nLab Lie category

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

The notion of Lie category goes back to

• Charles Ehresmann, Catégories topologiques et categories différentiables, Colloque de Géométrie différentielle globale, Bruxelles, C.B.R.M., (1959) pp. 137-150 (pdf, zbMath:0205.28202)

which also allowed for $C^r$ manifolds and structure maps.

A more recent study, including the case involving manifolds with boundary, is

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