Talk:Epimorphism

message: I've never encountered the use of "monomorphism" and "epimorphism" as synonymous with "1-1" and "onto" respectively, as apparently Bourbaki intended. I have always been thought to consider what category one was working in when encountering these terms. In other words, it makes no sense to say "f is an epimorphism from N to Z". If you mean in Set, then just say surjective. If you mean in monoid category, say so. I wasn't aware there was a confusion over the terminology. In other words, it makes perfect sense to me to say, "This epimorphism isn't surjective". Revolver 22:11, 25 January 2006 (UTC)
replies: text: I see, it's universal algebra that has different definitions. Revolver 22:15, 25 January 2006 (UTC)
thread_title:  Never encountered this use 

message: I'm not sure of the tone of recent changes. "Beginners often make the mistake..." seems needlessly judgemental, and there are plenty of textbooks that use "epimorphism", rightly or wrongly, as a synonym for surjective (such as Bourbaki, to go no further). If the person who added the stuff on the further concepts of epimorphisms could give their precise definition, I do not see why new pages should be added, except perhaps as redirects to this page. Magidin 05:15, 6 February 2006 (UTC)
replies: text: Well, I've added back the algebraic definitions. It's an unfortunately fact of life that the two concepts don't agree, but we can't ignore the fact the both meanings are in use. -- Fropuff 05:22, 6 February 2006 (UTC)
replies: text:  I fully agree. I do see that the comment on "beginners being often mistaken" is still there. While the comments following do seem appropriate, I still think the first sentence is needlessly judgemental. Magidin 05:24, 6 February 2006 (UTC)
thread_title:  Recent changes 

message: I have never seen "epimorphism" used for a surjection except in Bourbaki.  In my professional experience, epimorphism refers exclusively to the categorical notion.  I am familiar with the book of Bergman cited in the references and it does not include "algebraic" epimorphisms.
Please provide a citation for this if you insist on keeping it.  As it is, I believe the article to be incorrect and misleading.  141.211.62.20 22:23, 6 February 2006 (UTC)
replies: text: I cited Bergman's book with reference to the history and to Mac Lane's attempt at clearing up the ambiguities. As for references, I'm pretty sure Lang's Algebra uses epimorphism as a synonym for surjective, and will be happy to withdraw the claim or confirm it tomorrow morning when I can check in my office, as well as provide (or disprove) other references. Magidin 01:03, 7 February 2006 (UTC)

text: A quick perusal through my bookshelf revealed the following books in which epimorphism is defined simply as a surjective map: Lattice Theory by Birkhoff, Revised Edition; Homological Algebra by Cartan and Eilenberg; Rings and Categories of Modules by Anderson and Fuller (they give the categorical definition in an exercise); Algebra by Lang; A Basic Course in Algebraic Topology by Massey; and Basic Algebra I by Jacobson. All I can say is that my professional experience does not agree with yours: both notions appear to be widespread, and there are few books that even remark on the conflicting use. Magidin 15:37, 7 February 2006 (UTC)
replies: text: I will add Advanced Linear Algebra (2nd edition) by Steven Roman; the book includes rings, modules, and algebras, and defines epimorphism as a surjective morphism. Magidin 15:49, 7 February 2006 (UTC)

text: Well, in that case, I suppose it's worth keeping.  It seems odd to me, though--maybe it's because I've only ever heard it used in the categorical sense.  141.211.62.20 20:03, 7 February 2006 (UTC)

text: I have a feeling that in undergraduate algebra courses it can be quite common to use the terms monomorphism and epimorphism just to mean injective and surjective morphisms respectively.---  —Preceding unsigned comment added by 58.152.130.209 (talk) 03:19, 17 June 2009 (UTC)
thread_title:  "Algebraic" epimorphisms 

message: Paul writes
:I'm not sure why Axel removed the bit about every "surjecton is an epi"?
It's mentioned in the paragraph above, although not quite as explicitly. It probably can't hurt to mention it twice though. AxelBoldt 05:50, 9 March 2006 (UTC)
replies: text: Well not quite ;-)  The previous paragraph says that the surjective homomorphisms studied in univeral algebra are epis. But in fact all surjections are epis. Paul August ☎ 15:04, 9 March 2006 (UTC)
replies: text: Good point. So it's all good right now. AxelBoldt 20:38, 9 March 2006 (UTC)
thread_title:  Surjective implies epi 

message: Magidin writes:
:bimorphisms are homeos in Top; changed to Haus.
Not so. The map [0,1) → S1 which sends x to exp(2πix) is a bimorphism but not a homeomorphism in Top. The present example works as well. -- Fropuff 18:37, 10 March 2006 (UTC)
replies: text: Ah, I see my mistake. Of course, "bijective continuous map" is not the same thing as homeomorphism. Magidin 14:35, 13 March 2006 (UTC)
replies: text: Correct, the inverse has to be continuous as well. Paul August ☎ 16:15, 13 March 2006 (UTC)
thread_title:  Bimorphisms in Top 

message: :any map from a commutative ring R to one of its localizations is an epimorphism.
Surely this is only true for the natural map from R to a localization? Otherwise every endomorphism of a commutative ring would be an epi in Ring, but Z[X] → Z[X] which maps X to 0 isn't. AxelBoldt 17:38, 13 March 2006 (UTC)
thread_title:  Localizations 

message: I found your article to be useless and uninformative. It was too confusing to be useful. The catagory definition was too obscure. Rigor does not produce understanding for the user.71.251.189.131 11:55, 25 October 2007 (UTC)
thread_title: Dazed And Confused

message: There was a claim of error in the proof that epimorphisms in compact Hausdorff spaces are surjective. The new proof invokes normality (T4), but is actually using perfect normality (T6). In any case, it is poorly written. Could someone check it over and fix the write-up? Magidin (talk) 01:43, 5 January 2009 (UTC)
replies: text: There was a mistake. For example, the unit interval [0,1] is compact Hausdorff, and the subset [0,1/2] is closed, but the characteristic function of [0,1/2] is not continuous, since [0,1/2] is not clopen. The new proof uses Urysohn's Lemma, which holds for all normal spaces, not perfect normality.87.95.55.17 (talk) 09:45, 5 January 2009 (UTC)
replies: text: Fine, then it should state that it is using Urysohn; the formatting in the new version is poor in any case. I will rewrite. Magidin (talk) 15:50, 5 January 2009 (UTC)
thread_title:  Compact Hausdorff spaces 

message: The pedagogical value of Wikipedia is diminisioning as it seems destined to evolve to being merely the personal log of professionals in their rarefied fields well ensconced in their peculiar jargon and offering no clear on ramp for outsiders beyond, perhaps, professional training or decades of independent study. As part of this trend, there is a lack of subject origin or justifying context within the narrative of the articles themselves. A reader new to the subject is therefore left with jargon and no perspective on how it is being used or anyway to contextualize the often arbitrary names being introduced; links are a problem when a popup synopsis would do. Presumable better policies could be adopted to encourage contributors to include such things or encourage amendments to articles to introduce such provisions. Ideally, Wikipedia could be a classroom to the world or an even more valuable supplement.  — Preceding unsigned comment added by 99.226.237.229 (talk) 08:08, 19 March 2018 (UTC)
thread_title:  Link forest developing amidst sparsity of subject motivation in articles