From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5875 Path: news.gmane.org!not-for-mail From: jim stasheff Newsgroups: gmane.science.mathematics.categories Subject: Re: Straw man terminology Date: Thu, 27 May 2010 18:28:14 -0400 Message-ID: References: Reply-To: jim stasheff 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 1275151816 27004 80.91.229.12 (29 May 2010 16:50:16 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 29 May 2010 16:50:16 +0000 (UTC) To: Urs Schreiber Original-X-From: categories@mta.ca Sat May 29 18:50:14 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 1OIPEi-0007Kd-92 for gsmc-categories@m.gmane.org; Sat, 29 May 2010 18:50:12 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OIOdS-0005el-Kz for categories-list@mta.ca; Sat, 29 May 2010 13:11:42 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5875 Archived-At: Urs Schreiber wrote: > 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. > > > THIS IS MUCH BETTER - A DEFINITION - NOT AN EXAMPLE (AKA MODEL) OR APPROACH > [...] > > 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/ ]