From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9493 Path: news.gmane.org!.POSTED!not-for-mail From: Richard Williamson Newsgroups: gmane.science.mathematics.categories Subject: Re: V-included categories Date: Fri, 5 Jan 2018 00:28:20 +0100 Message-ID: <20180104232820.GA1326@richard.richard> References: Reply-To: Richard Williamson NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: blaine.gmane.org 1515170479 24396 195.159.176.226 (5 Jan 2018 16:41:19 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Fri, 5 Jan 2018 16:41:19 +0000 (UTC) Cc: Paul Blain Levy , categories@mta.ca To: Eduardo Julio Dubuc Original-X-From: majordomo@mlist.mta.ca Fri Jan 05 17:41:15 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 1eXV3S-0005yT-Ge for gsmc-categories@m.gmane.org; Fri, 05 Jan 2018 17:41:14 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:46685) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1eXV6L-00057c-Nz; Fri, 05 Jan 2018 12:44:13 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1eXV4z-0006tO-R8 for categories-list@mlist.mta.ca; Fri, 05 Jan 2018 12:42:49 -0400 Content-Disposition: inline In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9493 Archived-At: Hello, > The authors are Verdier and Grothendieck, I doubt they made > mistakes, and specially in the very basic definitions of the > whole theory. This made me smile, but I agree! > There is something odd here but I am not inclined to take time to > clarify it, I will sit and wait to see if some in the list come out with > an explanation. I think the explanation is rather simple. The first point is that the foundations here are Bourbaki set theory: everything is a set, including a category. When they say that C belongs to U, they literally mean that C, when rigorously formalised as a set, is an element of U. The same goes for a functor. The second point is that U-smallness of a set X is defined to be a set which is isomorphic to an element of U, not necessarily an actual element of U. The word 'isomorphic' is underlined. Let us consider (C2) first. The axioms of a universe only allow one to construct sets in the universe from other sets in the universe. Since U is not in U, there is no way that any definition/construction involving it can produce a set in U. Certainly one needs to involve U to be able to define the Hom sets of Func(C, U-Ens). But the Hom sets of Func(C, U-Ens) are isomorphic to elements of U, i.e. are U-small. This is because U-Ens is a U-category, i.e. the Hom sets are U-small. One can make the same point about the set of objects of Func(C, U-Ens). One needs U to define it, so it is certainly not an element of U. It is not even isomorphic to an element of U, because this would imply that the cardinality of U is strictly less than U. In (C1), it is not exactly this that is asked, but rather that the set of objects of Func(C, U-Ens) is a subset of U. This is impossible for the same kind of reasons: one will need to consider for instance a subset of the set C x U to define it, and there is no way that show that this is a subset of U (it is perfectly possible if instead of U we have some element u of U, because then the product of C and u belongs to U). Hence Func(C, U-Ens) is, as SGA claims, the prototypical example to illustrate why the definition of a U-category is exactly the way it is. Best wishes, Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]