From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6936 Path: news.gmane.org!not-for-mail From: F William Lawvere Newsgroups: gmane.science.mathematics.categories Subject: Re: Reference requested Date: Fri, 30 Sep 2011 18:31:05 -0400 Message-ID: Reply-To: F William Lawvere 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 1317471685 31885 80.91.229.12 (1 Oct 2011 12:21:25 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 1 Oct 2011 12:21:25 +0000 (UTC) To: , "Fred E.J. Linton" Original-X-From: majordomo@mlist.mta.ca Sat Oct 01 14:21:20 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 1R9yZE-0007zA-4U for gsmc-categories@m.gmane.org; Sat, 01 Oct 2011 14:21:20 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:43269) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1R9yWz-0003zn-Oo; Sat, 01 Oct 2011 09:19:01 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1R9yWy-00075i-7x for categories-list@mlist.mta.ca; Sat, 01 Oct 2011 09:19:00 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6936 Archived-At: Fred, you caught me ! Indeed my memory of JL Kelley was blurred by wishful thinking. Hence, dear friends, please be so generous as to ignore the attempt at historical justification and instead further elaborate the rational arguments as Fred has. Bill Lawvere On Fri 09/30/11 5:19 PM , "Fred E.J. Linton" fejlinton@usa.net sent: > Bill recalls: >=20 >> For at least 50 years, the two, words >> chaotic >> codiscrete >> were alternate terminology for a certain kind of > space. > My memory rather matches instead what I see in (3.2(d)) of Willard,=20 > that the topology with only the whole space and the empty set > "open"is called either 'trivial' or 'indiscrete'. >=20 > In my experience I've never encountered 'chaotic' as the > adjective used for that attribute -- indeed, 'chaotic' would have=20 > conflicted rather badly with the Chaos Theory arising out of > Ren=C3=83=C2=A9 Thom's Catastrophe Theory of the early '60s or so -- and = 'codiscrete'=20 > strikes me as what only a categorist hoping (as we many of us=20 > long did) to systematize terminology into dual camps of 'properties'=20 > and 'coproperties' (in the model of adjoint/coadjoint, limit/colimit,=20 > terminal/coterminal, etc.) could have come up with -- not a term any > self-respecting point-set topologist would have thought to use :-) . >=20 > 'Codiscrete' of course does, for just that reason, have its merits, > as Bill points out: >=20 >> I prefer "codiscrete" since it clearly > indicates something> opposite to discrete, the precise sense of > oppositeness being> that of inclusions adjoint to the same uniting > functor, often called> the "underlying". (In higher > dimensions, coskeletal and skeletal> are similar identical opposites, wit= h > "truncation" as uniter).>=20 >> In fact every groupoid is a colimit of > codiscrete ones, indeed> groupoids form a reflective subcategory of the > topos that classifies > Boolean algebras, and the latter has a site > consisting of codiscrete> groupoids. (The generic Boolean algebra 2^( ) > has as its natural> geometric realization the infinite-dimensional > sphere, containing > the ordinary interval as a generating > distributive lattice). > And I object not one whit to any of that :-) . >=20 >> More recently, "chaotic" has come to > have a different meaning, > although one also involving a right adjoint. = If > f:X->Y is a map > from a space equipped with an action of a monoid > T to another> space, then f is a chaotic observable if the > induced equivariant> map from X to the cofree action Y^T is > epimorphic. A classic "symbolic"> example has Y=3Dpi0(X), i.e. the obser= vation > recorded by f is merely of> which component we are passing through, but > almost any > T-sequence of such is obtained by a sufficiently > clever choice> of initial state in X. >=20 > This again suggests that 'chaotic' might not be the best choice of > adjective for that indiscrete/codiscrete topology, or the analogous > type of category, or groupoid, or topological category or groupoid. >=20 > Cheers, -- Fred >=20 >=20 >=20 >=20 >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]