Date: Mon, 22 Dec 1997 10:33:36 +0100 (MET) From: Anders Kock <kock@mi.aau.dk>
Concerning Colin McLarty's questions: The category of groups should really be seen as a 2-category, in fact as part of the 2-category of groupoids.
This gives an approach I had not thought of, and should have: take axioms for the category of categories, and change the description of the generator 2 to make it a groupoid. For the category of categories, 2 has no non-constant, non-identity endofunctors. For the category of groupoids it has exactly one, and that is an involution. (By a constant functor I mean one factoring through the terminal category 1.) A few other changes might be needed, depending on details of the axioms for the category of categories. But the key seems to be that the category of groupoids is cartesian closed and its insertion into the category of categories preserves exponentials--the prominent fact that a natural transformation with all components iso is a natural iso. So the center of the strength of the cat-of-cats axioms can be kept unchanged for the cat of groupoids. This is somehow orthogonal to the approach Mike Barr suggested, in that cartesian closedness of the category of categories is very close to associativity of composition. Mike lost associativity, but kept groups as opposed to groupoids. Anders's strategy gets associativity/cc-ness by foregoing uniqueness of objects. I have no idea how far this shows something objective, and how far it is an accident of the approaches we've thought of.