From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6942 Path: news.gmane.org!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: Reference requested Date: Mon, 03 Oct 2011 02:06:21 -0400 Message-ID: Reply-To: "Fred E.J. Linton" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1317645210 10539 80.91.229.12 (3 Oct 2011 12:33:30 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 3 Oct 2011 12:33:30 +0000 (UTC) To: Original-X-From: majordomo@mlist.mta.ca Mon Oct 03 14:33:25 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.30]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1RAhhy-0008Qd-Si for gsmc-categories@m.gmane.org; Mon, 03 Oct 2011 14:33:23 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:59097) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RAhgN-00036M-L8; Mon, 03 Oct 2011 09:31:43 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RAhgL-00064p-Sx for categories-list@mlist.mta.ca; Mon, 03 Oct 2011 09:31:41 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6942 Archived-At: I recently reported that > ... I sought Search-engine advice regarding the use of the = > 'chaotic topological space' lingo, and came up with ... =2E.. surprisingly little. Early, early this morning I tried that again, but enclosing the search string in double-quotation marks. Ah, now I found two other references, worth citing, perhaps: 1) Volker Runde's 2005 Springer Universitext, isbn=3D038725790X, = A Taste of Topology, includes a passage (page 72), beginning = "Let (X,TX) be a chaotic topological space (ie, TX =3D {=E2=88=85,X}), = let (Y,TY) be a Hausdorff space, and let f: X =E2=86=92 Y be continuous" = and deducing such f must be constant; links (to Google Books and a PDF): [long url omitted by moderator], ftp://210.45.114.81/math/2007_07_06/Universitext/V.Runde%20A%20Taste%20of= %20Topology.pdf (no hint, though, how 'standard' Runde thought his use of "chaotic" here = was :-{ ); and 2) Mat{=C2=B4=C4=B1}as Menni's 2000 Edinburgh PhD thesis {Exact Completio= ns and Toposes} makes multiple mention of chaotic structures, with frequent citations of Bill Lawvere's = interest in such things. All of Chapter 7 is about "Chaotic Situations", = with Section = 7.1 focused in particular on "Chaotic Objects"; and Section 8.4 returns t= o "Chaotic Situations". A hint of the flavor is given in the Introduction already: "In 1999, Longley introduced a typed version of the notion of a partial = combinatory algebra in [68] and described how to build a category of assemblies = Ass(A) over a [sic] such a structure A. Shortly after, Lietz and Streiche= r showed = that the ex/reg completion of Ass(A) is a topos if and only if the typed structure = A is equivalent, in a suitable sense, to an untyped structure. Their proo= f uses = the notion of a generic mono (a mono =CF=84 such that every other mono ar= ises as a pullback of =CF=84 along a not necessarily unique map) and of the constan= t-objects embedding of Set into the category Ass(A) which they see as an inclusion = of = codiscrete objects. Related to this, it should be mentioned that Lawvere = had = already advocated for a conceptual use of codiscrete or chaotic objects i= n = other areas of mathematics (see for example [59, 55, 61, 63])." No surprise, then, to see the right adjoint =E2=88=87 to the forgetful fu= nctor Set -> Top described (p. 23) as follows: " ... the functor =E2=88=87: Set -> Top ass= igns to each set = S the =E2=80=9Cchaotic=E2=80=9D topological space with underlying set S a= nd, as open sets, only S itself and the empty set." Cf. http://citeseerx.ist.psu.edu/viewdoc/download?doi=3D10.1.1.22.9817&rep=3D= rep1&type=3Dpdf --- BTW, those four Lawvere references are these: [55] F. W. Lawvere. Toposes generated by codiscrete objects in combinator= ial topology and functional analysis. Notes for colloquium lectures given at North Ryde, New South Wales, Australia on April 18, 1989 and at Madison USA, on December 1, 1989. [59] F. W. Lawvere. Categories of spaces may not be generalized spaces as= exemplified by directed graphs. Revista colombiana de matem=C2=B4aticas, 20:179=E2=80=93 186, 1986. [61] F. W. Lawvere. Some thoughts on the future of category theory. In Proceed- ings of Category Theory 1990, Como, Italy, volume 1488 of Lecture notes in mathematics, pages 1=E2=80=9313. Springer-Verlag, 1991. [63] F. W. Lawvere. Unit and identity of opposites in calculus and physic= s. Applied categorical structures, 4:167=E2=80=93174, 1996. --- All told, eight hits, all either these two, or references to them, or search-database errors :-) . Not very heavy evidence in favor of "chaotic= ". So: cheers -- and back to [co-|in-]discrete, I fear :-) , -- Fred = [For admin and other information see: http://www.mta.ca/~cat-dist/ ]