From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2971 Path: news.gmane.org!not-for-mail From: jean benabou Newsgroups: gmane.science.mathematics.categories Subject: Terminology again Date: Fri, 30 Dec 2005 17:29:15 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v543) Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019014 6670 80.91.229.2 (29 Apr 2009 15:30:14 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:30:14 +0000 (UTC) To: Categories Original-X-From: rrosebru@mta.ca Sat Dec 31 10:27:33 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 31 Dec 2005 10:27:33 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EshdD-00002D-KZ for categories-list@mta.ca; Sat, 31 Dec 2005 10:22:51 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 46 Original-Lines: 301 Xref: news.gmane.org gmane.science.mathematics.categories:2971 Archived-At: In a previous mail I said that I was strongly opposed to the=20 replacement of "cartesian" and "co-cartesian" maps by "prone" and=20 "supine" ones for linguistic, mathematical and ethical reasons which I=20= was ready to explain in detail if I was asked to do so. I have waited a few days to see what reactions I would get. So far, my=20= position has been supported by Peter May and Eduardo Dubuc on ethical=20 and/or linguistic grounds, and Keith Harbaugh has asked me what=20 "ethical" issue is involved, and more generally to clarify my position=20= on the mailing list. Although the ethical issues are for me the most important, I shall=20 postpone them to another mail, if the moderator of this list permits me=20= to do so, and I shall concentrate to-day on the mathematical an=20 linguistic reasons of my opposition. I am no linguist, but I am a mathematician, so THE MATHEMATICS WILL=20 COME FIRST, and I would like to make this text more "palatable" and=20 less "negative" by introducing some genuinely new ideas, which some of=20= you might find of interest, and which are relevant in this debate. =A71- DEFINITIONS Let P: C ---> B be a a functor. 1.1- A map v of C is VERTICAL (forP) if P(v) is an identity. I=20 denote by V(P) or simply V the subcategory of C which has the same=20 objects as C and as maps the vertical ones. Every identity is vertical, and if in a commutative triangle in C two=20 of the maps are vertical so is the third. 1.2- Let f be a map f of C. We shall say that f is : (i) CARTESIAN if for every pair (g,b) with g in C , b in B, and=20 P(g)=3DP(f).b there exists a unique map h in C such that: P(h)=3Db = and =20 g=3Df.h. (ii) PRECARTESIAN If the previous condition is satisfied only when b=20 is an identity, (iii) HORIZONTAL if every vertical map is orthogonal to f . (that=20 would probably be "prone" in the proposed new terminology) I shall denote by K(P), PreK(P) and H(P) the classes of maps of C=20 which are cartesian, precartesian and horizontal, and abbreviate by K,=20= PreK, and H if P is fixed. 1.3 - The functor P is a fibration (resp. a PREFIBRATION) if for every=20= pair (b,X) where b is a map of B, and X an object of C and=20 P(X)=3DCodom(b) there exists a cartesian (resp.precartesian) map f: Y=20= --->X such that P(f)=3Db 1.4 - If X is an object of C and b: J--->P(X) we denote Pl(b,X) the=20 category of P-LIFTINGS of b with codomain X with objects the maps f:=20 Y--->X of C such that P(f)=3Db, a morphism from f to f': Y'--->X is a=20= vertical map v: Y--->Y' such that f.v=3Df' . I shall say that P is a HOMOTOPY PREFIBRATION if all the categories =20 Pl(b,X) are connected (which of course imply non empty), the motivation=20= of the name is that any two liftings are "homotopic". 1.5 - Let P: C ---> B and P': C' ---> B be two functors and F; C=20= ---> C' be a functor over B i.e. such that P'.F=3DP. Such an F = obviously=20 preserves and reflects vertical maps for P and P'. Moreover if f: Y=20 --->X is in Pl(b,X) , F(f): F(Y) ---> F(X) is in P'l(b,F(X)) hence F=20 induces functors Fl(b,X): Pl(b,X) --->P'l(b,F(X)) , f l--->F(f) , for all=20 "compatible" pairs (b,X) I shall say that F is a CARTESIAN FUNCTOR if all the functors Fl(b,X)=20 are final. (No assumption is made on P or P'). =A72-REMARKS "EN VRAC" ABOUT THESE DEFINITIONS (or, let's do a little = bit=20 of mathematics!) 