] ## References * Žan Grad, _Fundamentals of Lie categories_, [arXiv:2302.05233](https://arxiv.org/abs/2302.05233). <a href="https://ncatlab.org/nlab/revision/Lie+category/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Lie+category">current</a>

   Created:

   Definition

   An internal category object in the category of smooth manifolds in which the source and target maps are submersions.

   Sometimes, the smooth manifold of morphisms is allowed to have a boundary, in which case the restrictions of the source and target maps to the boundary are required to be submersions themselves.

   Related concepts

   • Lie groupoid, internal category

   References

   • Žan Grad, Fundamentals of Lie categories, arXiv:2302.05233.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeFeb 15th 2023
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexAdded the reference to Ehresmann's 1959 paper <a href="https://ncatlab.org/nlab/revision/diff/Lie+category/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/Lie+category/2">v2</a>, <a href="https://ncatlab.org/nlab/show/Lie+category">current</a>

   Added the reference to Ehresmann’s 1959 paper

   diff, v2, current

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeFeb 15th 2023
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexEhresmann only required that the source and target maps have locally constant rank, rather than being submersions, IIRC. I've not seen any concrete examples that took advantage of this greater generality, though.

   Ehresmann only required that the source and target maps have locally constant rank, rather than being submersions, IIRC. I’ve not seen any concrete examples that took advantage of this greater generality, though.

4. • CommentRowNumber4.
   • CommentAuthorJoe Moeller
   • CommentTimeSep 10th 2024
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   Author: Joe Moeller
   Format: MarkdownItexadded link to [[Lie monoid]] <a href="https://ncatlab.org/nlab/revision/diff/Lie+category/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/Lie+category/3">v3</a>, <a href="https://ncatlab.org/nlab/show/Lie+category">current</a>

   added link to Lie monoid

   diff, v3, current