1- TYPOGRAPHY AND MODERATION In this mailing list, such fonts as "bold face", "italics", etc are =20 not accepted, only lower and upper case letters can be used. I =20 mention these trivialities because in february 2006 I received a =20 lesson of typography from Wood. As I receive more and more "lessons" =20 through this mailing list, I have made a file in my computer, with =20 these lessons, an a few other things, under the name of : " Kafka in =20 Category-land". I have consulted my "Kafka-file" thus I can quote =20 precisely Wood's lesson: "Jean Benabou...should also be told that ordinary words written in =20 upper case are understood to be SHOUTED ". I have not the slightest intention to shout in this mail, nor in any =20 other I send through this mailing list. But, in spite of Wood's =20 lesson, because of the drastic typographical limitations of this =20 list, I shall use upper case letters for titles, subtitles, or =20 whenever I want to emphasize a word or a sentence. And if Professor Wood disapproves my decision, I refer him to Freyd =20= and Scedrov who used quite liberally upper case letters in their =20 book, although they had huge typographical possibilities, including =20 very fancy "punctured diagrams". I'm getting a bit tired by the ever increasing number of lessons I =20 receive in this mail. About every subject. Except mathematics, which =20 I'd very much like to receive. Unfortunately in this domain I'm =20 usually on the "giving" side. As for example in my mail on locally =20 cartesian closed categories. Most of you know that English is not my mother language. I try to =20 write it as well as I can. If you find something wrong in my typing, =20 my spelling, my grammar, or even my "style", forgive me. But no more =20 "lessons' please! I shall try to answer to each of the persons who have participated to =20= this discussion. There are many, and there might even be more since I =20= received this morning mails from Janelidze and Hardbaugh. You'll =20 understand that, even if I work full time at it, and I do, it will =20 not be possible to "sort out" the order of my answers, so i'll give =20 each of them as soon as I have enough documentation to complete it. I =20= apologize if some of you will have to wait longer than others. I have been advised many times to use "moderation" in my answers. I =20 am usually a moderate person, but I shall try to be even more so. =20 Please try to make the the same "effort" when addressing to me. I =20 shall use in each individual answer the same "degree of moderation" =20 as the one I found in the mail (or mails) I'm answering to. for =20 example in my : 2- ANSWER TO DUSKO PAVLOVIC =46rom his mail I quote: "but now great jean benabou brought my little name into his argument, together with claudio hermida. i am not sure what he means, but it =20 sounds like claudio and i are the examples of unreliable sources." And I answer : Great Dusko Pavlovic, with all due respect of course, =20 little Jean Benabou would like to point out that it was never him, =20 but Marta Bunge who brought your great name, together with Claudio =20 Hermida's, into this argument. As for your reliability, and the relevance of your testimony in this =20 subject, I do not have the slightest doubt about them. It seems that Peter =20 Johnstone has some, because from one of his mails to Marta I quote: "Without wishing to be rude, I'm not sure that I would take Dusko as an authority on the history of category theory" I do not know what he meant by that, thus I'm afraid that 4 persons: =20 Claudio, Marta Peter, and great Dusko, will have to "sort out" their =20= big problems. They are no concern of little Jean Benabou. Respectfully yours,Jean. 3- ANSWER TO JIM STASHEFF I quote: "Perhaps some one took notes on paper and hasn't thrown them out? It would be great to ad them to Jon's small nachlass." A very good question Jim, even if your purpose is different from mine =20= I totally agree with it. I'm surprised nobody did come out with an =20 answer. And I ask you a straightforward question: Aren't YOU surprised that of this "mythical" result, there is not a =20 single written trace, no publication by Beck or by ANYONE who would =20 have used his result in his work. And no written notes? I won't be as choosy as Peter Johnstone, and will accept any =20 reference in a thesis, a prepublication in the most obscure =20 university, anterior to my joint paper with Roubaud. Fair enough? 