2.1- Let P: C --->B be a functor (we assume NOTHING on P) Then: (i) Every cartesian map is horizontal i.e. K(P) is contained in H(P). The converse need not be true, for example if all vertical maps are=20 iso's i.e. all the fibers of P are groupo=EFds, then all maps of C are=20= horizontal, and "co-horizontal" And one can construct a P where C and B=20= are finite posets where no map, except of course the identities, is=20 pre-cartesian or pre-cocartesian, let alone cartesian or cocartesian (ii) THEOREM. If P is a homotopy- prefibration, then cartesian=20 coincides with horizontal, i.e. H(P)=3DK(p). (It is not completely trivial) The definitions 1.4 and 1.5 are genuinely new, and might seem=20 surprising, the following remarks will give a very small idea of what=20 can be done with them 2.2- A cartesian functor preserves precartesian maps : Because f: Y --->X is in PreK(P) iff it is a final object of =20 Pl(P(f),X) , and final functors preserve final objects However if P and P' are are arbitrary functors such a preservation is=20 not enough to insure that F is cartesian because there might not be=20 "enough" precartesian maps in C. But cartesian functors have so far=20 NEVER been used except between prefibrations, and in that case our=20 definition coincides with the usual one because we have: 2.3- If P and P' are prefibrations, F is cartesian iff it preserves=20= precartesian maps.(It suffices in fact that P is a prefibration) I like to make the following "analogy" : if S and T are topological=20 spaces a continuous function f: S --->T preserves convergent=20 sequences, if X is metrisable, this is enough to insure the continuity=20= of f 2.4 - Cartesian functors are closed under composition, and every=20 equivalence over B is cartesian. In fact we have much better, namely. 2.5 - If a functor F over B has a left adjoint then F is cartesian 2.6 - Homotopy-prefibrations are stable by composition. This seems "harmless" and trivial, but it is neither. It is well known=20= that fibrations are stable by composition, but it is probably a little=20= less well known, because I have never seen a statement to that effect,=20= that prefibrations ARE NOT . 2.7 - Homotopy-prefibrations (h-p) are special cases of cartesian=20 functors, because P. C --->B is a h-p iiff it is a cartesian functor:=20= (C,P) --->(B,IdB). (This of course is no longer true if h-p is replaced by prefibration or=20= fibration) =46rom this it follows that if F:(C,P) --->(C',P') is cartesian and = P'=20 is a h-p so is P. 2.8 - An important feature of h-p is that pointwise Kan extensions=20 along such P's can be computed fiberwise. Moreover this property=20 characterizes h-p' s. In particular such a P is final iff all its fibers are connected, and=20 it is flat iff it's fibers are cofiltered. The previous results are special cases of properties true for arbitrary=20= cartesian functors. 2.9 - REMARK : Homotopy prefibrations are but ONE example of =20 MEANINGFUL generalizations of fibrations. I have considered many=20 others, all with important mathematical examples, here are some: (for a=20= functor P: C --->B) (i) The categories Pl(b,X) are filtered (ii) Each connected component of such a category has a final object (iii) Each connected component is filtered In (ii) and (ii) P is not even a homotopy prefibration, but in all=20 these cases the general definition of cartesian functor given in 1.5 =20 is the "correct" one and gives the expected results. =A73 LINGUISTICO-MATHEMATICAL REMARKS 3.1 - OK, let us try "prone" for "cartesian", what about the=20 precartesian maps, "preprone" ? They have nothing to do with a=20 weakening of orthogonality to V(P), which we shall examine in 3.3. What=20= about cartesian functors, "prone functors"? What about maps which are=20 both cartesian and cocartesian, such that e.g. the iso's, prone and=20 supine? A very uncomfortable position you'll grant me. I am no acrobat,=20= I tried it, I hurt my back and stomach, had to stand up, and ended=20 up...vertical! 3.2 - The proposed terminology is based ON A BIG MATHEMATICAL MISTAKE,=20= namely: confusing cartesian and horizontal, which in general do NOT=20 coincide, as shown in 2.1. Unless of course there no other functors but=20= fibrations, or if there are, the terminology should not be compatible=20 with them. Well I, and probably other persons, think that there are,=20 know that there are, and that they deserve to be studied, were it only=20= to have a better understanding of fibrations. In 2.9 I gave a few=20 examples of such functors. If there were ONLY fibrations, how would=20 one express the fact that a prefibration where all the fibers are=20 groupoids is a fibration? 