From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5825 Path: news.gmane.org!not-for-mail From: Urs Schreiber Newsgroups: gmane.science.mathematics.categories Subject: Re terminology: Date: Thu, 20 May 2010 13:58:17 +0200 Message-ID: References: Reply-To: Urs Schreiber NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1274385265 16709 80.91.229.12 (20 May 2010 19:54:25 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 20 May 2010 19:54:25 +0000 (UTC) To: Ronnie Brown Original-X-From: categories@mta.ca Thu May 20 21:54:24 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 1OFBp1-0000r7-Vs for gsmc-categories@m.gmane.org; Thu, 20 May 2010 21:54:24 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OFBKs-0000e4-0z for categories-list@mta.ca; Thu, 20 May 2010 16:23:14 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5825 Archived-At: Dear Ronnie Brown, you write:: > My own problem is with term `infinity groupoid' which > =A0is 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 tha= t > 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. > this distracts from studying the not so simple proofs that strict > =A0higher homotopical structures exist, I don't quite see why the term should distract from or otherwise diminish accomplishments made in the study of strict infinity-groupoids. On the contrary, to my mind the general theory of infinity-groupoids puts many of these constructions into the full perspective of higher category theory and thereby amplifies their relevance. > I know the term (\infty,n)-groupoid has been well used recently Not quite: the term (infinity,n)-category is used to denote an infinity-category in which all k-morphisms for k greater than n are equivalences. So an infinity-groupoid is an (infinity,0)-category. This terminology was not invented in order to hide anybody's previous accomplishments. On the contrary, I think, this terminology follows established use in higher category theory and serves to nicely organize past, present and future insights into higher category theory in a coherent picture. I am hoping that eventually we find a constructive way of thinking about these things, where past insights are seen as fitting into the beautiful picture of higher category theory that has recently been emerging, and are not seen to be in conflict with them. Here is an example I quite like: in the context of (infinity,1)-category theory Jacob Lurie gave an entirely intrinsic category-theoretic definition of (infinity,1)-sheaves, also known as infinity-stacks: these are infinity-groupoid valued presheaves satisfying a suitable descent condition. Working backwards from the abstract higher category-theoretic definition of these, one can work out how this notion matches related models that were previously investigated. Among these are two main strands: 1) the homotopical structures/model category structures on categories of ordinary presheaves with values in simplicial sets, as developed by Brown, Joyal, Jardine, Dugger and others. 2) The notion of presheaves with values in strict infinity-groupoids / strict omega-groupoids, as originally conceived by John Roberts and then formalized by Ross Street and others. One might worry that both these decade-old developments might not harmonize with the intrinsic higher-categorical notion of (infinity,1)-sheaf. But the opposite is true: one finds that they are neatly subsumed in the abstract definition and conversely provide concrete workable models for the abstract notion. For point 1) this is proven in Jacob Lurie's book on higher topos theory: the Joyal/Jardine model structure models precisely those (infinity,1)-sheaves which are "hypercomplete". More generally, the left Bousfiled localization of the model structure on simplicial presheaves at Cech nerves models (infinity,1)-sheaves. For point 2) this has been proven by Dominic Verity, following a conjecture I made: one can show that under mild conditons, under the inclusion of strict omega-groupoids / strict infinity-groupoids into all infinity-groupoids, the Roberts-Street notion of descent for such strict oo-groupoid sheaves does model the abstractly found (infinity,1)-sheaf condition. http://ncatlab.org/nlab/show/Verity+on+descent+for+strict+omega-groupoid+va= lued+presheaves So this means now not a diminishing but a considerable increase in value of the old work on presheaves with values in strict infinity-groupoids / omega-groupoids: since it embeds these constructions into a powerful abstract theory, we now conversely have all the abstract tools and insights available to study and use the former. I have been using this embeddingg of strict oo-groupoid valued oo-stacks into all oo-stacks quite a lot in my research, originally starting with the observation that the BCSS-model of the string 2-group realizes it as a strict 2-groupoid valued stack on Diff, which is quite useful for some applications. All along I have greatly benfitted by having your book and nonabelian algebraic topology next to me, together with Ross Street's articles on descent, and at the same time having Higher Topos Theory on the table. I find that that these two aspects interact very nicely, and I was therefore a bit saddened by hearing what sounded like accusations that new developments in higher category theory try to intentionally diminish previous development. I think math is a win-win game, not a competition: one person's insight does not dimish the other person's insight, but both increase each other's value. I am dearly hoping that those who practiced aspects of higher category theory for so long see the new developments not as in conflictt with their work, but as a beautiful blossoming of the theory that they started developing. Because it is true. Best regards, Urs [For admin and other information see: http://www.mta.ca/~cat-dist/ ]