From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9486 Path: news.gmane.org!.POSTED!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: V-included categories Date: Tue, 2 Jan 2018 11:15:34 -0800 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" X-Trace: blaine.gmane.org 1514992468 31014 195.159.176.226 (3 Jan 2018 15:14:28 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Wed, 3 Jan 2018 15:14:28 +0000 (UTC) Cc: "Categories list " To: Paul Blain Levy Original-X-From: majordomo@mlist.mta.ca Wed Jan 03 16:14:24 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 1eWkkH-0007bd-Nj for gsmc-categories@m.gmane.org; Wed, 03 Jan 2018 16:14:21 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:46031) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1eWkml-0006oB-Ua; Wed, 03 Jan 2018 11:16:55 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1eWklO-0004n3-7T for categories-list@mlist.mta.ca; Wed, 03 Jan 2018 11:15:30 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9486 Archived-At: I believe that in his paper "Notions of topos" Ross Street used the name "V-moderate category" for this or a closely related notion. There the point was that V-moderate categories have another advantage over locally V-small ones, namely that their objects can (assuming the axiom of choice) be well-ordered with all initial segments being V-small. On Mon, Jan 1, 2018 at 5:10 AM, Paul Blain Levy wrote: > > Hi, > > Let V be a Grothendieck universe.?? A "V-set" is an element of V, and a > "V-class" is a subset of V. > > Say that a category C is "V-included" when it has the following two > properties. > > (1) ob C is a V-class. > > (2) C(x,y) is a V-set for all x,y in ob C. > > The advantage of V-inclusion over local V-smallness (i.e. condition (2) > alone) is that V-included categories are W-small for every universe W > greater than V, whereas locally V-small categories are not, in general. > > Furthermore, all the standard categories constructed from V are > V-included.?? (Except for the ones that are not even locally V-small, > like the category of V-included categories.) > > Is there a standard name for V-inclusion? > > Paul > > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]