From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4530 Path: news.gmane.org!not-for-mail From: "Tom Leinster" Newsgroups: gmane.science.mathematics.categories Subject: Re: KT Chen's smooth CCC, a correction Date: Sat, 30 Aug 2008 01:14:26 +0100 (BST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241020007 13722 80.91.229.2 (29 Apr 2009 15:46:47 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:46:47 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Sat Aug 30 10:57:03 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 30 Aug 2008 10:57:03 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KZQuC-00023H-Ih for categories-list@mta.ca; Sat, 30 Aug 2008 10:54:20 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 66 Original-Lines: 173 Xref: news.gmane.org gmane.science.mathematics.categories:4530 Archived-At: On Tue, 26 Aug 2008, Bill Lawvere wrote: > fickle pedias and beckening bistros which, like the mythical > black holes, often regurgitate information as buzzwords and > disinformation. Disinformation is *deliberate* false information, false information *intended* to mislead. As I understand it, Bill's statement says, among other things, that disinformation often appears on the n-Category Cafe. I don't know whether Bill really meant to say this. I very much hope not. I can't think of a single instance where someone at the n-Category Cafe has intended to mislead. Best wishes, Tom On Tue, 26 Aug 2008, Bill Lawvere wrote: > Dear Jim and colleagues, > > By urging the study of the good geometrical ideas and constructions of > Chen and Froelicher, as well as of Bott, Brown, Hurewicz, Mostow, Spani= er, > Steenrod, I am of course not advocating the preferential resurrection o= f > the particular categories they tentatively devised to contain the > constructions. > > Rather, recall as an analogy the proliferation of homology theories 60 > years ago; it called for the Eilenberg-Steenrod axioms to unite them. > Similarly, the proliferation of such smooth categories 45 years ago wou= ld > have needed a unification. Programs like SDG and Axiomatic Cohesion hav= e > been aiming toward such a unification. > > The Eilenberg-Steenrod program required, above all, the functorality wi= th > respect to general maps; in that way it provided tools to construct ev= en > those cohomologies (such as compact support and L2 theories) that are > less functorial. > > The pioneers like Chen recognized that the constructions of interest (s= uch > as a smooth space of piecewise smooth paths or a smooth classifying spa= ce > for a Lie group) should take place in a category with reasonable functi= on > spaces. They also realized, like Hurewicz in his 1949 Princeton lecture= s, > that the primary geometric structure of the spaces in such categories m= ust > be given by figures and incidence relations (with the algebra of functi= ons > being determined by naturality from that, rather than conversely as had > been the 'default' paradigm in 'general' topology, where the algebra of > Sierpinski-valued functions had misleadingly seemed more basic than > Frechet-shaped figures.) I have discussed this aspect in my Palermo pap= er > on Volterra (2000). > > The second aspect of the default paradigm, which those same pioneers > seemingly failed to take fully into account, is repudiated in the first > lines of Eilenberg & Zilber's 1950 paper that introduced the key catego= ry > of Simplicial Sets. Some important simplicial sets having only one poin= t > are needed (for example, to construct the classifying space of a group)= . > Therefore. the concreteness idea (in the sense of Kurosh) is misguided > here, at least if taken to mean that the very special figure shape 1 is > faithful on its own. That idea came of course from the need to establis= h > the appropriate relation to a base category U such as Cantorian abstrac= t > sets, but that is achieved by enriching E in U via E(X,Y) =3D p(Y^X), > without the need for faithfulness of > p:E->U; this continues to make sense if E consists not of mere cohesiv= e > spaces but of spaces with dynamical actions or Dubuc germs, etcetera, e= ven > though then p itself extracts only equilibrium points. The case of > simplicial sets illustrates that whether 1 is faithful just among given > figure shapes alone has little bearing on whether that is true for a > category of spaces that consist of figures of those shapes. > > Naturally with special sites and special spaces one can get special > results: for example, the purpose of map spaces is to permit representi= ng > a functional as a map, and in some cases the structure of such a map > reduces to a mere property of the underlying point map. Such a result, = in > my Diagonal Arguments paper (TAC Reprints) was exemplified by both smoo= th > and recursive contexts; in the latter context Phil Mulry (in his 1980 > Buffalo thesis) developed the Banach-Mazur-Ersov conception of recursiv= e > functionals in a way that permits shaded degrees of nonrecursivity in > domains of partial maps, yet as well permits collapse to a 'concrete' > quasitopos for comparison with classical constructions. > > Grothendieck did fully assimilate the need to repudiate the second aspe= ct > (as indeed already Galois had done implicitly; note that in the categor= y > of schemes over a field the terminal object does not represent a faithf= ul > functor to the abstract U). Therefore Grothendieck advocated that to an= y > geometric situations there are, above all, toposes associated, so that = in > particular the meaningful comparisons between geometric situations star= t > with comparing their toposes. > > A Grothendieck topos is a quasitopos that satisfies the additional > simplifying axiom: > All monomorphisms are equalizers. > A host of useful exactness properties follows, such as: > (*)All epimonos are invertible. > The categories relevant to analysis and geometry can be nicely and full= y > embedded in categories satisfying the property (*). That claim arouses > instant suspicion among those who are still in the spell of the default > paradigm; for that reason it may take a while for the above-mentioned > 45-year-old proliferation of geometrical category-ideas to become > recognized as fragments of one single theory. > > There is still a great deal to be done in continuing > K.T. Chen's application of such mathematical categories to the calcul= us > of variations and in developing applications to other aspects of > engineering physics. These achievements will require that students pers= ist > in the scientific method of alert participation, like guerilla fighters > pursuing the laborious and cunning traversal of a treacherous jungle > swamp. For in the maze of informative 21st century conferences and > internet sites there lurk fickle pedias and beckening bistros which, li= ke > the mythical black holes, often regurgitate information as buzzwords an= d > disinformation. > > Bill > > > On Sun, 17 Aug 2008, jim stasheff wrote: > >> Bill, >> >> Happy to see you contributing to the renaissance in interest in >> Chen's work. >> >> It would be good to post your msg to the n-category cafe blog >> whee there's been an intense discussion of `smooth spaces' i various >> incarnaitons. >> >> jim >> >> http://golem.ph.utexas.edu/category/2008/05/convenient_categories_of_s= moot.html ----- The University of Glasgow, charity number SC004401