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I am also interested.
One way to think about this is: Injections that are epimorphisms.
For instance, in the category of Hausdorff topological spaces, inclusions of dense subspaces are epimorphisms. This captures your first example.
A similar example is: In the category of rings, the inclusion of the integers into the rational numbers is an epimorphism.
In order to capture your second example from a category-theoretic perspective, one would have to first cook up a suitable category where both vector spaces as well as plain sets are objects, in the first place. Once some such choice is made, we could again speak about injections that are epi.
In speaking of injections here, I am implicitly talking about concrete categories. More generally one could say that you are asking for examples of morphisms that are monomorphisms and epimorphisms, but not isomorphisms.
From a different point of view, one could simply regard these as examples of universal properties: a vector space is free on its basis, the real numbers are the completion of the rationals, and both of these are universal properties (of a kind fitting into an adjunction, one half of which is a forgetful functor).
That’s of course a good point and maybe closer to what “ml” might be looking for.
In any case, I have taken the occasion to start a list of examples of monos that are epi but not iso, here. Maybe it inspires somebody to add more examples.
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