Historical terminology,.. and a few other things. 0. Prologue 0.1. I want to thank all the colleagues who answered my questions on "historical terminology". I shall try to answer all of them, and shall greatly appreciate any remarks or comments about the present mail 0.2. I type very slowly, all the more so because I decided to respect the constraints of "Outypo" in this mail.(See the appendix for the meaning of this word). These constraints explain the unusual presentation of this text 0.3. I shall abbreviate by (c), (cc),(lcc) and (lcct) the following properties of categories: "cartesian", "cartesian closed", "locally cartesian closed" , and "locally cartesian closed with a terminal object". 0.4. I shall denote by El the book of P. Johnstone: Sketches of an Elephant, Vol.1 and, and for references I shall use the monumental bibliography of El in the following manner: (i) If the reference is there, I use El then the number, in bold face between brackets, e.g.(El 911) means : S.C. Nistor and I. Tofan, On the category ab(E) (Romanian), Bul.Inst.Politechn. Iasi (I) (1985) 205-207 ; MR 87g:18001 (ii) If the reference is not there, I shall write the symbol (El ?) followed by the minimum information permitting to localize it, e.g. (El ?) P.J. Cohen, Set theory an the Continuum Hypothesis 0.5. The notations shall be either standard or self explanatory with the following exception: I shall abbreviate "x is an element of y" by "x@y". For example in the following definition: A category X is lcc if it satisfies: forall x (x@Ob(X) => X/x is cc) 0.6. A first draft had convinced me that my mail would be abnormally long, so I have decided to break it in two parts. The first, almost totally devoted to (lcc), already very long, which I send immediately. I shall then wait a few days for the second part, devoted to (c) and (cc). This will permit me to receive any remarks or comments you would want to make about this first part, and to answer them as well as I can in the second part. I hope I won't be too boring, and apologize in advance if I am 1. Locally cartesian closed categories (lcc) From Prof. Peter Johnstone's answer to the question: Why did he impose a terminal object in his definition of lcc, I quote: "I did that because it seemed the appropriate convention to adopt in the context of topos theory. I wasn't trying to dictate to the rest of the world what the convention should be. On the other hand, there seem to be remarkably few `naturally occurring' examples of locally cartesian closed categories which lack terminal objects: the category of spaces (or locales) and local homeomorphisms is almost the only one I can think of." Since "he can think" only of very few or almost only one "naturally occurring" lcc's which lack terminal objects, let me provide "a lot" more ones. 1.1 First examples of categories which are l.c.c but have no terminal object (i) The smallest "I can think of" namely O, the empty category. (ii) A little bit bigger, but not too big: 1+1, the discrete category with two objects. (iii) Any discrete category ( except 1 of course) (iv) Any group (again except 1). (v) Any groupoïd (non trivial, i.e. non equivalent to 1). The reason for (v) is the following characterization of groupoids, which does not seem to have found its way in standard texts on category theory: A category X is a groupoid iff all its slices are equivalent to 1 Moreover, if X is a groupoid, the following are equivalent : (a) X has a terminal object (b) X is non empty and has binary products (c) X is equivalent to 1 Since (i), to (v) are groupoids, I shall count the whole previous list as one single type of example, and give others which don't fit in this pattern. (vi) The ordered set Omega of natural numbers. (viI) More generally any limit ordinal . (viii) The ordered category On of all ordinals. (ix) Any coproduct of l.c.c. (Even if they all have terminal objects, provided there is more than one such category) (x) Any rooted tree, with the canonical ordering obtained by taking the root as smallest element. In particular the binary tree. And of course, by (ix) any "planted forest", i.e. coproduct of rooted trees. (xi) the ordered set R_+ of non negative real numbers. (xii) A little bit bigger than Omega, but still countable namely: the category with objects the finite cardinals and maps the injections. (xiii) The following one might be "handy" for logical purposes: If C is a topos, the category Mono(C) having the same objects, and as maps the monos of C is lcc without terminal object. Question: Which, among the previous list, is so "unnaturally" occurring" that it should be "banned" from category theory? And moreover at the cost of a "linguistic violation" of the commonly adopted meaning of "local"? 1.2. First "stability" results The following theorems "explain" and generalize many of the previous examples, and permit to construct many more important lcc's without terminal objects. 1.2.1 Theorem If C is a lcc category, so is Mono(C). 