3.3 - Even for fibrations there are interesting maps which are neither=20= vertical nor cartesian and that one might want to study. Let me give an=20= example. Both cartesianness and horizontality assume the existence and=20= uniqueness of maps satisfying certain conditions. What about those=20 where we drop existence and keep uniqueness. Following Peter Freyd's suggestion, let me call them=20 quasi-cartesian(QK), and quasi-horizontal(QH),and see what they are. A=20= map f: Y --->X is QK (rep. QH) iff for every parallel pair =20 (g,g'): Z=3D=3D=3D>Y coequalized by f, if P(g)=3DP(g') (resp.if = g.v=3Dg'.v for=20 some vertical map v) then g=3Dg' Even in the case of fibrations, where K=3DH, QK is only contained in QH=20= but not equal.This can be seen in the most trivial case, where B=3D1, = and=20 all maps are vertical. A map f is QK iff it is a mono, it is QH iff =20 for every pair of maps (g,g') WHICH CAN BE EQUALIZED, fg=3Dfg' implies = =20 g=3Dg'. Now if cartesian=3Dprone, QK will have to be "quasi-prone", a strange=20 position again, but never mind. However, how should we call QH ? 3.4- I can speak, read, and write a little bit of English, but I am=20 French and might someday have the preposterous idea to lecture on=20 fibered categories in French. Of course only in France, and to an=20 audience uniquely composed of french persons. Perhaps MM Taylor and=20 Johnstone, could suggest adequate french translations for prone and=20 supine, which I can't seem to find. And they should be ready to do the=20= same thing for German, Italian, Spanish, and many other languages. No such problems with cartesian of course, because cartesian.... is=20 cartesian is cartesian is cartesian! 3.5- By now many thousands of pages have been written in various=20 languages using "cartesian", and many hundreds are being prepared, or=20 ready to be published, using the same word. What should be done with=20 all that past or future rubbish, now that we have received THE LIGHT=20 and the WORD(S)? =A74 TEMPORARY CONCLUSION I apologize for such a long mail, but I wanted also to show, among=20 other things , that it is possible to handle new and relevant=20 mathematical notions by introducing a SINGLE new word, namely;=20 "homotopy prefibration" , which has a clear intuitive content, and=20 moreover is easy to translate in most languages. I have given many arguments to explain my position, and I have many=20 more. But for the moment, I'd like to know the arguments of the persons=20= in favor of these changes, PRINCIPALLY, of course, those of Paul Taylor=20= and Peter Johnstone. If it is only the "joke" aspect, I want to add=20 that I do also like jokes, very much, perhaps not the same as theirs..=20= I even used to compete with Sammy, who was an expert, about who'd know=20= some jokes the other didn't. When this mail was almost completely finished, I found the reaction of=20= Vaughan Pratt from which I quote: "Has the adoption of frivolous nomenclature for quarks ("strange," "charm," "beauty" and even "quark" itself) diminished in any way the world's respect for quarks and their investigators?" I want to be clear on that matter. I have no objection to "frivolous"=20 naming of NEW concepts by the person or persons who DISCOVERED or=20 INVENTED them. But I object VERY STRONGLY to "renaming" well=20 established concepts, used for more than 40 years by the mathematical=20 community, even if the new names were NOT frivolous, and especially if=20= such a renaming is made by persons who have made no MAJOR contribution to the development of the field of FIBERED CATEGORIES. As a side remark, I have no problem whatsoever to translate in French :=20= "strange", "charm", "beauty", "quark", "sober" or "bottom". And to be=20 "frivolous", even if it's not so easy in a foreign language,=20 "homotopy's bottom" came ages before Scott's, and "Galois connection"=20 ages before the "french" one. Since my english is not too good, in particular I knew only "the other"=20= meaning of "supine", I'll borrow, a bit freely, from "a good author" I=20= admire a lot, and remind that: Men gave names to many animals In the beginning, in the beginning Men gave names to many animals In the beginning, long time ago. Best wishes to all, Jean