From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5870 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= Newsgroups: gmane.science.mathematics.categories Subject: Re: Straw man terminology Date: Wed, 26 May 2010 23:31:29 -0400 Message-ID: References: Reply-To: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= 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 1275151547 26176 80.91.229.12 (29 May 2010 16:45:47 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 29 May 2010 16:45:47 +0000 (UTC) To: "Urs Schreiber" Original-X-From: categories@mta.ca Sat May 29 18:45:40 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 1OIPAJ-0005Uw-64 for gsmc-categories@m.gmane.org; Sat, 29 May 2010 18:45:39 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OIOVr-0005QO-4w for categories-list@mta.ca; Sat, 29 May 2010 13:03:51 -0300 Thread-Index: Acr8/4Zc/oiVlsAKQJ2aDp0tObcPvQASfGo+ Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5870 Archived-At: Dear Urs, I agree that Lurie is using the infinity-n-category terminology. I am not questioning that. I am observing that he is calling a quasi-category an infinity-category. In his terminology, an infinity-category is a special kind of = infinity-one-category.=20 I believe that the name infinity-category should apply to all "infinity" = categories,=20 inculding the infinity-one-category. No? Best, Andre -------- Message d'origine-------- De: Urs Schreiber [mailto:urs.schreiber@googlemail.com] Date: mer. 26/05/2010 13:59 =C0: Joyal, Andr=E9 Cc: categories@mta.ca Objet : Re: RE : categories: Re: Straw man terminology =20 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=3D31 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/ ]