From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6411 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Terminology of locally small categories without replacement Date: Wed, 8 Dec 2010 11:48:59 +1100 Message-ID: References: Reply-To: Richard Garner NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1291776327 30817 80.91.229.12 (8 Dec 2010 02:45:27 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 8 Dec 2010 02:45:27 +0000 (UTC) Cc: Categories To: JeanBenabou Original-X-From: majordomo@mlist.mta.ca Wed Dec 08 03:45:22 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1PQA1y-0000es-Gl for gsmc-categories@m.gmane.org; Wed, 08 Dec 2010 03:45:22 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:42811) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PQA1n-0007O9-LS; Tue, 07 Dec 2010 22:45:11 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PQA1k-0006pK-27 for categories-list@mlist.mta.ca; Tue, 07 Dec 2010 22:45:08 -0400 In-Reply-To: <3280C591-6F02-454A-A1A4-FA8AC7FD0086@wanadoo.fr> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6411 Archived-At: Dear Jean, > It is incorrect because Fib(S) does not have pullbacks or equalizers hence > it not finitely complete Yes, that's true; however Fib(S) does have PIE limits (or even just bilimits) which is enough for Ross's definition to make sense. In fact, one need not even assume the existence of any limits at all: an object may be defined to be locally small just when, for every cospan of arrows with small domain, the comma object exists and is again small. > It is incomplete for two reasons: > (i) When I asked for significant mathematical examples, I meant apart from > locally small fibrations. In fact there are no other examples; the two notions are essentially equivalent. Given the 2-category K with a class of small objects therein, we can consider the full sub-2-category of K spanned by those objects, and the underlying ordinary category C of this. Now each object x in K induces a fibration p: E --> C whose total category E has as objects, morphisms f: c --> x in K with small domain; and as arrows (f, c) --> (f',c'), pairs of a morphism h: c --> c' and a 2-cell f => f'h in K. It's now not hard to show that this fibration will be locally small just when the object x is locally small in the sense described by Ross (under the additional, and reasonable assumption that the class of small objects is dense in K---which in particular is the case when K = Fib(S) and C are the representable fibrations). Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]