Hmm. I suppose that restricting to subcategories which respect the group structure on the Hom-sets would be enough to render the problem harder (the group structure of course can be recovered canonically from the underlying category, so this merely refines the class of subcategories we are considering). I would imagine this restriction would also have more repercussions for algebra, anyway... Josh On Sun, 2 Mar 2008, Colin McLarty wrote:
Now let R be a ring and M an R-module. Is there a minimal abelian subcategory of Mod-R containing M? If so, is there a canonical way to describe it?
This question, as posed, is too easy: Just take M and its identity arrow. It will be a zero-object in that subcategory. There may be a better question here guiding Walter Mazorchuk's intuition, but it will have to require something more than just containing the one object.
Colin