From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9489 Path: news.gmane.org!.POSTED!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: V-included categories Date: Thu, 4 Jan 2018 20:47:18 +0000 Message-ID: References: <918B0A9E-DFD0-4033-AB7A-1A8A364DB8A9@cs.bham.ac.uk> <20180104110046.GA24344@mathematik.tu-darmstadt.de> Reply-To: Steve Vickers NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 (1.0) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1515168903 6631 195.159.176.226 (5 Jan 2018 16:15:03 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Fri, 5 Jan 2018 16:15:03 +0000 (UTC) Cc: Paul Blain Levy , categories list To: streicher@mathematik.tu-darmstadt.de Original-X-From: majordomo@mlist.mta.ca Fri Jan 05 17:14:59 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 1eXUdx-00011b-G8 for gsmc-categories@m.gmane.org; Fri, 05 Jan 2018 17:14:53 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:46651) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1eXUgd-0003th-BW; Fri, 05 Jan 2018 12:17:39 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1eXUfH-0006fV-3a for categories-list@mlist.mta.ca; Fri, 05 Jan 2018 12:16:15 -0400 In-Reply-To: <20180104110046.GA24344@mathematik.tu-darmstadt.de> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9489 Archived-At: Dear Thomas, What I'm curious to understand is how inevitable is a certain practical beha= viour of universes. In many situations we know they are foundationally neces= sary, but we still seek ways to disregard them in everyday mathematics. If we try to use generalized spaces as our prime notion of collection, inste= ad of sets, it begins to look different. We still have universes of a kind. = For example the object classifier is a universe whose elements are sets (dis= crete spaces). But one would not make the mistake of thinking it is a set it= self, as it has a non-discrete topology - it even has non-identity specializ= ation morphisms in functions between sets. There are various other universes= for various other kinds of spaces, such as the Boolean algebra classifier f= or Stone spaces. The different universes represent qualitative distinctions,= not just one of "size". As you know, I'm trying to do something along those lines but based on arith= metic universes instead of Grothendieck toposes.=20 All the best, Steve. > On 4 Jan 2018, at 11:00, streicher@mathematik.tu-darmstadt.de wrote: >=20 > Dear Steve at al. >=20 > I think in one or the other form universes are inevitable be they type > or set theoretical. Of course, you can do category over fairly general > bases like finite limit categories (as Benabou developed some time ago). > But then you are bound to work externally since one can speak about a > fibration in the internal language of its base in an only very > restricted way. >=20 > Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]