From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7709 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Jean_B=E9nabou?= Newsgroups: gmane.science.mathematics.categories Subject: "Terminolgy" re-visited Date: Tue, 7 May 2013 10:23:34 +0200 Message-ID: <91D4EC91-1FDC-45E6-9A36-7132291436C1@wanadoo.fr> Reply-To: =?iso-8859-1?Q?Jean_B=E9nabou?= NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Apple Message framework v1283) Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1367931424 4920 80.91.229.3 (7 May 2013 12:57:04 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 7 May 2013 12:57:04 +0000 (UTC) To: Categories , Original-X-From: majordomo@mlist.mta.ca Tue May 07 14:57:03 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1UZhS0-0005Wt-Jv for gsmc-categories@m.gmane.org; Tue, 07 May 2013 14:57:00 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:46687) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1UZhQS-0004F2-BL; Tue, 07 May 2013 09:55:24 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1UZhQR-0001kc-A2 for categories-list@mlist.mta.ca; Tue, 07 May 2013 09:55:23 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7709 Archived-At: Dear all, I cannot type any form of LaTeX, and do not know the "standard" ways to = introduce indices,exponentials and so on, using only the typing which is = admitted on this list. Thus I shall use "non standard" notations, very = simple, which I shall explain precisely. If A is a category an object a of A can be identified with a functor = "name of a" which I denote by "a": 1 --> A . If F: A --> C and G: B --> C are functors I denote by F/G the comma = category they define, and by F//G their 2-pull-back sometimes called their pseudo pull-back. I shall call "weak equivalence" a functor F: A --> B full and faithful = and essentially surjective (ff-es) and say that A is weakly equivalent = (we) to B if there is such an F. This defines a preorder relation which = I denote by=20 W(A,B). It is symmetric iff the Axiom of choice (AC) holds. A strong equivalence between A and B is a pair of adjoint functors F: = A --> B and F': B --> A such that the adjunction morphisms are isos. = I shall say that A and B are strongly equivalent if such a pair exists = and denote by SE(A,B) this equivalence relation. If S is a category I denote by Fib/S the 2-category of fibrations over S I denote by S=B0 the dual of S . An S indexed category C is a pseudo = functor C: S=B0 --> Cat . For each map =20 f: s --> t of S , I denote by f*: C(t) --> C(s) the value of C on f. = If g: t --> u is another map of S, I denote by c(f,g) the isomorphism = f*g* --> (gf)*. I shall assume C normalized i.e. for each object s of = S, id(s)* =3D id(C(s)) For an indexed functor F: C --> C' i use the following notations: If s = is an object of S, F(s): C(s) --> C'(s) ,=20 if f: s --> t is a map of S is the iso of functors F(s)f* --> f*F(t) (I'm aware of the ambiguity of denoting by the same f* the "re-indexing = functors" defined by C and C' but it is the best I can do with the = limited typographic means I use, without having too cumbersome = notations)=20 If S is has pull-backs, I denote by Cat(S) the category of categories = internal to S =20 Answer to Toby Bartels: (i) You propose to compose the spans by pullbacks. Here is an example of = two spans A <-- X --> B and=20 B <-- Y --> C such that the functors A <--X and Y --> C are isos, the = functors X --> B and B <-- Y have unique quasi inverses, and if Z is = the pullback none of the functors A <-- Z and Z --> B is a weak = equivalence: Take A=3DX=3DY=3DC=3D1, take for B the coarse category with two objects = a and b , for X --> B and B <-- Y the functors "a": 1 --> B and B = <-- 1: "b" . The pullback Z is 0=20 How stronger a counter example do you need? And using zig-zags will make = the situation even worse. (ii) you define equivalences by spans, not "up to anything". With this = definition an equivalence between 1 and 1 is any non empty coarse = category. Every non empty set determines up to isomorphism such a = category. thus there are at least as many equivalences from 1 to 1 in = your sense as there are non empty sets. A bit much don't you think? Answer to David Roberts. I quote you: "Also, for the purposes of further discussion, the terminology "evil" = has been demoted to a mere footnote at the nLab, as it was probably = always meant to be by its coiners, and replaced by the morally neutral = and more informative name 'principle of equivalence'. Interestingly, = Voevodsky's Univalence Axiom is a way of ensuring the principle is = always respected (and without awkward acrobatics)." I am a very thorough person, so I have looked at the instances of "evil" = in nLab and found hundreds of them, not mere footnotes. I shall concentrate on the article: "Grothendieck fibrations in nLab". = It has been revised by Mike Shulman on october 11, 2012, i.e. not very = long ago, and it has many interesting features. I shall comment only = those of them which have some bearings to the present discussion. Although it is supposed to deal with