From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5831 Path: news.gmane.org!not-for-mail From: Urs Schreiber Newsgroups: gmane.science.mathematics.categories Subject: Re terminology: Date: Fri, 21 May 2010 20:48:36 +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 1274542872 5806 80.91.229.12 (22 May 2010 15:41:12 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 22 May 2010 15:41:12 +0000 (UTC) To: Ronnie Brown Original-X-From: categories@mta.ca Sat May 22 17:41:04 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 1OFqou-00039m-2P for gsmc-categories@m.gmane.org; Sat, 22 May 2010 17:41:00 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OFqIv-0002Mc-AU for categories-list@mta.ca; Sat, 22 May 2010 12:07:57 -0300 In-Reply-To: <4BF6BC2C.2000606@btinternet.com> Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5831 Archived-At: Dear Ronnie, > You mention the process from category to infinity-category. Actually that > was why we introduced the term infinity-category This is why I am thinking you could embrace the way the term is used these days: because it follows precisely your use back then, only removing the restriction of strictness. And algebraicity can be restored. See below... > 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.=A0 These considerations led K= eith > Dakin to the notion of T-complex=A0 for his 1976 thesis; somehow `T-compl= ex' > 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 compl= ex > still has not captured something about the basic example; but how to repa= ir > that is quite unclear. This has recently been clarified by Thomas Nikolaus in his work on algebraic Kan complexes (which are essentially simplicial T-complexes!) and algebraic quasi-categories: http://ncatlab.org/nlab/show/model+structure+on+algebraic+fibrant+objects He shows that the model category/quasi-category/(oo,1)-category (check preferred term) of all Kan complexes is equivalent to that of all Kan complexes with all horn fillers chosen. And analogously: that the model category/quasi-category/(oo,1)-category (check preferred term) of all quasi-categories is equivalent to that of all quasi-categories with all inner horn fillers chosen. This says that while a Kan complex or quasi-category is not directly an algebraic model for an oo-groupoid or (oo,1)-category, respectively, you can immediately turn it into an algebraic model by making choices, and up to equivalence, the resulting algebraic model does not depend on these choices. Best, Urs [For admin and other information see: http://www.mta.ca/~cat-dist/ ]