4- ANSWER TO MICHAEL BARR =46rom the second answer of Barr I quote "Jon had a precise formulation of a rather simple special case that =20 did not mention either fibrations or indexed categories (the latter =20 didn't exist at the time" This, for me, settles the problem on two counts: (i) I learnt the Chevalley conditions from Chevalley in 1964, in a =20 public course, he is dead, but many people who attended this =20 course,in particular Jacques Roubaud, but many others I know, will be =20= quite ready to testify if you "force me to ask them". The Chevalley condition is about FIBRATIONS, it can of course be =20 reformulated in terms of "indexed categories" which, according to =20 Barr, didn't exist at the time, neither did fibrations, the first =20 mention of them in "Category-land" is in John Gray's paper in La =20 Jolla (1965). Thus Beck couldn't have discovered this condition in =20 1964. And I promise to rise a public debate, outside of this list, if =20= the name of Jon Beck is mentioned about them in the future. Some people seem to believe that there were no categories outside of =20 "Category-land". Typical of this attitude, sorry Mike, is your =20 statement that indexed categories DIDN'T EXIST at the time. They were invented by Grothendieck in 1961, NOT by Lawvere in 1971. The main mistake of this "fertile brain" (El; preface) was to have =20 called them "pseudo-functors" which isn't such a bad name after all. =20 At the same period he did a much much more INEXCUSABLE mistake, =20 namely: not realizing that: The Times They Are A'changin' (Bob Dylan, =20= 1964). And that "prone" and "supine" were obviously more "chic" than =20 the "overworked" "cartesian" and "cocartesian". (I shall examine this question of "terminology" in more detail in my =20 answer to Peter Johnstone) Thus the Benabou-Roubaud result cannot have been proved by Beck. It's =20= a result about Chevalley bi-fibrations, which can be,but why should =20 they be?, reformulated in terms of indexed-categories. But none of =20 the two "existed" according to Barr, and not in "category-land" =20 according to Benabou. (For more details about this "non-existence", I also refer to my =20 answer to Bill Lawvere, which will be come next) 5- ANSWER TO WILLIAM LAWVERE I quote you: "I would like to urge the readers of our categories list to study =20 Jon's 1967 thesis which is now available via TAC Reprints. There one =20 finds Jon's tripleability theorems which were used for example by =20 Benabou and Roubaud in their 1970 paper on descent" Sorry Bill, don't CONFUSE THE ISSUES: We are talking about =20 Chevalley's condition, and my joint result with Roubaud about the =20 relation between "descent data" and algebras for a monad. ABSOLUTELY =20 NOT about Beck's tripleability theorems, which we used, with due =20 reference, in our note. Is there ANY mention in this "now available" =20 thesis of fibrations, of ANY KIND of condition on fibrations, and of =20 "descent data". If there is none, mentioning this thesis is totally =20 IRRELEVANT to the present discussion. One of the major interests of our result, was that it gave the =20 possibility TO USE Beck's theorems, in a totally "new" domain, namely =20= "descent", and that IN ORDER TO DO IT, Chevalley's condition was =20 sufficient. It is also necessary, which means our result CANNOT be =20 improved! Can anyone after 37 years, give a better result, of course =20 in the general case of fibrations (or indexed categories, if you =20 still prefer them!) Of course, in special cases you can get "better" results. Going to =20 the extreme, I can "prove" that, for the identity fibration, every =20 map is a descent map! Our result can be GENERALIZED, if one generalizes the notion of =20 descent. I have had such a huge generalization for many years, in a =20 paper called "what is descent in year 2000?" I am NOT going to talk about it, or about many new results of mine =20 concerning not only fibered categories, but the much more general =20 FOLIATED ones, I have defined and studied, because in these sad, sad =20 times, I'm SURE I would find these results in some other "Elephant", =20 re-named, duly "re-indexated", and with NO reference to me. For =20 "space-saving" reasons, of course! I quote you again: "Ross remembers that Jon arrived in Tulane in 1969 prepared to lecture on triples and descent" First big mistake Bill. Ross mentioned that Jon arrived in Tulane =20 after I left. And I was there in early 1970. So the earliest Jon =20 could have been there was spring 1970, not 1969! Although I have many =20= answers to give, in each case I check all the "details"; and in this =20 case it is NOT a detail. Ross also "remembers" that Jon DID NOT speak. Neither in a formal =20 lecture, nor in INFORMAL DISCUSSIONS, with ANYBODY. I was in Tulane =20 at about the same period, there was quite a concentration of category =20= theorists, and "I remember", that apart from the formal lectures, =20 there were many informal discussions, as there are in any category =20 theory meeting, or in ANY mathematical meeting. Quite surprising, isn't it, that Jon spoke to NOBODY who can =20 "remember" ANYTHING Quotation again "I heard Jon lecturing on that topic already in late 1967 at a =20 meeting of the American Mathematical Society in Illinois. In =20 particular, he explicitly stated the condition that