Grothendieck's fibrations there is = a long paragraph,almost a whole page, with the title: "Non evil = version". The very first line is: "There is something evil about the = notion of fibration.." and the word "evil" is even underlined. Another interesting feature is the definition of internal fibration in a = strict 2-category K (why "strict"? a 2-category is a 2-category. = Probably because the notion of 2-category is unbearably evil). If you use this definition for K =3D Cat, it is easy to see that such = internal fibrations are Grothendieck fibrations equipped with a = cleavage. The situation is even worse if K =3D Cat(S)=20 Last but not least, I recommend reading the deep discussion between = Shridar Ramesh and Mike Shulman about "indexing" the fibration Z --> = Z/2Z. (but hurry, because after this mail there might very well be a = quick revision of the article). I happen to like very much Grothendieck's fibrations, this would = probably qualify me as an evil person, let me say that i'm proud of this = kind of evilness. =20 Preliminary answer to Marta Bunge: Many thanks for your mail. Before I give a complete answer, I have some = questions: (1) - What do you mean precisely by a full, faithful essentially = surjective indexed functor. Does this notion depend on the = indexings?(this is why I took the pain to give precise notations for = such indexed functors) (2) - I suppose that if C is internal to S, your [C] is the canonical = (split) indexing defined by [C](s) =3D S(s,C) thus your answer to (1) = will tell me what you mean by [C] and [D] are weakly equivalent. Before getting your answers, I can already make a few comments about = your mail. (a) Assuming you have a notion of equivalence E(A,B) reflexive = transitive and symmetric, your answer to (ii) is a bit surprising. Given = any A, suppose someone asks how many B's are equivalent to A? If B and = B' are such B's, they are equivalent, hence if you work up to = equivalence, the answer is 1. What do stack completions have to do with = this answer?=20 (b) You say that you could have worked with fibered categories instead = of indexed ones, and I believe you. But why didn't you do it? What is at stake in this this discussion is = the equivalence of categories with or without AC. You know that the = relation between indexed categories and fibered ones depends essentially = on AC. I have seen very strange things about this relation. The strangest one can be found in the Elephant: p.4, one can read: "we should make the smallest possible demands on the metatheory within = which we interpret the theory of categories (and in particular we shall = not assume that it satisfies any form of the axiom of choice)" True to this very strong statement, the author gives the proof, due to = Grothendiek, that indexed categories are "the same thing" as fibered = categories equipped with a cleavage. And that is O.K. for me. But then, without any transition or warning he starts saying: "Let P: C --> S be a fibration and CC be the associated indexed = category .." It is not an accidental slip. It can be found in Example 1.3.13, Lemma = 1.4.1, Lemma 1.4.5, Lemma 1.4.9, Lemma 1.4.10, Lemma 1.4.11, Theorem = 1.4.12, etc.. He does not say "an" associated, but " the" associated. This means not = only that every fibration has a cleavage (which is equivalent to AC when = restricted to small fibrations) but also comes equipped with a cleavage, = which is equivalent to huge form of AC for classes. Not bad considering = the strong statement I have quoted. But even assuming this huge AC, "the" associated indexed category is = astounding. It well known, and was remarked by Grothendieck more than 50 years ago, = that a group homomorphism=20 P: E --> B is a fibration iff it is surjective, and that a cleavage is a = section S of (the underlying map of) P. You won't find this fact in the Elephant. "The" associated indexed = category means that such a P comes equipped with a section. Imagine the reaction of the students if, in order to explain periodic = functions via the surjection P: R --> R/Z , you had to assume that a = section S of P is already given, and that S has any influence on the = notion of periodic function! I believe that indexed categories have been a disaster for the = mathematicians, some of whom very serious, who have adopted them. Let me give two more arguments. (a) They forbid the study of functors P: E --> B more general than = fibrations, but which keep enough of the features of the fibrations to = be able to prove about them many of the results of fibered category = theory. The simplest case is pre-fibrations. (b) The theory of fibrations is a first order theory, therefore they can = be internalized, e.g. in a topos (this is much too strong) Can anybody = give a definition of internal indexed categories and functors?To Excuse me for such a long mail, there are many more things to say about = equivalences of categories, but they will have to be postponed to = another mail. =20 (a) You work with S-indexed or=20= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]