Hi Tom, Silly observation, but wouldn't the contractible category consisting only of M and its identity morphism constitute an abelian subcategory by this definition, albeit one that is trivial? It would seem that the question for full subcategories is more interesting (and harder). Best, Josh On Sun, 2 Mar 2008, Tom Leinster wrote:

My colleague Walter Mazorchuk has the following question.

Being abelian is a *property* of a category, not extra structure. Given an abelian category A, it therefore makes sense to define a subcategory of A to be an ABELIAN SUBCATEGORY if, considered as a category in its own right, it is abelian. Note that a priori, the inclusion need not preserve sums, kernels etc.

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?

Any thoughts or pointers to the literature would be welcome. Feel free to assume hypotheses on R (it might be a finite-dimensional algebra etc), or to answer the question for full subcategories only.

Thanks, Tom

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