From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5839 Path: news.gmane.org!not-for-mail From: Ronnie Brown Newsgroups: gmane.science.mathematics.categories Subject: Re: terminology Date: Sat, 22 May 2010 22:43:15 +0100 Message-ID: References: Reply-To: Ronnie Brown NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1274623560 7002 80.91.229.12 (23 May 2010 14:06:00 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 23 May 2010 14:06:00 +0000 (UTC) To: =?ISO-8859-1?Q?Andr=E9?= Original-X-From: categories@mta.ca Sun May 23 16:05:59 2010 connect(): No such file or directory Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OGBoU-0005N1-R0 for gsmc-categories@m.gmane.org; Sun, 23 May 2010 16:05:59 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OGBZs-0002XA-L5 for categories-list@mta.ca; Sun, 23 May 2010 10:50:52 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5839 Archived-At: Dear Andr=E9 There seems to me to be a tremendous amount of great work going on=20 higher category theory, but when you write ----------------------------------------------------- One lies in the fact that equivalent categories are considered to be the = "same",=20 even if [or] when they are not isomorphic. ----------------------------------------- this seems to go against the grain of what I have been doing in groupoids= since I decided they were valuable in about 1965! It sounds like the old= canard `groupoids reduce to groups', so there must be some confusion in = my mind on what you are saying.=20 One thing that took me a while to realise was that it was not enough to s= tudy the fundamental groupoid or a fundamental group but one needed to co= nsider intermediate cases, namely the fundamental groupoid on a set of ba= se points chosen according to the geometry at hand. (`Algebraic topology'= has not understood this it seems.) The vertices of a groupoid give a spa= tial component to group theory, a kind of geography, and sometimes, even = often, that is needed to model the geometry. So for example it is useful= to replace the trefoil group which has 2 generators x,y and one relation= x^2=3Dy^3 by the trefoil groupoid which is the double mapping cylinder (= homotopy pushout) in groupoids of the two maps Z \to Z, given by squaring= and cubing. So we add an extra groupoid generator iota on different vert= ices which turns x^2 into y^3. This corresponds to the double mapping con= struction to give a CW-complex.=20 So groupoids give the strict algebra of keeping the information which mak= es things the same.=20 In higher dimensions we want not just commutative diagrams but control of= the ways of filling these diagrams. If the diagram is a pentagon (as we = all know does happen) I would want a pentagon as part of the geometry, an= d the only question is how to deal with multiple compositions of various = such objects, and that was the aim of David Jones thesis on Polyhedral T-= complexes. The point is that the pieces to be composable have to be all f= aces but one of a general poyhedral `horn', the process of composing them= is the filler of the horn, and the composite of the pieces is the remai= ning face of the filler. (It was not attempted to do this in category rat= her than groupoid terms, and that is still a mystery!) So you can see I h= ave long been very sympathetic to using the Kan condition for describing = algebraic or structural objects, but find the simplicial approach too awk= ward (for me, of course; I found the way Nick Ashley coped with that was = amazing).=20 I do not want to consider equivalent groupoids the same, as I may want to= use the spatial components to describe how they might be glued together.= It is partly the old tag of not throwing away information till the last = possible moment.=20 On the other hand, some computations are best done at the strict level, r= ather than the weak one. I mention here the rotations in my paper: ``Higher dimensional group theory'', in {\em Low dimensional topology}, = London Math Soc. Lecture Note Series 48 (ed. R. Brown and T.L. Thickstun= , Cambridge University Press, 1982), pp. 215-238. (see also a fuller exposition in the new book on Nonabelian algebraic top= ology), which would seem to be more difficult to write out at the lax lev= el. The fact that the strict calculations imply the existence of certain = homotopies is part of the interest.=20 So in the work with Higgins a Kan fibration - from the singular filtered = complex of a filtered space to the quotient to give a strict structure - = ties in the lax and the strict in a necessary way for the theory and calc= ulations.=20 I am really searching for points of agreement.=20 Best regards Ronnie [For admin and other information see: http://www.mta.ca/~cat-dist/ ]