From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5862 Path: news.gmane.org!not-for-mail From: Urs Schreiber Newsgroups: gmane.science.mathematics.categories Subject: Re: Straw man terminology Date: Wed, 26 May 2010 19:59:42 +0200 Message-ID: Reply-To: Urs Schreiber NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1274968289 461 80.91.229.12 (27 May 2010 13:51:29 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 27 May 2010 13:51:29 +0000 (UTC) Cc: categories@mta.ca To: =?ISO-8859-1?B?Sm95YWwsIEFuZHLp?= Original-X-From: categories@mta.ca Thu May 27 15:51:27 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 1OHdUc-0004Az-Cb for gsmc-categories@m.gmane.org; Thu, 27 May 2010 15:51:26 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OHdDy-0006Ey-J4 for categories-list@mta.ca; Thu, 27 May 2010 10:34:14 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5862 Archived-At: Dear Andre, > I agree that the terminology (infinity,1)-terminology can be useful. Okay. > Can I point out that Lurie is calling a quasi-category an infinity-category? Okay, let's look at Lurie's use of terminology then. Notice that just a little later in On the classification of TFTs http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.0465v1.pdf#page=31 In the remark 2.1.26 he speaks of "the various models of the theory of (oo,1)-categories" referring to Julie Bergner's article which shows that quasi-categories, sSet-categories, Segal categories and complete Segal spaces give four equivalent such models. Then still a bit later in (oo,2)-Categories and the Goodwillie calculus http://www.math.harvard.edu/~lurie/papers/GoodwillieI.pdf he uses terminology exactly as I have been suggesting in my previous messages: starting in the third sentence: "Let us use the term (oo,n)-category to indicate a higher category in which all k-morphisms are assumed to be invertible for k> n. [...] The theory of (oo,1)-categories is also quite well understood, though in this case there is a variety of possible approaches. [...] These are known as quasicategories in the literature; we will follow the terminology of [HTT] and refer to them simply as oo-categories." So, for what it's worth, Lurie adopts the convention that I was talking about, it seems to me: to say (oo,n)-category for the general concept and use other terms for concrete models. He just happens to have the extra convention that "oo-category" (without the ",1") is his term for the model that you called quasi-category. Maybe in this context it is noteworthy that in this last article alone, there is presented literally a dozen of different and equivalent models for (oo,2)-categories. Best, Urs [For admin and other information see: http://www.mta.ca/~cat-dist/ ]