From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7688 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Jean_B=E9nabou?= Newsgroups: gmane.science.mathematics.categories Subject: Terminology: Remarks Date: Wed, 1 May 2013 07:17:16 +0200 Message-ID: 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=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1367457177 16308 80.91.229.3 (2 May 2013 01:12:57 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 2 May 2013 01:12:57 +0000 (UTC) To: Categories Original-X-From: majordomo@mlist.mta.ca Thu May 02 03:12:57 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 1UXi4u-0002i3-1f for gsmc-categories@m.gmane.org; Thu, 02 May 2013 03:12:56 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:44149) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1UXi2w-0003r2-MN; Wed, 01 May 2013 22:10:54 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1UXi2y-0004HT-3b for categories-list@mlist.mta.ca; Wed, 01 May 2013 22:10:56 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7688 Archived-At: Dear all, Let me first thank the persons who have answered my questions on = terminology. Since the answers were different, it seems that the terminology is not = standard and that my questions made sense. Preliminary remarks.=20 By now, everybody understands what kind of categories I was talking = about. they are very simple, one might be tempted to say trivial. But in = many highly non trivial questions they appear either as "building = bricks" of more complex constructions (see e.g. fibrations such that all = the fibers are of that kind), or as special cases, unavoidable, of more = general situations (e.g. Freyd's notion of "equivalence kernels"). Of course such categories can be "internalized ", say in a topos, (this = is much too strong), and it would be nice that the terminology should = fit also the internal case, and in particular any reference, explicit or = implicit, to AC should be avoided. 1- Discrete versus indiscrete, or coarse, etc. The categories 0 and 1 = are both discrete and indiscrete So each name night pose a problem. The = terminology "discrete" is by now well established. Thus I think one = should avoid "indiscrete" and use "coarse" instead. Thus 0 and 1 are = discrete and coarse and that is admissible. In the internal case every = sub object of 1 is both discrete and indiscrete. So "subterminal" is a = nice name to cover both cases. 2- "Essentially". One has to be careful about this word. It seems to = mean "up to equivalence". But that depends on what you call "equivalence = of categories". There is a very strong 2-categorical notion, namely a = pair of functors f: A --> B and g: B --> A with fg and gf isomorphic = to identities (with or without the adjunction axioms). But it is useless = for our purpose. The one which might serve here is f full and faithful = and essentially surjective. But unless we have AC it is not symmetric, = even for A and B small.=20 Thus with AC we might adopt the suggestions of Thomas and use = essentially discrete and essential sub terminator. But do we really need = AC or even any notion of equivalence at all? 3- Elementary remarks. Let S be a category with finite limits. I want to "internalize" the two = notions for which I'd like a name, suitable not only when S=3DSet, but = in general. (i) if X is an internal category I denote by Ob(X) and Map(X) the = objects of objects and maps of X and by d: Map(X) --> Ob(X)xOb(X) with projections Dom and Codom. When S=3DSet = if f: x -->y is a map of X, df is the "direction" of f. X is a preordered object iff d is a mono.=20 It is easy, using the multiplication of X, to express in terms of finite = limits, the fact that X is a groupoid. Hence equivalence relations are = definable in any category with finite limits, although composition of = arbitrary relations is possible only when S is regular. Moreover if F: S = --> S' is a functor which preserves finite limits and X is an = equivalence relation so is F(X) No essentiality no AC is needed (ii) the functor X --> 1 is full and faithful iff the direction map d = is an iso. This again is preserved by finite limits preserving functors = F. (iii) If F preserves finite limits and is faithful it reflects = equivalence relations and property (ii). In particular the Yoneda = embedding preserves and reflects the previous properties. this enables = us to work with them as if S=3DSet. No AC is involved . For all these reasons I'm not too keen about using "essentially" I have many more remarks about Fred Linton's and David Robert's postings = but this mail is already a bit long. So I shall wait a few days before i = send another mail. By that time I hope to have some reactions to the = present posting and I shall do my best to answer them. Best regards, Jean [For admin and other information see: http://www.mta.ca/~cat-dist/ ]