The following is a little irrelevant now that PTJ has piped up with his beautiful lemma, but just for the sake of propriety I should correct yet another minor error in what I wrote. (I'm not posting this to the list.) On Tue, Dec 18, 2007 at 01:08:07PM +0000, Robin Houston wrote:
For the implication 2 => 3, consider the following sequence of isomorphisms natural in A \in C:
What I wrote isn't quite right, of course. You should really consider the following sequence of isomorphisms natural in A and B \in C: C(A, B) -> C(B -o D, A -o D) ;apply the functor (- -o D) = C((B -o D) @ A, D) ;closure = C(A @ (B -o D), D) ;symmetry of tensor = C(A, (B -o D) -o D) ;closure again with the rest of the argument as I said. But PTJ's argument is much more elegant! Robin