From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10071 Path: news.gmane.org!.POSTED.blaine.gmane.org!not-for-mail From: Ross Street Newsgroups: gmane.science.mathematics.categories Subject: Re: well powered categories, categorically? Date: Fri, 6 Dec 2019 06:21:21 +0000 Message-ID: References: Reply-To: Ross Street Mime-Version: 1.0 Content-Type: text/plain; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable Injection-Info: blaine.gmane.org; posting-host="blaine.gmane.org:195.159.176.226"; logging-data="252313"; mail-complaints-to="usenet@blaine.gmane.org" Cc: "categories@mta.ca list" To: Paul Taylor Original-X-From: majordomo@rr.mta.ca Fri Dec 06 15:49:58 2019 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.55]) by blaine.gmane.org with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.89) (envelope-from ) id 1idEvd-0013IR-Jx for gsmc-categories@m.gmane.org; Fri, 06 Dec 2019 15:49:57 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:55152) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1idEvD-0003m0-8T; Fri, 06 Dec 2019 10:49:31 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1idEuS-0001R9-CQ for categories-list@rr.mta.ca; Fri, 06 Dec 2019 10:48:44 -0400 In-Reply-To: Accept-Language: en-GB, en-US Content-Language: en-US Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:10071 Archived-At: On 5 Dec 2019, at 7:38 AM, Paul Taylor > wrote: - I would like the homs of C to be objects of E, so it is an E-enriched category. Dear Paul If, rather than C E-enriched, you give meaning to E-indexed families of C o= bjects by taking C to be a category (parametrized) over E, then what I think you w= ant is in the works of B\'enabou, Par\'e-Schumacher [SLNM 661], and Section 2 of 32. (with D. Schumacher) Some parametrized categorical concepts, Communications in Algebra 16(1988) 2313--2347 where you will find precise references to other works. For example, let C just be the topos E. We know how to internalise ("classify") subobjects - using the subobject classifier Omega. Taking C to be the the arrow category of E, over E via the codomain functor= , wellpowered is about the existence of a subobject classifier in E. All the best, Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]