From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9488 Path: news.gmane.org!.POSTED!not-for-mail From: Paul Blain Levy Newsgroups: gmane.science.mathematics.categories Subject: Re: V-included categories Date: Wed, 3 Jan 2018 06:44:02 +0000 Message-ID: References: Reply-To: Paul Blain Levy NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: 8bit X-Trace: blaine.gmane.org 1515026897 32240 195.159.176.226 (4 Jan 2018 00:48:17 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Thu, 4 Jan 2018 00:48:17 +0000 (UTC) To: edubuc@dm.uba.ar, "Categories list\" " Original-X-From: majordomo@mlist.mta.ca Thu Jan 04 01:48:12 2018 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1eWthX-0007eN-FP for gsmc-categories@m.gmane.org; Thu, 04 Jan 2018 01:48:07 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:46210) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1eWtjk-0003m6-9S; Wed, 03 Jan 2018 20:50:24 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1eWtiM-0006Zg-QK for categories-list@mlist.mta.ca; Wed, 03 Jan 2018 20:48:58 -0400 In-Reply-To: <03876a66-f7ee-a161-091c-32944a0d8556@dm.uba.ar> Content-Language: en-US Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9488 Archived-At: On 03/01/18 00:20, edubuc@dm.uba.ar wrote: > On 1/1/18 19:46, Paul Blain Levy wrote: >> Dear Eduardo, >> >> On 01/01/18 21:14, edubuc@dm.uba.ar wrote: >>> Have you considered the following at the very begining of SGA4: >>> >>> Remarque 1.1.2. ??? Soit D une cat??gorie poss??dant les propri??t??s >>> suivantes : >>> (C1) L???ensemble ob(D) est contenu dans l???univers U . >>> (C2) Pour tout couple (x, y) d???objets de D, l???ensemble HomD(x, y) est >>> un ??l??ment de U . >>> >>> (Les cat??gories usuelles construites ?? partir d???un univers U poss??dent >>> ces deux propri??t??s: U -Ens, U -Ab,. . .). >>> >>> Soit C une cat??gorie appartenant ?? U . Alors la cat??gorie >>> Fonct(C, D) ne poss??de pas en g??n??ral les propri??t??s (C1) et (C2). Par >>> exemple la cat??gorie Fonct(C,U-Ens) ne poss??de aucune des propri??t??s >>> (C1) et (C2). >> Thanks for your reply but I am mystified by this statement in SGA4.?? It >> appears to me Fonct(C,U-Ens) does satisfy both (C1) and (C2), so I must >> be missing something. > > It is clear that U-Ens^C satisfy (C1) and (C2) (see the practice of > category theory by any mathematician). OK, I mistakenly assumed you were endorsing the statement you quoted.?? Sorry for boring you with this obvious proof. > Now, it is necessary to see what exactly means "Fonct(C,U-Ens)" > and/or?? "appartenant ?? U" in SGA4. :-)???? Alternatively: the authors just made a mistake.?? And evidently, had they not made this mistake, they would have defined "U-category" by (C1)--(C2), since they regard these conditions as a priori natural.?? That's good to see. Paul [For admin and other information see: http://www.mta.ca/~cat-dist/ ]