What is known about monomorphisms in the following categories:

- Schemes
- Complex manifolds
- $C^\infty$--manifolds

and any other kinds of geometric objects that you might think of.

How do we choose a subobject to call it a submanifold (or a subscheme...)? What is the motivation?

Please post all the weird examples you know!

everyLie subalgebra to correspond to a (connected) Lie subgroup. The theory of foliations gives another reason. – BCnrd Oct 8 '10 at 15:29