I therefore =20 called the "Beck condition" in my work on Hyperdoctrines (presented =20 to an AMS meeting in NYC in early 1968) On what "topic"? If it is "the" Chevalley condition, Chevalley =20 lectured about it in 1964, i.e. four years BEFORE the 1968 meeting =20 you mentioned (He probably had it before 1964, but 1964 is good =20 enough for me!) You probably remember that I spent the academic year 1966-67 in =20 Chicago. We even shared the SAME office. There were many meetings of =20 the Midwest Category Seminar, And "I remember", with precision, that =20 I was AMAZED by the fact that fibered categories and descent were not =20= mentioned A SINGLE TIME during that whole year. My surprise was all the greater since in Paris, the Grothendieck. =20 school was devoting a lot of time and energy on this "topic"; I might =20= remind to some of those who have small problems with their memory, =20 that Giraud's thesis was published in 1964 and the title was "Methode =20= de la Descente". And of course, of the fundamental paper of =20 Grothendieck, "Categories fibrees et Descente" (1961). As a side remark, you are of course aware that "Hyperdoctrines" are =20 an important, but very special case of fibrations . And I am still =20 waiting for an answer to the question I asked in my previous mail: =20 Has ANYBODY defined a notion of morphism of Hyperdoctrines, and, if =20 nobody has, WHY NOT? Thus sorry Bill, with all due respect, I don't think Jon Beck, =20 "invented" the Chevalley condition, and if he proved ANYTHING about =20 Monads and descent, it must have been in a very, very special case, =20 since as Michael Barr stated it a bit curiously, "indexed =20 categories", let alone fibrations, "did not exist" when we published =20 our note. One more "detail" , I greatly admire your deep mathematical insight, =20 therefore I am convinced that if you had heard Beck speaking about =20 our joint theorem, you would immediately have understood its meaning, =20= and its possible consequences, and either you, or some of your =20 students, or both, would IMMEDIATELY have put the theorem "at work". =20= And it would have found its way very quickly in many papers and =20 books. Just as Beck's tripleability theorems did, and were frequently =20= quoted, and "refined". e.g. in a long paper by Duskin in the Reports =20 of the Midwest Category Seminar (RMCS) , Springer Lecture Notes (LN) =20 106 (1969). No mention of fibrations or descent in Duskin's paper, although Jack =20 quoted in his bibliography: BECK , J., untitled manuscript, Cornell, =20 1966. No mention in this whole (LN). No mention in the next RMCS, LN137,1970. WHERE ARE THE PAPERS BILL? Was all "Category-land" ASLEEP between =20 1967 and 1971? Again a quotation "Later I saw this condition referred to as the Chevalley condition in =20= a paper of J-L Verdier. I do not know whether Jon was familiar with =20 that work of Chevalley" Later than what? Again I insist on the fact that Chevalley's =20 condition is about fibrations. You "do not know", but ,concerning the =20= question of priority of Chevalley, with all due respect, what YOU =20 know is not the issue. Neither is what I know, but at least, I was in =20= Paris in the years 1960-1965, where all these things happened, I =20 participated in some of them, thus what I know, I know "first hand". "Final" quotation: "The abstracts for the talks at that meeting were published in the Notices of the AMS Volume 14 (1967). On page 938 one finds Jon Beck's abstract: 652-8. Jon Beck, Cornell University, Ithaca, New York. Descent and standard constructions (triples). There is a close relationship between descent theory in algebraic geometry and the theory of categories which are =20 definable by means of standard constructions (tripleable categories). The "tripleableness theorem" sheds some light on descent criteria. The form of Cech cohomology used in descent theory is an appropriate triple cohomology theory. Its interpretation is discussed from the triple point of view. (Received October 2, =20 1967.) It is possible that someone still has notes of that lecture 40 years =20 later" Again sorry Bill. There is A HUGE difference between noting that in =20 one, or perhaps a few, SPECIAL CASES, such as Cech cohomology, there =20 was a "close relationship", which "sheds some light" on descent =20 criteria, and a precise GENERAL theorem, which, in order to be =20 stated, needs fibrations, AND the Chevalley condition! As I already mentioned,Beck's tripleability theorems found their =20 place IMMEDIATELY in countless books and papers,WHY NOT "his" other =20 theorem? WHEN is the first precise mention of it before, say, 1975. There is A WHOLE CHAPTER in Mac Lane's CWM book (1971) devoted to =20 monads, no mention of Beck's "contribution" to descent. Not in the =20 exercises also, not even in the informal historical note, where he =20 could have mentioned it without