I want to comment on Juergen's question. It is possible, I suppose, that Grothendieck used the phrase "dualizing object" somewhere, but as far as I am aware, he never did anything with them. It is not, after all, a difficult concept. He certainly never talked about "*-autonomous categories", although Grothendieck (and everyone else who ever gave it a thought) was surely aware that finite dimensional vectors spaces and finite abelian groups were such. If he ever isolated the concept as an interesting one, I am unaware of it and, in any case, I don't believe he ever pursued it (someone would surely have let me know by now). This mad insistence on giving Grothendieck credit for every fleeting idea he may (or even may not) have mentioned somewhere is a perfect example of how the star system (no pun intended) has permeated our consciousness. I am not, of course, blaming Grothendieck for any of this. In another instance, one of the best ideas I ever had has been named after Euler, who never heard of cohomology groups. Now, as I have said elsewhere, I was very much aware of the "pairs" of vector spaces used by the topological vector space theorists when I created the Chu construction. I believe that the sources I had seen stuck to the separated extensional case. They did not mention that this construction was originally due to Mackey, although I eventually (fairly recently) tracked it down. It was in his PhD thesis, in fact. I believe he did not stick to the separated extensional case. But neither he, nor anyone else I read talked about morphisms of pairs. In the se case, the morphisms are obvious. In the general case a bit less so. The duality was, of course, obvious and the main raison d'etre for the pairs. But, although it was obvious how to make the morphisms between two pairs into a vector space, no one seems to have even raised the question of making it into a pair. That it is possible and even easly struck me--and still strikes me--as an amazing bit of magic. Michael ------------------------------------------------------------------- History shows that the human mind, fed by constant accessions of knowledge, periodically grows too large for its theoretical coverings, and bursts them asunder to appear in new habiliments, as the feeding and growing grub, at intervals, casts its too narrow skin and assumes another... Truly the imago state of Man seems to be terribly distant, but every moult is a step gained. - Charles Darwin, from "The Origin of Species"