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What is known about monomorphisms in the following categories:

  1. Schemes
  2. Complex manifolds
  3. $C^\infty$--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!

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For complex and differentiable manifolds, they are the injective maps. There are lots of those. For maps if finite type of schemes, they are the geometrically injective unramified morphisms. I really don't see the point of the question. –  Angelo Oct 8 '10 at 12:53
    
Could you provide a little motivation here. Why are you interested in monomorphisms and in those particular categories? How is this going to help with your research? –  Andrew Stacey Oct 8 '10 at 12:58
    
I was just curious why people like immersed submanifolds and embedded submanifolds for $C^\infty$--manifolds . One could also try to say open, closed, locally closed...ideal sheaves...like for schemes...but wikipedia does not. Do people use these terms and sheaf theory for $C^\infty$--manifolds? Prof. Vistoli wrote above that an immersed submanifold is not a subobject if I understand correctly. –  Maxim Oct 8 '10 at 13:05
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Maxim: "I was just curious" is not a great way to motivate a question. We call a submanifold a "submanifold" because we've found that it's a useful concept to give a name to because it keeps coming up in differential topology. That suggests that the best strategy for you is to go and find where submanifolds are used and see why each result needs submanifolds and doesn't work for mere injections. Saul: I don't understand that remark. Did you have a particular thing in mind? –  Andrew Stacey Oct 8 '10 at 13:33
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Dear Maxim: Angelo didn't say that immersed submanifolds are not subobjects. Immersions in differential geometry are monomorphisms. (This is nicely explained in the discussion of Frobenius integrability in Warner's book "Foundations of Diff'ble Manifolds and Lie groups".) One reason to care about immersed submanifolds rather than just locally closed submanifolds is seen in Lie theory, to define a concept of (connected) Lie subgroup that allows for every Lie subalgebra to correspond to a (connected) Lie subgroup. The theory of foliations gives another reason. –  BCnrd Oct 8 '10 at 15:29
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