From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9839 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: Fri, 22 Feb 2019 23:02:21 +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="127889"; mail-complaints-to="usenet@blaine.gmane.org" To: , categories Original-X-From: majordomo@mlist.mta.ca Sat Feb 23 17:36:40 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 1gxaI4-000X92-BP for gsmc-categories@m.gmane.org; Sat, 23 Feb 2019 17:36:40 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:45759) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1gxaHZ-0000AZ-GZ; Sat, 23 Feb 2019 12:36:09 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1gxaGF-0001MG-5U for categories-list@mlist.mta.ca; Sat, 23 Feb 2019 12:34:47 -0400 In-Reply-To: Content-Language: en-US Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9839 Archived-At: On 09/02/2019 21:43, ptj@maths.cam.ac.uk wrote: > 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. The paper John Bourke, 2017, Equipping weak equivalences with algebraic structure. https://arxiv.org/abs/1712.02523. has a terminology for this that is appealing: an algebraic injective object, with respect to a class J of arrows, is an object D equipped with extensions c(j,f) : Y -> D for each j:X->Y in J and f : X -> D. (Then you can consider the obvious morphisms of algebraic injective objects that commute with the designated extensions c(j,f).) Thanks to Mike Shulman for this reference. Martin > > 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/ ]