1.2.2 Theorem Let P: D-->C be a discrete fibration, if C is l.c.c. so is D 1.2.3 Corollary Let C be a category and D be a sieve of C. If C is lcc so is D. I shall leave to your "fertile brains" (Preface of El , p.viii) the pleasure to use 1.2.1 and 1.2.2 to construct lcc's without terminal objects, and I shall use only the much weaker 1.2.3 to give more examples. They all fit in the following general pattern: Start with a C which is lcc, and may or may not have a terminal object. Suppose is is equipped with a notion of "boundedness",and take for D the sieve of bounded objects. Here are a few important examples: (xiv) Take for C an elementary topos, and call an object of C bounded if it is contained in a (Kuratowski)-finite object. (xv) Let A be any small category. Take for C the category A^ of presheaves over A and call a presheaf bounded if it is contained in a representable one. (xvi) Let X be a metric space. Take C=Opn(X) , the locale of its open subsets, and call an open set U bounded if it is contained in a sphere. (xvii) Let X be a topological space, and again C=Opn(X), and call U bounded if it is contained in a compact. (xviii) one can "sheafify" (xvi) and get the lcc of sheaves with compact support. How many more examples does one need? Let me mention that the direct "proof" of 1.2.3 is so simple you could cry: If D is a sieve of C, all its slices are slices of C, thus any local property, not only lcc, satisfied by C will be shared by D. But, even if C has a terminal object, D does not need to have one. 1.2.4 Remarks "en vrac" 1- Even if we assume in the previous results that C has a terminal object, neither Mono(C) nor D need have one. Thus (lcc) has many more "stability properties" than (lcct). See also 1.1 (ix) 2- It is even possible to have an lcc category , with binary products but no terminal object. This is the case in the examples (xv), (xvi) and (xvii) if the whole space X is not "bounded" (partial answer to Dubuc) 3- There even exits an lcc which has binary products and is not connected, namely 0 . (Dubuc again) 4 - There are much stronger " stability results" than the previous ones permitting to construct lcc's but it would take too much space to build the the "set up" where they can all be stated, let alone proved. I shall say a few words about parts of this set up in the next section, and show how it can be used for "general" local properties, not just lcc. Question: What important theorems hold for (lcct), but not for (lcc)? If you ask the question in the other direction my answer is easy: All the stability theorems I mentioned Remark: In the theory of fibered categories we work with fibers and slices and the introduction of unneeded terminal objects weakens the results or confuses some issues, and sometimes does even both. 1.2.5 Terminology again. There is by now an unwritten but (almost) unanimous "linguistic consensus" on the following terminology: If P is a "property" of categories, a category satisfies P locally iff all its slices satisfy P. I do not want to "dictate" anything to anybody, but I think we could all easily agree on this "unifying" terminology and, if for some mathematically imperative reason, we need a different notion, we should use our imagination to give it another name. There is already an unfortunate exception, namely: locally small categories, which I never liked, (I would have preferred something like: piecewise small). But "locally small" has been used by countless mathematicians in countless texts, and in such a case I am not very prone to change the terminology, even if I don't like it. 2 . Local properties of functors We extend the previous definitions to properties of functors in the following manner. 2.1 Basic definitions 2.1.1. If F: X-->Y is a functor, and x is an object of X, the slice of F at x is the obvious functor F/x: X/x-->Y/Fx induced by F We remark that, for G: Y-->Z , we have: (GF)/x = (G/F(x))(F/x) . 2.1.2. A property (of functors) is a class P of functors satisfying: Every iso F: X-->Y is in P, and P is stable under composition 2.1.3. If P is a property, a functor is locally in P if all its slices are in P , thus we get a new property called the localized of P and denoted by L(P) 2.1.4. If P is a property we say that a category X is in P if the functor X-->1 is . In that case every category isomorphic to X is also in P. 2.1.5. Properties are ordered by inclusion, we shall write P=>P' for " P is contained in P' ". This implies L(P)=>L(P') 2.2 Remarks, again "en vrac" 1- One can define more general notions of "localness" than those I gave in 2.1, but I shall not mention them in this mail 2- From a categorical point of view the definitions I gave are "natural" . They are also very simple and, because of this simplicity, one might think that not much can be derived from them. That it need not be so, comes from the fact that very simple properties may have much "richer" localizations. 