From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6399 Path: news.gmane.org!not-for-mail From: JeanBenabou Newsgroups: gmane.science.mathematics.categories Subject: Locally small categories without replacement, or anything Date: Fri, 3 Dec 2010 09:07:55 +0100 Message-ID: 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 1291427188 4852 80.91.229.12 (4 Dec 2010 01:46:28 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 4 Dec 2010 01:46:28 +0000 (UTC) To: Colin McLarty , Categories Original-X-From: majordomo@mlist.mta.ca Sat Dec 04 02:46:21 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 1POhCe-0000NP-TN for gsmc-categories@m.gmane.org; Sat, 04 Dec 2010 02:46:21 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:51825) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1POhCU-0005tX-A9; Fri, 03 Dec 2010 21:46:10 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1POhCQ-0002Pi-EJ for categories-list@mlist.mta.ca; Fri, 03 Dec 2010 21:46:06 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6399 Archived-At: Dear Colin, I gave 36 years ago a definition of locally small fibration over an =20 arbitrary base category S. Now look at the following very special =20 case: S =3D Set, C is a category, and P: Fam(C) --> S the canonical =20 fibration where the fiber over the set I is the category C^I. To say =20 that this P is locally small in my sense coincides exactly with the =20 "more general" notion of C being locally small that you suggest. Note that my definition of locally small fibration does not suppose =20 that S has a terminal object 1, let alone that 1 is a strong =20 generator in S. To show that this definition is equivalent to the =20 "usual" one, you need not only a replacement scheme in Set, but also =20 the fact that 1 is a strong generator in Set. Thus I think that the correct general definition of "local smallness" =20= is the one I gave for fibrations. As a side important remark, the identity fibration Id(S): S --> S is =20 always locally small without any assumption on S, in particular S =20 need not have a terminal object, pull-backs or any kind of limit. =20 None of this is true with any of the "variants" of my definition you =20 can find e.g. in the Elephant, where you have to assume that S has =20 finite limits. Thus "evil" fibrations can be interesting after all. Best to all, Le 1 d=E9c. 10 =E0 23:00, Colin McLarty a =E9crit : > Locally small categories are always defined as categories such that: > > LS) for any objects A,B there is a set of all arrows A-->B. > > When the base set theory includes the axiom scheme of replacement that > is equivalent to a prima facie stronger property: > > ??) for any set of objects there is a set of all arrows between them. > > These two are not equivalent in the absence of the axiom scheme of > replacement. There the second is much stronger, but it remains > important. Is there a good term for it? > > thanks, Colin > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]