After giving more thought to my question about Set vs. Ab I decided that it was hopeless to try to resolve as a theorem, and that the only way out was to resolve it with a definition. Definition. A category C is *definable in* a closed category D when it embeds fully (as a category) in every self-dual closed category embedding D (as a closed category). This definition settles by fiat the question of whether *finite* sets can be defined from finite groups, since FinAb is self-dual, as I mentioned previously. Extending this to Ab is a matter of checking that Set does not embed in Ab\op x Ab, which would surprise me greatly. If Set actually does embed in Ab, as some have been telling me privately, this does not contradict my definition, it merely answers my original question in the other direction. To convince me that Set embeds in Ab it will suffice for you to give me representatives of 2 and 3 whose homobjects between them are the representatives of 4, 8, 9, and 27. If you have a more productive notion of "definable in", meaning one that lets you define categories not possible with the above, there is a possibility that some trace of Set has leaked into your definition. If, after convincing yourself that all traces of Set have been eliminated, it is *still* more productive, then this would be a notion of "definable in" well worth discussing. Vaughan Pratt