3- Although this may seem "strange", I do not assume that: if F: X-->Y is in P, and F': X-->Y is isomorphic to F, then F' is in P, let alone that every equivalence functor is in P. e.g. P could very well be the property of functors which are surjective on objects. .2.3 Local Iso's I denote by Iso the smallest property: the only functors in it are iso's, the functors in its localized are thus local iso's. Quite "surprisingly" we have: 2.3.1 Theorem A functor F: D-->C is a local iso iff it is a discrete fibration As a consequence we have following "ubiquity" of discrete fibrations, if F: D-->C is such a fibration, then for any property P, F is in L(P). In particular, If C is in L(P) is in L(P) so is D. Since for any category X the functor 0-->X is a discrete fibration, it will be be in L(P), and in particular so will be the initial category 0. More than 2/3 of the examples of §1 follow from the previous remarks applied to the very special case of lcc's 2.4 Local surjective equivalences The smallest property "I can think of", after Iso of course, is SEq , surjective equivalences, which I define as:functors which are full,faithful and surjective on objects. (They are also surjective on maps). Thus its localization L(SEq) is defined. Even more "surprisingly", we have: 2.4.1 Theorem Let F: D-->C be a functor. The following are equivalent : (i) F is locally a surjective equivalence, i.e. F is in L(SEq) . (ii) Every map of D is F-Cartesian, and F is a fibration. (iii) F reflects isos and F is a fibration. (iv) F is a groupoid - fibration (i.e. a fibration where all the fibers are groupoids. Almost every property P contains all equivalences, and in particular the surjective ones. Thus we get an "almost ubiquity" of groupoid fibrations :For any such P we have: (i) Every groupoid fibration F: D-->C is in L(P). In particular: (ii) If C is in L(P) so is D (iii) If G is a is a groupoid it is in L(P) This explains, and considerably generalizes 1.1 (iv), and it might interest Ronnie Brown. 2.4.2 Remarks Surjective equivalences are much better than mere equivalences because : (i) They are stable under pull backs whereas equivalences are not, and they can even be characterized as the only equivalences stable under pull-backs. (ii) They are fibrations and, as fibrations , they have simple characterizations, namely: Let F be a fibration: F is an equivalence <=> each fiber is equivalent to 1 <=> F is in SEq (iii) They can be internalized in any regular category, and "behave" there as well as one might want. (iv) I don't know who observed first that equivalences are not stable under pull-backs. It is mentioned explicitly in Grothendieck's first paper on fibrations, together with the fact that, for fibrations, they are stable: (El ?) A. Grothendieck , Categories fibrees et descente; SGA 1961 . (v) Much later, Freyd and Scedrov (loc. cit. 1.361 p.19) have considered surjective equivalences under the name of inflations, but they were mainly interested, as most people, in "mere" equivalences and the inflations served only to decompose equivalence functors. In particular they never mentioned that inflations were fibrations, nor that they were stable under pull-backs, both facts which are important to me. 2.5. The local game Given time and space,I could have added to 2.3. and 2.4 a very long list of localized properties, I shall just give an idea of the "local game" I have been playing, with interruptions,for more than 25 years. Chose simple properties P and find out what L(P) is. In the other direction, chose some important (for you at least) property Q and try to see if it is of the form L(P) for some P . Such a P need not be unique, so try to find one "as simple as possible". 2.5.1. Example If P is the property of functors which are surjective on objects and have a right adjoint, we have: A functor is in L(P) iff it is a fibration. Thus, if F:X-->Y is a fibration, for every property Q such that P=>Q , we have: If G:Y-->Z is in L(Q) so is GF, and in particular, if Y is in L(Q) so is X. 2.5.2. The "game" can be made more complex, and more difficult, by imposing further "constraints" (see appendix) of the kind: For which P's is L(P) stable under pull-backs, or pseudo pull backs, or has "adapted" calculus of fractions, etc. Before I got used to the game, I had many "surprises". Each of them brought a new result or at least a better understanding of old ones. 3. A few comments on the answers I received 3.1 General picture There seems to be a general agreement in all the mails about the fact that (lcc) should not include terminal objects, except of course for Prof. Peter Johnstone, but I have already made long comments about his mail, and if he is not convinced by my purely mathematical arguments, I'd greatly appreciate if he could tell me why he isn't. Of course again: only for mathematical reasons. The mails which agreed with my definition explained this agreement by two kind of arguments: (i) Purely linguistic and coherent use of "local". This includes Fred Linton, Eduardo Dubuc and Phil Scott's second mail. As I totally agree with them on this basis I shall make no more comments (ii) Arguments coming from "logical systems" such as dependent types, lambda calculus, etc. This includes Phil Scott, Vaughan Pratt,and Paul Taylor. I must confess i am not convinced by these types of arguments, and I shall explain why. I expect strong reactions to some of my statements, and I shall be very happy to to hear them, and try to answer them. 3.2. First comments I'm absolutely sure, even if I don't know some of them very well, that the "logical systems" mentioned in the previous sub-section, provide very important examples of lcc or lcct categories. Nevertheless I'm tempted to say So What? I won't say it because some people might think there are limits