Dear colleagues, You will probably be surprised to receive this mail directly, and not through the category list. It has been sent to the list almost one month ago, and rejected by the "moderator" who did not give any reason except the fact that I refused to change a single word in my mail, which, of course I wouldn't. I had received more than 30 answers to a previous mail, none of which had been censored, in particular no less than 15 mails from Marta Bunge in less than one week! I had begun to answer them in detail, and the mail which has been censored, which I send now without changing a single word, contained a few more answers. I still have a few more persons to answer to. I shall do it very soon. It has taken me quite a long time to find the e-mail addresses of some colleagues, and many are probably missing, but I'm patient and I shall complete my own mailing list. Hence the delay. I have added to these answers a PDF file which contains part of what I call my "Kafka file" where you'll see how the "moderator" had already stopped me once. I let everybody judge his methods Mister moderator, I always answer to my mails. You have stopped me once, you will never stop me any more. The category list does not belong to you! SECOND ANSWERS ANSWER TO DISKIN Thank you for this very interesting historical information, which I quote: "You may find it interesting that a particular case of the condition (in the particular context of hyperdoctrines-as-logical-theories) was formulated by Paul Halmos in his Algebraic Logic (1962). Halmos considered trivial algebraic theories with variables as the only terms and their substitutions as morphisms. A direct comparison is not straightforward because Halmos' formulation was for FOL without equality. Yet with the help of Robert Seely's paper "Hyperdoctrines, natural deduction and the Beck condition" (1983), one can find what part/form of the condition was stated by Halmos." I shall make a few remarks (i) What is at stake here is not the the Chevalley condition, even in a very special case, but the possibility, using this condition, to identify descent conditions to algebras over a triple. Is there anything similar in the paper of Halmos, even in this special case? (ii) It has taken all of 21 years, from 1962 to 1983, to work out a similarity between the paper of Halmos and a special case of the Chevalley condition, thus, as you say, "a direct comparison is not straightforward". When i first read Freyd's Adjoint Functor Theorem, it took me less than 10 minutes to find a very straightforward similarity between Freyd's theorem, and a theorem of Bourbaki's "Theorie des ensembles". All was there, even the "solution set condition". It sufficed to change a few words: Use the words "categories" and "functors", replace "structure" by "object", substructure" by "subobject", and of course, last but not least, to translate Bourbaki in English. I pointed out this "similarity" to Mac Lane the very first time I met him, in 1965. He remembered my remarks, and mentions this similarity in a small historical note p.54 of his CWM. With a small, quite understandable, error on the dates: He mentions 1957 for Bourbaki's result. I'm sure it was earlier, but Mac Lane had consulted the 1957 edition of Bourbaki without being aware that the result had figured in previous editions. I think it was due to Samuel, a member of the Bourbaki group at the time, and if my memory is correct, it must have been obtained as early as 1948. As I do not have the first editions of Bourbaki,I cannot tell with more precision. And of course I cannot tell either If Freyd new the result of Bourbaki when he published his theorem. My guess is that he did not, because Mac Lane seemed quite surprised when I mentioned Bourbaki. ANSWER TO JANELIDZE Thank you for having mentioned Benabou-Roubaud in your paper. Thank you also for having avoided to "give me a lesson" of moderation, or anything else. But, I quote you: "Thanks to an important observation by Bénabou and Roubaud and by Beck, the monadic description covers descent also in the abstract context of a bifibred category satisfying the Beck-Chevalley condition. ...,and the precise reference on your paper with Roubaud is given, while Beck's name is also mentioned but without any reference to a paper (since as far as we knew he did not publish such a paper)". I'm totally sure of your honesty in crediting also Beck. But I'd greatly appreciate precise answers to the following three questions: (i) Since there was no published paper by Beck on the subject, what led you to think that he had, independently, obtained PRECISELY the same result as we had? (ii) You mention that this result, in all generality, requires bi- fibations satisfying the Chevalley condition (which you call Beck- Chevalley, and which I have seen frequently called the Beck condition) I recall that I, together with Roubaud, Verdier, and many others, learnt the Chevalley condition in a public course, given by Chevalley in 1964. The audience was familiar with fibered categories, which according to Barr "didn't exist at the time. Could you give me a reference to ANY paper, Beck's or anybody else's, mentioning fibrations published by a member of "Category land" before 1965? (iii) Do you SINCERELY believe that Peter Johnstone had never heard of the Benabou-Roubaud note, almost 30 years after it appeared, when he published "The Elephant"? I do not care for "polite" or "diplomatic" answers. I want "honest" answers, and I trust you enough to expect from you THAT kind of answer. Best regards, Jean ANSWER TO MARTA BUNGE (PART 1) (I urge