giving details. So WHERE ARE the =20 publications, by Beck or ANYBODY, mentioning "his" theorem, or USING it? Sorry Bill you have totally failed to convince me. If you have =20 convinced other persons, when I answer them, I shall carefully study =20= their arguments and shall try to answer them. But I won't even bother =20= to answer to "religious" arguments of the form "I believe that" Beck =20 had the theorem because Bill Lawvere "said so". Absolutely no offense =20= meant to you Bill, "I believe" in your deep insight in mathematics, =20 "I believe" in the importance of your contributions to category =20 theory. But, please Bill, don't ask me to believe in more. Maybe it is high time that some persons realize and/or admit that, =20 NOT ALL of category theory was born in "category-land", and some of =20 it still lives "abroad". 6- ANSWER TO ROSS STREET I quote : "the "indexed categories " of Pare-Schumacher" !? Sorry Ross, the "indexed categories" are not Pare-Schumacher, and not =20= Lawvere. They are, as I mentioned to Barr, also due to the "fertile =20 brain" of "you know who". I have repeated this for more than 35 =20 years. Vainly it seems. And since you are kind enough to mention my =20 1967 paper on bicategories that many a'member of "Category land", and =20= quite a few "outlandish" mathematicians have read, they are in this =20 paper, in detail, page 47, under the name of pseudo-functors and and =20= in the bibliography, clearly attributed to you know who. Can't most =20 category-landers read? That paper was written in English if my memory =20= doesn't " fail me" . By the way, "descent data" were also in this paper, "profunctors" =20 were announced page 49, and the fundamental theorem extending =20 Grothrndieck's construction to morphisms: E=B0-->Prof was clearly stated, together with the applications I had in mind. Thank you for having "forced" me to re-read that 40 years old paper =20 which i had not looked at for more than 30 years. It hasn't aged at =20 all, and I can still recommend it for the variety of important =20 examples it contained. Some of which I didn't remember I had =20 mentioned at that "prehistorical" age. I quote you again "I learnt from Benabou's lectures about the "Chevalley condition" for =20= fibrations and how descent data were Eilenberg-Moore algebras. Jean =20 gave me a copy of his Comptes Rendus article with Jacques Roubaud" "Very soon after Jean Benabou left, Jon Beck arrived ...When I told =20 him about Benabou's lecture on descent, he said that that was what he =20= had planned to talk about ("triples" and descent). I encouraged him =20 to do so but he decided to change his topic." Thus, you didn't hear Beck speak, not even in a private conversation, =20= on a "mythic" theorem he MIGHT have had about fibered or indexed =20 categories, which "DIDN'T EXIST". And that, of course, fully justifies Jonhstone not only for giving =20 Beck full credit, but "forgetting" to mention in his bibliography our =20= joint note. Which I gave you personally, and lectured publicly about. =20= And, by a strange "coincidence", the theorem stated by Johnstone, was =20= PRECISELY, the one in our note! I said I wouldn't shout, so I don't. =20 I'm not sure that Roubaud won't when he learns about the very =20 "special" sense of history and honesty which seems to prevail in =20 "Category-land"! And "operads", comma objects, and other 2-categorical "whatnots" =20 won't hide the real issue of Johnstone, KNOWING WHAT HE WAS DOING, =20 falsified "history" as you call it. But he did that SO MANY TIMES, =20 with MY work in particular, and with the approbation of many members =20 of the establishment of "Category-land", that he thinks he can get =20 away with "anything". On a totally different question, I quote you again: "Now, as much as I would love SIX bottles of GOOD champagne, I am not =20= going to submit a suggestion for Jean's challenge. Composition of =20 fibrations is a wonderful thing as is composition of homomorphisms of bicategories; =20= but they do different jobs. It is hard enough to say fibrations are =20 composable from the homomorphism viewpoint! There is a thing about this that requires a mixture of the two views. Regard one fibration p : E --> A as a homomorphism E_ : A --> Cat. Keep the other q : A -- > B as a fibration. Then the homomorphism =20 corresponding to the composite q p is a generalized left Kan extension of E_ along q." In my paper on bicategories, I gave a lot of examples. One of which =20 was pseudo-functors, now called indexed categories, and attributed to =20= Lawvere. But I NEVER mentioned fibrations in that paper, although I =20 knew about them, and about Chevalley's condition, which I remind you, =20= I learned in 1964. Because I KNEW, already at that time, that in =20 spite of what "The Elephant" says: FIBRATIONS AND "INDEXED CATEGORIES", ARE NOT THE SAME THING. Because: (i) The