From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6410 Path: news.gmane.org!not-for-mail From: JeanBenabou Newsgroups: gmane.science.mathematics.categories Subject: Re: Terminology of locally small categories without replacement Date: Wed, 8 Dec 2010 01:11:03 +0100 Message-ID: References: Reply-To: JeanBenabou NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v753.1) Content-Type: text/plain; charset=ISO-8859-1; delsp=yes; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1291776284 30661 80.91.229.12 (8 Dec 2010 02:44:44 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 8 Dec 2010 02:44:44 +0000 (UTC) Cc: Categories To: Richard Garner Original-X-From: majordomo@mlist.mta.ca Wed Dec 08 03:44:40 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 1PQA1I-0000Tq-8j for gsmc-categories@m.gmane.org; Wed, 08 Dec 2010 03:44:40 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:42805) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PQA0z-0007EQ-2h; Tue, 07 Dec 2010 22:44:21 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PQA0q-0006nm-OM for categories-list@mlist.mta.ca; Tue, 07 Dec 2010 22:44:13 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6410 Archived-At: Dear Richard, Thank you for your quick answer, which unfortunately is both =20 incorrect and incomplete. It is incorrect because Fib(S) does not have pullbacks or equalizers =20 hence it not finitely complete It is incomplete for two reasons: (i) When I asked for significant mathematical examples, I meant apart =20= from locally small fibrations, because I do not believe in abstract =20 nonsense "generalizations" which have no genuine examples except well =20= known special cases. (ii) In my mail there was another question which you seem to have =20 forgotten namely: What can you prove about the locally small objects =20 of K, especially since you assume nothing on C ? To complete my remark (ii) I would mind a little less the lack of =20 genuine examples of this generalized notion if at least under the =20 mere assumptions of Street on could prove a few non totally trivial =20 results. I would like to point out for example that, with Street's definition, =20= one cannot even prove that a small object of K is locally small. I'm sure that Ross, who gave this definition, will very soon give a =20 correct and complete answer to the three questions I asked him in my =20 previous mail. Best regards, Jean Le 7 d=E9c. 10 =E0 23:22, Richard Garner a =E9crit : > Dear Jean, > >> Could you please tell me: >> (i) Given a category S how does one chose the finitely complete 2- >> category K and the class C of small objects so that the locally small >> objects of K in your sense, are the locally small fibrations over S ? > > Take K =3D Fib(S) and take C to be the representable fibrations. For = me > this is actually the easiest way to remember the definition of locally > small fibration. > >> (iii) What significant mathematical examples can you give of your >> notion ? > > See (i). > > Best regards, > > Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]