everybody, and in particular Marta, to read Part 2 before drawing any "conclusion" from Part 1. On the other hand I'm really too tired to re-write some passages of Part 1, and delete some others, in order to make a single, coherent, answer.) Dear Marta, I was tempted to write "Marta Dear", but my English isn't so good, and I didn't relish the idea of receiving many lessons in "good manners", so i'll let it be Dear Marta. Dear prolific Marta, You have given me a lot of work, de you know Marta, I had to write almost completely 3 versions of my answer to you. I have kept the first drafts, but I shall not reproduce them completely. The first one, after your first 3 mails, was very pleasant to write. I thanked you, with profusion, for the very clear precisions contained in your third mail, and I made a few comments about, I quote you :"the ongoing controversy about fibrations versus S-indexed categories" The second was quite another story. I wrote it after I received 6 more mails from you, the last ones contradicting the first ones, where you obviously had rallied the "politically correct category landers". You even called in support Hermida and Pavlovic. And even Johnstone realized you were a little bit overdoing it. Thus the first words of my second draft were: "Tu quoque, Marta?" And, among other things I asked you to be kind enough to send 3 more mails, in order to reach a nice round total of a dozen. Obviously I was too modest, because, as of to-day the number is 14. Seems you'll soon be in the "Guinness book of Records". Although I have read carefully, many times, all your mails, I hope you'll excuse me if i answer only to the last ones. I suppose they describe your present position. Or have you yet a new one? I quote you (November 6) "It is just that, to me, and inj connection with my recent work on n- stacks, it makes more sense to have discovered the descent theorem first and then having given tripleability conditions in the general case, than the other way around" Thank you for bringing in the fabulous new historical argument : "it makes more sense" I'm sure that Siasheff, Hermida, Pavlovic, Duskin, etc, and even Johnstone and Lawvere, will be delighted by such a proof. And of course Benabou and Roubaud! You are, again, overdoing it a little, Marta, I can understand you must be very tired, what with all the mails you've been sending me! I shouldn't even care to answer such an argument, but by now you know that I am a thorough person. And a polite one, in-spite of what some people on this list may say. If "it makes more sense" it should have been stated in Beck's thesis BEFORE his tripleability theorems, and people who had access to his thesis, e.g. Duskin, or who mentioned his tripleability theorems, or also Mac Lane, would have mentioned "his" preliminary descent ones. Now Beck's thesis is available. Is there any trace in it of the Chevalley condition, or of descent theorems? Sorry Martha, no go! I guess you are in for a new "no rude" correction by Johnstone. I quote you again "As I said before, in order to prove the Benabou-Roubaud theorem (which they did not in the Comptes Rendues, but which we did in Bunge- Pare), one does not need the Beck Tripleability theorem (as they say so themselves). I add that to use it for that purpose would be akin to trying to use a hammer to kill a fly, quite ineffectively too. It is the *combination* of the two (quite distinct) theorems that constitutes a powerful tool in applications -- in order to prove that a certain fibration is a stack (say), one may resort first to the Benabou-Roubaud theorem which relates descent with tripleability and then, in order to prove tripleability, one may resort next to Beck's Tripleability theorem, or to any variation of it (such as Duskin's, as we did)". Thank you even more, on my behalf and on Roubaud' s, for doubting we had a proof before we published our note, and not doubting that Beck had both the theorem, and the proof, when he "unpublished" any of them! This is a serious accusation Marta, and I'm sure that Roubaud, who is understandably very angry, will appreciate it greatly! For the sake of the "young ones" who have not lived in these prehistorical times, I give the following precisions: A Compte Rendus note cannot be longer than 4 pages, where we had not only to state "the" theorem, but also a few applications to descent. A full public proof was given by me at the "Seminaire Benabou" which I have directed for more than 20 years, and where some fairly important parts of category theory were elaborated, presented and discussed. While researching documents for this controversy, I found in an old folder a hand written letter by a certain Marta Bunge dated: May 30, 1973, which I quote: "Dear Jean, Just back from Aarhus for the open space on Topoi. I heard your name mentioned more than once in this connection. However, I did not get hold of anything written by you, although Kock said there was something about the language of Topos theory. Could you send me whatever you have? Please send it to Montreal (Mc Gill University) rather than here, as I will be there shortly. And thanks very much! Forgive me for not writing French; but I have really forgotten the little I used to know. Best regards, Marta Bunge P.S. I am not coming to Oberwohlfach this year as we must be back in Montreal to arrange the house we bought. But what I really regret is having missed April in Montreal, as it seems it was rather exciting. That was bad luck, indeed." Thus the "Seminaire Benabou" existed! It's reputation reached much beyond Paris, or France, since my name was mentioned "more than once" in Aarhus, where I have NEVER BEEN INVITED, either before or after your letter; Thus I think that a public proof, given by me, in such a seminar, is certainly more reliable that a "non proof", given nowhere, that some people, including you, seem to favor. Incidentally the "something" about the language of topos theory, is "nothing but" the internal language of toposes, which I defined in 1972, long before Mitchell, although it is called now "the Mitchell- Benabou" language". I never protested about this fact, although I knew whose the priority was. "April in Montreal" was when I presented this, and other, languages; And I remember clearly the two persons who were most opposed to these languages, and wanted to stick to "diagrammatic" definitions and proofs namely: Joyal and Reyes You probably didn't miss much, what with Benabou speaking. Can you imagine, he hadn't even been capable to give a proof of the, by then well known, Beck's descent Theorem! If you want, provided you express this desire PUBLICLY ON THIS LIST, I'll be quite happy to call "Beck-Bunge-Pare theorem" a result that I thought naively, had been stated, and proved, by Benabou-Roubaud in 1970,many many years before your astounding proof with Pare. I quote you again (November 11) about Johsntone: "In fact, I do not think that Peter Johnstone "supports" himself in this matter -- he has admitted the error. What more do you want him to say?" and I quote Johnstone before commenting "Since their work was independent, I should of course have credited Benabou--Roubaud as well as Beck at this point in the text of the Elephant, and I apologize for not having done so. The reason, I must confess, was that I had simply not come across the Benabou--Roubaud paper clearly, this was a failure of due diligence on my part." I want you to note that, apart from being a thorough and polite person, I'm also a DECENT one. I have not made public the 2 mails you sent me directly. They are in my "Kafa file". I shall ask you straightforward questions for which I'd like straightforward answers, not diplomatic ones; (i) Even if you have sincerely been convinced for some time that Beck had the Chevalley condition before 1964, and the Benabou-Roubaud result before 1970, after all the arguments I gave, do you still believe any of these two things? (ii) Do you think Benabou or Roubaud will ever believe that Johnstone had just "not come across" their joint paper when he gave full credit to jon Beck, and didn't even mention their note in his 1262 references bibliography? Please note the precise formulation of (ii). I could have phrased it either as: (ii)' If you were Benabou or Roubaud, would you have believed that...? , or as: (ii)'' Do you, Marta, believe that ...? I didn't, because I didn't want to embarrass you and create "frictions" between you and Johnstone (decent again, don't you think?) ANSWER TO MARTA BUNGE (PART 2, after receiving the 15th mail) Marta, Dear Marta, If I was a bit sarcastic, in the first part of my answer, please forgive me! Remember that the beginning of of my "second draft" was "Tu quoque, Marta?", and the "?", not in the original text, could mean many things. For me it meant that I have for a long time respected you as a mathematician, and even more important to me, as a person, and I was both surprised and sad, that, I quote you, in your mail (November 06): "Alternatively, one should just quote Benabou-Roubaud, and then add (in parenthesis) that it is believed that Beck had discovered this theorem independently, either as a motivation for his tripleableability theorem or as an application of it. The fact is -- we do not even know which. I will of course abide by the general consensus and say no more on this delicate issue." If "it is believed" by whom, and with what proof?, and if "we don't even know which", what forced you to "abide by the general consensus"? You have corrected this impression in your last mail, and I cannot thank you enough for that. I quote your "last mail" (last, for the time being, but I very much hope it won't be the "absolute" last!) "even you dismiss me as not important enough to reply to. Believe me,Jean, this does not concern me at all. I am past personal pride." No, Marta, I don't "dismiss" you, and the length of the first part of my answer where I was "half angry" and "half sad" proves that for me you were, and of course still are, "important enough to reply to". Don't be silly Marta, you deserve a lot of "personal pride", for your maths, your scruples, and your honesty. And I'm afraid in these sad times you'll need all the pride you can gather. And I, Jean Benabou, will be very proud if you still think of me as your friend. You have already, in this mail, answered in the best possible manner to question (i) of the first part, remains (ii), and also, in a private mail, I'd very much appreciate an answer to (ii)''. I look forward to your next mail, "off record" if you prefer, meanwhile, best regards, and, if you allow me, a big big hug, Jean A QUESTION TO JACK DUSKIN Dear Jack, Quite surprisingly, you have not, so far, participated to this discussion. So, for the sake of a very old friendship, which hasn't changed on my side, I ask you: Come on Jack, be a "mensch", tell us when you, personally, heard Beck mention "his" result, and therefore decided to talk about the Benabou- Roubaud-Beck theorem? Before Tulane, where you were, when I spent a few weeks? But then, why didn't you tell ME during one of the many evenings we spent enjoying good drinks, good mathematical conversations, good "many things" ? Tres amicalement, Jean