From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9829 Path: news.gmane.org!.POSTED.blaine.gmane.org!not-for-mail From: Martin Escardo Newsgroups: gmane.science.mathematics.categories Subject: Re: Terminology regarding injectivity of objects Date: Sat, 9 Feb 2019 23:43:56 +0000 Message-ID: References: Reply-To: Martin Escardo Mime-Version: 1.0 Content-Type: text/plain; charset="utf-8"; format=flowed Content-Transfer-Encoding: 8bit Injection-Info: blaine.gmane.org; posting-host="blaine.gmane.org:195.159.176.226"; logging-data="256480"; mail-complaints-to="usenet@blaine.gmane.org" Cc: categories To: Original-X-From: majordomo@mlist.mta.ca Sun Feb 10 14:49:35 2019 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([198.164.44.56]) by blaine.gmane.org with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.89) (envelope-from ) id 1gspUB-0014W9-3m for gsmc-categories@m.gmane.org; Sun, 10 Feb 2019 14:49:31 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:40769) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1gspTx-0007S0-5r; Sun, 10 Feb 2019 09:49:17 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1gspT4-0003E5-My for categories-list@mlist.mta.ca; Sun, 10 Feb 2019 09:48:22 -0400 In-Reply-To: Content-Language: en-US Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9829 Archived-At: Thank you, Peter. I am in a similar situation, where weak and strong injectivity agree for all objects, and some distinguished objects are injective in the four senses. Martin On 09/02/2019 21:43, ptj@maths.cam.ac.uk wrote: > Dear Martin, > > I encountered this situation when I considered injectivity in Top: > see my paper in SLNM 871, and also pages 738-9 in?? the Elephant. > I used the terms `weakly injective' and `strongly injective' (not > very imaginative, but they did the job), and also `completely > injective' for the case where the `extension along j' operation can be > taken to be right adjoint to restriction along j (you could of > course use `cocompletely injective' for the case where it's left adjoint). > Fortunately, in Top the notions of weak injective, strong injective > and complete injective coincide. > > Peter Johnstone > > On Feb 9 2019, Mart??n H??tzel Escard?? wrote: > >> >> (1) An object D is called injective over an arrow j:X->Y if the >> "restriction map" >> >> ???????? hom(Y,D) -> hom(X,D) >> ???????????????? g???? |-> g o j >> >> is a surjection. This is fairly standard terminology (where does it come >> from, by the way). >> >> (2) I am working with the situation where the restriction map is a >> *split* surjection. >> >> I though of the terminology "D is split injective over j", but perhaps >> this is awkward. Is there a standard terminology for this notion. Or, >> failing that, a terminology that at least one person has already used in >> the literature or in the folklore. Or, failing that too, a good >> suggestion by any of you? >> >> Thanks, >> Martin >> >> >> [For admin and other information see: http://www.mta.ca/~cat-dist/ ] >> -- Martin Escardo http://www.cs.bham.ac.uk/~mhe [For admin and other information see: http://www.mta.ca/~cat-dist/ ]