Timeline for What are some natural examples of "bimorphism" classes?
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 25, 2023 at 11:42 | comment | added | Tian Vlašić | @AndrewStacey Maybe my answer to this post would be of interest to you. | |
| Sep 8, 2010 at 4:29 | comment | added | Mike Shulman | @Andrew: it's true that we do encounter such situations, but the question was specifically about bimorphisms, not injective+surjective morphisms. | |
| Sep 3, 2010 at 12:37 | comment | added | Andrew Stacey | @Jose: "Why" is probably a bit broad for such a question. There are probably lots of possible reasons why, but not all will apply in all situations. It might help to think of it the other way around: some proto-functors only work for monomorphisms - think about constructions on inner product spaces, for example. If you have a specific example in mind and don't want to ask it as an MO question (too discussiony?), then try it over at the nForum. Also, you're reaching the limits of my categorical expertise (as Mike as pointed out) so the nForum may suit you better. | |
| Sep 3, 2010 at 12:28 | comment | added | José Figueroa-O'Farrill | @Andrew: yes, sorry -- wish I could edit comments!!! What I meant is that there are constructions which do not give rise to a functor F, say, because F(fg) = F(f)F(g) does not hold for general morphisms, but only, say, when f,g are epis. | |
| Sep 3, 2010 at 8:11 | comment | added | Andrew Stacey | I agree with both of Mike's comments. I'd forgotten exactly what "extremal epi" meant (should've looked it up on the nlab!) so went for "regular" as I was happier with that and my comment to Yemon should be changed, but as it's just a comment I'll not bother. Also, I agree that we don't tell our students about 'mono' or 'epi', though I think that this misses the point of what I was trying to say. That point was that we do encounter situations where we show a formally weaker condition than "there is an inverse" but which implies it in the particular case. | |
| Sep 3, 2010 at 6:09 | comment | added | Mike Shulman | I don't think I agree with your example of vector spaces. Injective + surjective = iso is almost always true in algebraic situations; the hard question is the relationship between surjective and epimorphic. And we surely don't tell our students learning linear algebra what an epimorphism is, nor do I think we very frequently use mono+epi=iso for vector spaces (as opposed to inj+surj=iso). | |
| Sep 3, 2010 at 6:06 | comment | added | Mike Shulman | Actually, extremal epi + mono = isomorphism is already a standard categorical theorem; it falls right out of the definition of extremal epi. | |
| Sep 1, 2010 at 7:44 | comment | added | Andrew Stacey | @Jose: Not sure I understand the question. "Functorial under arbitrary morphisms" doesn't parse for me. Do you mean that you have something that looks a bit like a functor, but is only a functor when you only consider epis? Can you give an example? (If you don't want to ask an MO question on this, consider taking it to the nForum). @Yemon: Not off track at all! Yes, "extremal epimorphism" is what you're looking for. What the theorem says categorically is "all extremal epimorphisms are regular", but "regular epi + mono = isomorphism" is a standard categorical theorem (hope I'm right!). | |
| Sep 1, 2010 at 0:11 | comment | added | Yemon Choi | To get slightly off track, functional analysts are particularly fond of the fact that a morphism which is monic and "a bit stronger than epic" is often one half of an isomorphism -- here I mean something like extremally epic, if memory serves correct. Anyway, of course what I have in mind is the Banach isomorphism theorem. which in plebeian non-categorical language says that a continuous linear bijection from one Banach space onto another has continuous (linear) inverse. | |
| Aug 31, 2010 at 22:26 | comment | added | José Figueroa-O'Farrill | Speaking of epimorphisms, and rather than ask yet another potentially embarrassing question... I've noticed more than one constructions which is not functorial under arbitrary morphisms but is under epis. Any reason why? | |
| Aug 31, 2010 at 12:12 | comment | added | Andrew Stacey | Some of my best friends are epimorphisms. | |
| Aug 31, 2010 at 11:58 | comment | added | Martin Brandenburg | I don't think that "bimorphism" is really easier to check than "isomorphism" in practice. Epimorphisms don't behave well. | |
| Aug 31, 2010 at 7:28 | history | answered | Andrew Stacey | CC BY-SA 2.5 |