theory of fibrations is a FIRST ORDER theory, the "theory" of =20= indexed categories IS NOT. It is not even a "higher order theory" =20 however "high" you are ready to go. (ii) Fibrations can be INTERNALIZED in a topos (a logical category =20 suffices), indexed categories CANNOT. (iii) To go from an "indexed category" to a fibration, by the =20 Grothendieck construction, DOES NOT require the axiom of choice (AC) =20 whereas in the other direction YOU NEED AC. For sets, if you restrict =20= to small categories, and for classes (whatever that means) if you =20 deal with big ones, even if they are locally small. In most definitions, constructions and proofs with fibered =20 categories, all we need is FINITE DIAGRAMS involving vertical and =20 cartesian maps (sorry, "prone" maps if you understand those better). =20 We almost never need a cleavage of the fibration, which requires AC. =20 And in the very, very few cases where we do need it, a special =20 emphasis should be put on this necessity. (iv ) Last, but not least, fibrations DO compose, and "indexed =20 categories" DON'T And because of this last fact, I knew I was taking no big risk by =20 offering champagne! But of course, MY OFFER IS STILL STANDING. Sorry Ross, you don't qualify for even A GLASS of champagne. =20 Nevertheless I'll be very happy to offer you more than a glass if you =20= visit me in Paris, for old friendship's sake, but not for having come =20= any close to answering my question. And your so called "mixture of two views", requires, if you start =20 with two "indexed categories", FIRST TO REPLACE one of them by a =20 fibration, and THEN, to use generalized left Kan extensions. You =20 CANNOT avoid the first step, can you? All this complication compared =20 with the well known, 5 lines proof or no proof at all, result : =20 fibrations are stable under composition. Quite (un) surprisingly, I could not find any "trace" of this result =20 in the "monumental" Elephant. Am I wrong professor Johnstone? Are you =20= going to explain, AGAIN, as for my Louvain Paper, Celeyrette's =20 thesis, and the Comptes Rendus note that it was because of a "space-=20 saving decision"? Well, I've tried to explain, as I have done now for more than 30 =20 years, why I thought, and think more and more now, especially after =20 reading some "surprising" pages of the Elephant, that "indexed =20 categories" were to put it mildly, a WRONG manner to view fibered =20 categories.(See again (i) (ii) and (iv) even if you want to assume =20 any form of AC). But as we say in France: "Il n'est pire sourd que qui ne veut entendre" Even Peter Johnstone, who has been coeditor with Bob Pare of LN 611 =20= (1978) "Indexed Categories and their applictions" has been" forced" =20 to introduce fibered categories in his Elephant. Very badly I must =20 say, And, of course, with the same "space-saving" attitude concerning =20= good references.Especially to my work! As for me, I have never been "forced" to change my position. "Fibered =20= man" I was, from the beginning, and "fibered man" I remain. I don't =20 even have to compromise, and become "half-fibered and half-indexed". I"ll interrupt this mail now because: (i) I don't want it to be rejected because it is too long (ii) I really need some sleep! (iii) Although many other answers are almost finished, I need a =20 little more documentation to be absolutely sure about a few facts. I shall continue my answers on tuesday, meanwhile any further =20 "testimonies" or "comments" from any of you, even from the very =20 "prolific" Marta, will be welcome! Before I stop, I'd like to add a very special 7- ANSWER TO RONNIE BROWN Dear Ronnie, Please forgive me for not having answered earlier to your very kind, =20 and nice message of october 31. I appreciated it all the more because =20= it was the only mathematical "reaction" I received after my long mail =20= concerning locally cartesian categories, and "a few other things". I want to thank you publicly. And I want also to tell you that in the =20= "help!" discussion about what to tell to absolute beginners about =20 Category Theory, your contribution was, by far, the one I appreciated =20= most. You'll understand why, if I tell you that in 1999, in a colloquium on =20= history and philosophy of mathematics I gave a talk with the title =20 "Une analogie en theorie des categories". Moreover in the =20 introduction of this talk I explained that I had intended initially =20 to speak about : "Analogies et theorie des categories". And a first =20 draft of more than 120 pages, covering only a part of what I wanted =20 to say, convinced me that "une analogie" would suffice. I hope when this unpleasant phase is finished, if it is ever =20 finished, that we'll have more time to compare our ideas. They will =20 probably not coincide, but I'm sure we will very quickly agree on =20 many many points. Meanwhile, many thanks again, Jean=