From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6396 Path: news.gmane.org!not-for-mail From: "F. William Lawvere" Newsgroups: gmane.science.mathematics.categories Subject: RE: Terminology of locally small categories without replacement Date: Thu, 2 Dec 2010 10:18:15 -0500 Message-ID: References: Reply-To: "F. William Lawvere" 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 1291344067 15574 80.91.229.12 (3 Dec 2010 02:41:07 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 3 Dec 2010 02:41:07 +0000 (UTC) To: , categories Original-X-From: majordomo@mlist.mta.ca Fri Dec 03 03:41:03 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 1POLa3-00088M-1u for gsmc-categories@m.gmane.org; Fri, 03 Dec 2010 03:41:03 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:45643) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1POLZr-0002UD-DO; Thu, 02 Dec 2010 22:40:51 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1POLZo-0002it-3s for categories-list@mlist.mta.ca; Thu, 02 Dec 2010 22:40:48 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6396 Archived-At: Dear Colin=2CI think your stronger definition is the correct one=2C by anal= ogy withother categories (where properness and other manifestations ofinten= sive and extensive objective quantities come up). If your definition of 'c= ategory' itself is equivalent to 'internal category C3=3D>C2->C1 x C1 in t= he category of classes '=2C then your notion seems to be a case of the cond= ition (on E ->B) that the pullback of any small S ->B is small.The AXIOM of= replacement would perhaps be the extra condition on the universe that the = pullbacks of S =3D 1 suffice to test the above=2C a condition that is perha= ps appropriate for abstract constant discrete sets but not for cohesive var= iable ones.It would not be the 'scheme' of replacement that is relevant her= e since the category of objective classes (not their sometimes representing= subjective formulas) is directly under consideration.I presume that you ar= e here trying to extend the Bernays-Mac Lane framework.It is not clear what= would result if we alternatively considered that a category C itself is ju= st a formula=2C i.e objectively=2C a subset naturally defined in every mode= l. Bill > Date: Wed=2C 1 Dec 2010 17:00:51 -0500 > Subject: categories: Terminology of locally small categories without repl= acement > From: colin.mclarty@case.edu > To: categories@mta.ca >=20 > Locally small categories are always defined as categories such that: >=20 > LS) for any objects A=2CB there is a set of all arrows A-->B. >=20 > When the base set theory includes the axiom scheme of replacement that > is equivalent to a prima facie stronger property: >=20 > ??) for any set of objects there is a set of all arrows between them. >=20 > These two are not equivalent in the absence of the axiom scheme of > replacement. There the second is much stronger=2C but it remains > important. Is there a good term for it? >=20 > thanks=2C Colin >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]