to heresy, but very deep in my mind I'll continue to think it. Why? 3.2.1. Other important examples can arise from different domains of Category Theory or more generally from mathematics. I gave a long list in §1 , which I could easily have made much longer, and even if someone could prove that, say, Omega , R_+ , any groupoid, sheaves with compact support , could be described in terms of "dependent types" or "lambda-calculus" I would still not be convinced! Because each description would require a different ad-hoc "logical system", which would certainly appear artificial to the specialists of the domain, whereas for a categoricist these examples are all easy and meaningful. In particular I ask the question: Which of the previous examples has ever appeared in the context of such logical systems? 3.2.2. I shall go a bit further. Even If someone did come up with a (meta) theorem of the kind: "Every lcc category can be described as a suitable model of such a forma system" , (Which is probably true, and probably easy to prove), even in that case, I would still not be convinced! Because: How would one interpret the various stability theorems I mentioned? For example: Suppose I know that a specific lcc category C is a model of some formal system (F). By 1.2.1 and the, yet unproved, meta-theorem I alluded to, I shall know that Mono(C) is a model of another formal system of the same kind, say (F'). But then how do I deduce syntactically (F') from (F) ? I contend that Category Theory by asking such natural questions, might inspire some interesting formal constructions in various "logical systems" 3.2.3. Category Theory is "irrigated" by many mathematical fields. An important one is so called "categorical logic", but it is not the only important one. And if we are tempted to think that some axioms we assume on categories make them "the embodiments" of some kind of logical system, they are almost never only that e.g. : Think of a topos as "semantics for intuitionistic formal systems" (El. Preface) what is the "syntactic counterpart" of the following well known and important result: If C is an internal category of a topos E, the category E^C is a topos. Such a syntactic description could perhaps be given, but at what cost? And would it clarify or obscure this basic result? 3.2.4. There is yet another question to specialists of "logical systems" Any category with a terminal object is connected as a category. lcct categories are connected. Now an elementary topos or a locale are lcct categories , where there is another important notion of connectedness, namely 1 is a connected object (I apologize for such trivialities). Here is the question: Is there a similar notion of connectedness for the kind of "logical systems" I mentioned earlier? 3.2.5. Suppose E is an elementary topos.(the assumption is much too strong, but I make it to be on the safe side) One might want to define internal categories in E which are lcc. Incidentally I did it . It was easy. And in order to do it, I didn't have to "internalize" (whatever that might mean) "Dependent Type Theory", or any other logical system. 3.3. The main objection 3.3.1 I have given in 3.2. many reasons why arguments coming exclusively from "categorical logic" did not convince me, but there is a fundamental one, namely: Viewing some categories as embodiments of "logical systems", most of the time does not give any indication about what the morphisms between such categories ought to be, and sometimes even suggest wrong directions. When we need these morphims,and in general we do need them, the ultimate choice comes from mathematics, not logics. 3.3.2. First examples (i) What is the notion of morphism, if any, suggested by formal systems such as : "Dependent Type Theory", or "Typed Lambda calculus" ? (ii) Has anybody defined a notion of morphism between "Hyperdoctrines", and, if nobody has, why not? (iii) In §2.5. of their very nice book (El 381) Freyd and Scedrov define the notion of congruence on an Allegory. This a natural definition of "syntactic type". But very quickly they restrict their attention to "amenable congruences" which are no longer "syntactic". This is a typical illustration of the fact that "ultimately, the choice comes from mathematics". (I shall come back to this notion of congruence in the second mail) 3.3.3. A morphism of toposes is ...? In the preface of El one can find a very illuminating list of "descriptions" beginning by "A topos is.." and numbered from (i) to (xiii). I shall use the same numbering if I want to refer to some of them. I was very much impressed, especially since it took more than 20 years to complete that list, and the contribution of, I quote: "the category-theory community", and, "the theoretical computer scientists" . Since there was not a single description of geometric morphisms, I studied carefully that list, in the light of 3.3.1, to see which of the 13 descriptions were most suitable to give indications about how to describe these morphisms. Obviously (v) and (viii) are too "sophisticated" and require too much preliminary knowledge of Topos Theory, and many other domains, to serve my purpose, so I dropped them and concentrated on the 11 remaining descriptions. And there, I had a big "surprise" : I am no linguist, and moreover English is not my mother language, but I remarked that the "A" in "A topos is" had different meanings, e.g. 1. In (xii) "A topos is" means "some toposes are". 