14
$\begingroup$

I know that a consequence of the Gabriel-Popescu theorem (i.e., every Grothendieck category is a torsion-theoretic localization of a full category of modules) is that any Grothendieck category (which by definition is cocomplete) is complete. I guess that this is not true for general abelian categories, so here is the question:

is it true that a cocomplete abelian category is complete? If no (as I suppose) is there some canonical counterexample?

  • 1
    The question was answered on mathoverflow, giving an example of a cocomplete but not complete Abelian category.2014-10-23

1 Answers 1

6

I think it's a good idea to take questions off the unanswered list whenever possible. Here is a direct link to the MO answer: https://mathoverflow.net/a/184486/2926.