From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6398 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: RE: Terminology of locally small categories without replacement Date: Thu, 2 Dec 2010 20:15:55 -0800 Message-ID: References: Reply-To: Michael Shulman 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 1291427059 4389 80.91.229.12 (4 Dec 2010 01:44:19 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 4 Dec 2010 01:44:19 +0000 (UTC) Cc: colin.mclarty@case.edu, categories To: "F. William Lawvere" Original-X-From: majordomo@mlist.mta.ca Sat Dec 04 02:44:14 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 1POhAc-0008C6-33 for gsmc-categories@m.gmane.org; Sat, 04 Dec 2010 02:44:14 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:50162) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1POhAR-0005iz-8m; Fri, 03 Dec 2010 21:44:03 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1POhAN-0002MX-HS for categories-list@mlist.mta.ca; Fri, 03 Dec 2010 21:43:59 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6398 Archived-At: A slightly different way of formulating this answer is in terms of indexed categories / fibrations. The standard definition of "locally small indexed category" is, for a "naively Set-indexed category," precisely the stronger definition referring to all families (or sets) of objects. On Thu, Dec 2, 2010 at 7:18 AM, F. William Lawvere w= rote: > > Dear Colin,I think your stronger definition is the correct one, by analog= y withother categories (where properness and other manifestations ofintensi= ve and extensive objective quantities come up). =A0If your definition of 'c= ategory' itself is equivalent to 'internal category =A0C3=3D>C2->C1 x C1 in= the category of classes ', 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= =A0replacement would perhaps be the extra condition on the universe that t= he pullbacks of S =3D 1 suffice to test the above, a condition that is perh= aps appropriate for abstract constant discrete sets but not for cohesive va= riable ones.It would not be the 'scheme' of replacement that is relevant he= re since the category of objective classes (not their sometimes representin= g =A0subjective formulas) is directly under consideration.I presume that yo= u are here trying to extend the Bernays-Mac Lane framework.It is not clear = what =A0would result if we alternatively considered that a category C itsel= f is just a formula, i.e objectively, a subset naturally defined in every m= odel. Bill >> Date: Wed, 1 Dec 2010 17:00:51 -0500 >> Subject: categories: Terminology of locally small categories without rep= lacement >> From: colin.mclarty@case.edu >> To: categories@mta.ca >> >> 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. =A0There the second is much stronger, but it remains >> important. =A0Is there a good term for it? >> >> thanks, Colin >> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]