2. In (i), (vi) and (xi) "A topos is" means "every Grothendieck topos is" 3. In (ii),(iii), (iv),(vii),(ix),(x) and (xiii) "A topos is" means "every topos is" I shall not insist on this "logical ambiguity",but obviously, if we seek a general description of morphisms of toposes, we won't find it in 1 because only "some" toposes fit in this description. In the "sublist" 3, (iii), (iv), (vii), and (xiii) come from various logical systems. I have tried to figure out, thinking only in terms of such systems, what a morphism should be. I confess I couldn't find a natural definition of such morphisms between, say: two ".. (embodiments of) an intuitionistic higher-oder theory" , (iii), let alone between one such embodiment and "..a setting for synthetic domain theory" (xiii). I'm sure a helpful colleague will supply a "bridge" between the two. The best I could do was to "describe", very vaguely, logical morphisms between two toposes, and only when their two "descriptions" were given by the same number on the list. When the "fertile brain" of Grothendieck (El preface, p.viii) gave the definition of geometric morphism, he knew only (i) in "the list", because he happened to have invented it. The definition was given for purely mathematical reasons. And as all very deep mathematical definitions, it has resisted time. It has even anticipated time, because it is suitable for elementary toposes which didn't even exist when he gave his definition! There is much more one can say about based toposes than about "mere" toposes, and I'd be curious to know how specialists of "logical systems" or "computer scientists" would have, even in a "descriptive" manner, answered the question: A based topos is ... ? I have some comments, questions, and even a few answers, about "La Lista", which is supposedly the fruit of: the category-theory community and the theoretical computer scientists. But I shall postpone them until "better times" 3.3.4. Logical categories and categorical logic I think that the scope of "categorical logic" should be much wider than the mere study of the categories which "embody logical systems", which I propose to call by the "generic" name of logical categories. (But of course I would never dream of trying "to dictate" anything to anybody, let alone to "the rest of the world") It could include in particular: (i) the study of "local properties" , a "flavor" of which has ben given in §2, (ii) Calculus of fractions, adapted to various "properties" of categories and functors , a first part of which, with clear motivations, can be found my paper (El 103) . But that was in 1989, almost prehistory. Some of you may have doubts about the relevance of this calculus of fractions to "categorical logic". In the second part, if I don't have to answer too many questions or objections about the present mail, in the same spirit as in §2 I shall give a few mathematical results to try to convince them (iii) abstract notions of homotopies in categories but also of categories, i.e. in Cat, as defined by Grothendieck in his "Pursuing stacks". A group of mathematicians, mostly French, are developing his ideas, and for those who might be interested, apart from their numerous papers, I recommend volumes 301 and 308 of "Asterique" : G. Maltsinotis, La theorie de l'homotopie de Grothendieck D.C. Cisinski, Les prefaisceaux comme modeles de l'homotopie This list is very far from complete, and I'm sure that many of you have in mind some important parts of category theory which could be added to it 3.3.5. I have worked, on and off, for more than 20 years, on some aspects of this "categorical logic". I have talked two or three times about my ideas and my very very first results, but I met only "polite but indifferent" reactions. Maybe my work didn't deserve much more. I don't care. Because, if you allow me to be a bit "personal", this work has given me a lot of pleasure. In particular because it has permitted to deepen my relation with old and dear friends such as fibered categories, cartesian maps and functors, categories of fractions, etc, and to improve my understanding, and knowledge, of their "qualities". And, last but not least, to prove new mathematical results about them. 3.4. The special case of Mr. Paul Taylor The answer to Paul Taylor deserves a special treatment. Although I deeply regret it, it will not be only mathematical, but such a choice was his to begin with. I quote his mail: "I am sorry to say that I have seen papers emanating from respectable universities in which the authors have appeared to believe that this is the definition. (One of the papers that I have in mind cites many eminent categorists, who may perhaps have an opinion about having their names appear alongside a lot of complete nonsense.)" 