From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8240 Path: news.gmane.org!not-for-mail From: =?utf-8?Q?Jonathan_CHICHE_=E9=BD=8A=E6=AD=A3=E8=88=AA?= Newsgroups: gmane.science.mathematics.categories Subject: Re: Final objects in 2-categories Date: Sat, 26 Jul 2014 15:40:09 +0200 Message-ID: References: Reply-To: =?utf-8?Q?Jonathan_CHICHE_=E9=BD=8A=E6=AD=A3=E8=88=AA?= NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Apple Message framework v1257) Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1406553373 4135 80.91.229.3 (28 Jul 2014 13:16:13 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 28 Jul 2014 13:16:13 +0000 (UTC) Cc: Categories mailing list To: Adam Gal Original-X-From: majordomo@mlist.mta.ca Mon Jul 28 15:16:08 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1XBkmZ-00057B-Gs for gsmc-categories@m.gmane.org; Mon, 28 Jul 2014 15:16:03 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:48671) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XBkls-0005Oi-Jn; Mon, 28 Jul 2014 10:15:20 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XBklr-0002kP-94 for categories-list@mlist.mta.ca; Mon, 28 Jul 2014 10:15:19 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8240 Archived-At: Dear Adam,=20 I don't have much time right now and have not read your paper carefully. = However, to elaborate on David Roberts's answer, in my work about the = homotopy theory of 2-categories, the property which I have found the = most useful (and which shows up in many natural circonstances) is the = following. Given a 2-category A, let us say that an object z of A has a = terminal object if Hom(a,z) has a terminal object for every object a of = A. This terminology was suggested to me by Jean B=E9nabou. It is of = course compatible with the usual definition if the 2-category happens to = be Cat. It can be shown that, if a small 2-category A admits such an = object, then the map from A to the point is a weak equivalence, i.e. its = nerve is a simplicial weak equivalence. I have some papers around = related stuff, which I could communicate when they are in their final = version.=20 This property of having such an object has already been considered in = the literature, for instance in Bunge's "Coherent extensions and = relational algebras", Jay's "Local adjunctions" and Betti-Power's "On = local adjointness of distributive bicategories". Thanks to Steve Lack = for having pointed out my attention to this last paper. You may also = find useful this question and its answers on Math Overflow: = http://mathoverflow.net/questions/160765/whats-an-initial-object-in-a-pose= t-enriched-category/161118#161118.=20 Best wishes,=20 Jonathan Le 24 juil. 2014 =E0 16:52, Adam Gal a =E9crit : > Hi all, >=20 > Has someone studied the notion of final objects in a 2-category? I > know that we can define it using the classifying space, and that in > this sense Quillen's theorem A holds and tells us this is equivalent > to some fibers being contractible. > This seems to be a bit too coarse though. For instance in our recent > paper (with E. Gal) we wanted to prove that something is final, and > what we showed is that these fibers have an initial and final object. > So they definitely have contractible classifying spaces, but it seems > that we can say something more precise than this. >=20 > The question is if this fits into some finer notion of final object in > 2-categories which has been studied. >=20 > Thanks, > Adam [For admin and other information see: http://www.mta.ca/~cat-dist/ ]