A couple of people have pointed out to me - in private, I think - that the question has a trivial answer (namely, the subcategory consisting of just M and its identity map). Sorry. I probably misinterpreted what Walter said to me. Tom
-----Original Message----- From: cat-dist@mta.ca [mailto:cat-dist@mta.ca] On Behalf Of Tom Leinster Sent: Monday, March 03, 2008 5:37 AM To: categories@mta.ca Subject: categories: Minimal abelian subcategory
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