3.3.1- Why? Why such petty and spiteful attacks on unnamed mathematicians, without any proof or justification, in a purely "historical", non polemic discussion? (c.f. The answers of all the other participants) Why didn't anybody react to such attacks, or to previous ones, by the same "Mr" Taylor? Does he enjoy some "special status", or shall we have to consider in the future such behaviors as "normal"? I quote him again: "My footnote refers to "other authors" who said that LCCCs should have binary products; I think I may have had Thomas Streicher in mind, but I don't recall what he may have said or in what paper." 3.3.2- Streicher &... others? Why mention Thomas Streicher without at least trying to find out what he said or wrote precisely on the question? Why not mention P. Johnstone's "Elephant" where this is precisely written, long after Taylor's world famous "footnote" was published. Lack of courage? Fear for future promotions? Why not mention two other " eminent categorists" who made the same mistake in a published paper that he certainly knew, namely Phil Scott and...Paul taylor himself who wrote in their joint paper (El. 977) at the very first page in the fist proposition: "Let C be a locally cartesian closed category( that is, C has finite limits and for each object X in C, the slice category category C/X is cartesian closed)..." Lack of memory? Lack, again, of elementary courage or decency to "confess past errors"? When was Mr. Taylor "struck by the light" ? When did he abandon the finite limits before writing "his footnote", and as all new zealots, started condemning very strongly his former "sinning colleagues"? Except the powerful ones, of course! 3.3.3- "Consensus"? I quote him again: "I confess that I'm a bit surprised to find that the consensus agrees with me, so to set matters straight I should also point out that my argument applies equally to elementary toposes and other familiar structures of categorical logic." I am greatly honored to find that I agree with Mr. Paul Taylor's footnote in "his book", which I have not read, and have no intention to read, about matters I had completely settled more than 20 years before "the" book was published ! Mr. Taylor was answering me. Thus I very gratefully thank him for teaching me a few things that presumably I didn't know such as: "The simplest formulation is that an LCCC is a category every slice of which is a CCC. In particular, every slice has binary products, which are pullbacks in the whole category." "Objects of an LCCC and the slices that they define correspond to objects of a base category and the fibres over them in a fibred or indexed formulation of logic," I certainly do agree, except on a "minor detail": I do not like the idea that my name could be in any manner whatsoever associated with "indexed categories". I never used the term, I said and wrote countless times that I considered the notion as wrong. Maybe I am wrong, the future will decide. But I want no part of responsibility in the propagation of this notion. This is my choice as a mathematician. Incidentally, I am in very good company, most mathematicians, some of them outstanding use fibrations. Of course, I am quite ready, if I am asked, to give,once again, purely mathematical reasons for this choice. But I'm afraid it will, again, be in vain, because : "Il n'est pire sourd que qui ne veut entendre". 4. Appendix: Outypo (for my friend Jacques Roubaud, a poet, a mathematician and an innocent victim) 4.1 Many of you have probably heard of "Oulipo", Ouvroir de Literature Potentielle, a literary group created in 1960 by Raymond Queneau and Francois Le Lionnais. It proposed to create literary texts submitted to well chosen but otherwise arbitrary "constraints" , of various nature: linguistic, syntactic, combinatorial, and even topological. (One of the best known examples is due to Georges Perec who managed to write a whole,and good, novel without ever using the letter "e" which is by far the most frequently used in French). Jacques Roubaud, a member since 1966, has invented dozens of such constraints, some of them quite sophisticated, using e.g non trivial groups of permutations or topology "a la" Moebius strip or Klein bottle. He is world wide known as the author of more than a dozen novels, many thousands poems, and one of the best specialists of the sonnet. (For more details, you can consult Wikipedia, about two "items": Oulipo, and Jacques Roubaud) 4.2. In all my life I have written only three joint papers. The first was: ( El ? ) J.Benabou, J.Roubaud, Monades et descente 4.3. Oulipo has "swarmed" from literature to many other domains : painting, music, photography,etc. New groups have been created, all over the world, in these domains. And, to "remember their filiation", they have chosen their name according to the following "constraint " : Ou X po , i.e; three syllables, the first "Ou", the last "po", the "X" in the central one being an abbreviation of the name of their domain. e.g. in the domain of painting, "peinture" in french, there has existed for almost 20 years now "Oupeintpo" as: Ouvroir de peinture potentielle. Before formally adopting "Outypo", I have consulted my "expert", J.Roubaud who confirmed that the name was correctly formed, and that my constraints were genuinely of "oulipian" nature. 4.4. Easy "oulipian" questions (i) Complete the reference of our joint paper (ii) Why, and of whom is Jacques Roubaud a victim? (iii) What does "Outypo" stand for, and what are its constraints ?