From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5833 Path: news.gmane.org!not-for-mail From: Ronnie Brown Newsgroups: gmane.science.mathematics.categories Subject: Re terminology: Date: Fri, 21 May 2010 18:00:28 +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: 7bit X-Trace: dough.gmane.org 1274542908 5892 80.91.229.12 (22 May 2010 15:41:48 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 22 May 2010 15:41:48 +0000 (UTC) To: Urs Schreiber , "categories@mta.ca" Original-X-From: categories@mta.ca Sat May 22 17:41:45 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 1OFqpd-0003Xm-Js for gsmc-categories@m.gmane.org; Sat, 22 May 2010 17:41:45 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OFqI8-0002M3-HG for categories-list@mta.ca; Sat, 22 May 2010 12:07:08 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5833 Archived-At: Dear Urs, Thanks for your friendly and detailed reply. I should say that I also feel responsible for defending and advertising the work of my long time collaborator, Philip Higgins, without whom of course much of the work would not have got done, certainly not so quickly. His last contribution to maths was in 2005; I helped with the presentation of his TAC paper, but insisted that it showed `you know the lion from his claw', as all the ideas were his. He is happily playing the violin and making string instruments from bare blocks of wood! (That shows his craftmanship.) He remembers the project as very hard work but a lot of fun! You mention the process from category to infinity-category. Actually that was why we introduced the term infinity-category in 34. (with P.J. HIGGINS), ``The equivalence of $\infty$-groupoids and crossed complexes'', {\em Cah. Top. G\'eom. Diff.} 22 (1981) 371-386. See also: 33. (with P.J. HIGGINS), ``The equivalence of $\omega$-groupoids and cubical $T$-complexes'', {\em Cah. Top. G\'eom. Diff.} 22 (1981) 349-370. The paper [34] also gives a definition of what is now called (rightly) a globular set. I am curious to know whether there are earlier definitions of these terms. Joachim Lambek once asked me at a category theory meeting: `Why don't people from xxxx refer to these papers?' What could I say? People do write about 2-groupoids without referring to analogous work on crossed complexes. The points Andre makes are very interesting. In the 1970s we were very puzzled by the Kan condition, and still are, for the following reason. The fact that the simplicial singular complex of a space is a Kan complex is due to a fact about the models, namely a geometric simplex retracts onto all faces but one. So we can have in effect fillers for S(X) natural in X, by making choices on the models. Also these fillers are clearly related to multiple compositions of the remaining faces. The problem is that there is no unique choice of such retractions, nor is it clear what might be the relations between iterates of such fillers. These considerations led Keith Dakin to the notion of T-complex for his 1976 thesis; somehow `T-complex' has more recently become `complicial set', but nobody asked me. (Groan! Groan!) So it seems that the notion of quasi category as a weak Kan complex still has not captured something about the basic example; but how to repair that is quite unclear. As I have said before, we found it necessary for certain aspects to work cubically, as expressing in a manageable way `algebraic inverses to subdivision', and also to get monoidal closed structures. Again, it is not clear how to capture axiomatically the properties of the cubical singular complex, as some kind of weak cubical infinity groupoid. I have been unable to cope with the complications of multiple compositions in globular or simplicial terms. Is there an operad view of the cubical case? I have no wish to hold things up or disparage work developing these ideas in different ways, just the contrary, and indeed I wrote that I was thinking of higher dimensional group theory, rather than category theory. The contrast and relations between such views could, perhaps should, be illuminating. We are putting a photo from Macquarie of John Robinson's sculpture `Journeys' as a frontispiece to the new book. Best wishes to all for the future of this great adventure. Ronnie Urs Schreiber wrote: > Dear Ronnie Brown, > > you write:: > > >> My own problem is with term `infinity groupoid' which >> is used to describe something which is not even a groupoid, >> > > It seems to follow the well established terminology in higher category > theory, which proceeds: category, 2-category, 3-category, .... > infinity-category and groupoid, 2-groupoid, 3-groupoid, ... > infinity-groupoid. > > >> It seems to be an example of these confusions is the way the simplicial >> singular complex of a space is called an infinity-groupoid, even the >> `fundamental infinity groupoid', when what seems to be referred to is that >> it is a Kan complex, i.e. satisfies the Kan extension condition, studied >> since 1955. >> > > The notion of Kan complex is one model for the notion of > infinity-groupoid. There are other, equivalent models. And there are > models that model stricter subclasses of infinity-groupoids, such as > those you are famous for having studied. Part of the point of saying > "infinity-groupoid" instead of "Kan complex" or else is to amplify the > general notion over its concrete implementation. > .... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]