From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9490 Path: news.gmane.org!.POSTED!not-for-mail From: Thomas Streicher Newsgroups: gmane.science.mathematics.categories Subject: Re: V-included categories Date: Thu, 4 Jan 2018 22:20:47 +0100 Message-ID: References: Reply-To: Thomas Streicher NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: blaine.gmane.org 1515168945 17128 195.159.176.226 (5 Jan 2018 16:15:45 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Fri, 5 Jan 2018 16:15:45 +0000 (UTC) User-Agent: Mutt/1.5.23 (2014-03-12) Cc: Paul Blain Levy , "Categories list\" " To: "Eduardo J. Dubuc" Original-X-From: majordomo@mlist.mta.ca Fri Jan 05 17:15:41 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 1eXUeb-0003ZA-80 for gsmc-categories@m.gmane.org; Fri, 05 Jan 2018 17:15:33 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:46658) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1eXUhP-00040t-1j; Fri, 05 Jan 2018 12:18:27 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1eXUg3-0006gH-2B for categories-list@mlist.mta.ca; Fri, 05 Jan 2018 12:17:03 -0400 Content-Disposition: inline In-Reply-To: <5c1ec079-335b-5609-9cb7-ae4e519f6716@dm.uba.ar> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9490 Archived-At: Dear Eduardo, thanks for your attempts to clarify the situation! > Aparently the universes are not closed in the sense that: > > 1) X ~ Y, Y belongs U ===> X belongs U > > which is currently accepted (as we see in Thomas posting above) in the naive > practice of category theory with universes. I certainly did not incline this (despite my partial involvement into HoTT :-)) I rather tacitly assumed that a locally U-small means that all homsets are elements of U. This I find natural though I read that Grothendieck and Verdier formulated it much more liberally. However, using AC every locally U-small category is isomorphic to some category where all hom-sets are elements of U. Moreover, every category equivalent to such a category is locally U-small in the liberal sense of Grothendieck and Verdier. Now, for a category C which is locally U-small in the restricted sense [C^op,U] is locally U-small in the restricted sense as shown by Paul's argument. But if C' is locally U-small in the liberal sense then [C'^op,U] \cong [C^op,U] and the latter is locally U-small in the restricted sense and thus [C'^op,U] is locally U-small in the liberal sense. Thus, I think in SGA4 they made a mistake. No question that both guys were great mathematicians but that doesn't prevent them from making little mistakes in logic. Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]