From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6737 Path: news.gmane.org!not-for-mail From: Steven Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: size_question_reloaded Date: Tue, 05 Jul 2011 10:37:00 +0100 Message-ID: References: Reply-To: Steven Vickers NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1309969434 7729 80.91.229.12 (6 Jul 2011 16:23:54 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 6 Jul 2011 16:23:54 +0000 (UTC) Cc: Categories list To: "Eduardo J. Dubuc" Original-X-From: majordomo@mlist.mta.ca Wed Jul 06 18:23:50 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.30]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1QeUt6-0006nG-MA for gsmc-categories@m.gmane.org; Wed, 06 Jul 2011 18:23:45 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:40573) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QeUr7-0004yQ-FV; Wed, 06 Jul 2011 13:21:41 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QeUr6-0005R3-Px for categories-list@mlist.mta.ca; Wed, 06 Jul 2011 13:21:40 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6737 Archived-At: Dear Eduardo, That's right. Specifically, Set_f here has N for its object of objects, and something more complicated (but geometrically definable) for its obje= ct of morphisms. That way the correct object classifier is defined for any base topos. (I'm thinking of Grothendieck toposes here, but I guess it works for any elementary topos with NNO. I even conjecture it does something useful for arithmetic universes.) That's a very strong notion of finiteness constructively. It requires not only Kuratowski finiteness and decidable equality, but even a decidable total order. Then the category of such finite sets is essentially small, equivalent to the Set_f I described above. It is the notion of "finite" needed in finitely presentable algebras, for example in the theorem that for a finitary algebraic theory T, the T-algebra classifier is Set^(T-Alg_fp^op), the topos of Set-values functors from the category of finitely presented algebras. Again, we want T-Alg_fp to be small. In my paper "Strongly algebraic =3D SFP (topically)" I was interested in = the situation where, for a geometric theory T, the classifying topos for T is= a presheaf topos in the form of the topos of Set-valued functors from the category of finite T-models (and I gave some sufficient conditions for th= is to happen). Again, the notion of finite model is this strong notion of finiteness. However, my main example also involved Kuratowski finite sets= , so the paper discusses in some detail the interplay between the different notions of finiteness. Regards, Steve Vickers. On Sun, 03 Jul 2011 19:59:16 -0300, "Eduardo J. Dubuc" wrote: > ... > For example, people which consider the presheaf category=20 > Set^((Set_f)^op) (object classifier) often do as if Set_f were=20 > canonical and small. >=20 > Now, if you work with a Grothendieck base topos =E2=80=9Cas if it were= the=20 > category of sets=E2=80=9D, you are forced to specify which small catego= ry of=20 > finite sets you are using, or not ?. >=20 > Cheers e.d. >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]