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I can't resist to advertise this result that I find really cool:
The following are equivalent for a topological field K:
(From Caicedo, Mantilla-Soler: On a characterization of path connected topological fields)
That's very pretty! It's nice to see how 2 <-> 4 appear to get the particularity of the real numbers entirely out of the Gelfand duality story.
Does there exist a path-connected topological field that's not an algebra over the real numbers?
The paper says that there exist path-connected topological fields of positive characteristic. If is such a field, it can't be an algebra over the real numbers. Let . If is an algebra over the real numbers, we get a ring homomorphism and so is invertible with inverse . So which implies and thus which is false by definition of a field.
Now the question is what are these path-connected topological fields of positive characteristic!!
We should take a look at the paper they cite to discover this.
Yes, such fields sound pretty weird.
Apparently yes, there exist such fields with positive characteristic, constructed in Waterman/Bergman: Connected fields of arbitrary characteristic
Roughly speaking, you just adjoin indeterminates for every to whatever integral domain you like, then create a norm that makes close to and manage to extend that to a whole topological ring structure and then to the field of fractions.
Could you just work in the category of topological rings, starting with a field of characteristic with the discrete topology, adjoin a family of indeterminates indexed by the space (which should make sense in the category of topological rings), quotient by the ideal , and then take the topological-field of fractions? Is there some reason some of those operations don't make sense in the topological world?
Ooh, interesting, so the universal property of this guy is that topological ring maps into a -algebra correspond with continuous paths ? Interesting! Maps out of give an continuous functor on topological rings satisfying the solution set condition, so that's definitely fine, and topological rings should be topological over rings, creating colimits there, so you've made it to the correct integral domain...It's less obvious to me how to get general category theory to help construct the topological field of fractions, since topological fields are not complete or accessible, so I feel very short on tools here. Is there an adjunction lifting theorem that will apply to lift along the pair of solid functors forgetting the topologies on rings and fields, maybe?
Couldn't you take the ordinary field of fractions and equip it with the finest field topology making the inclusion continuous?
I guess the sticking point in any abstract-nonsense approach like this would be proving that the topology you get isn't indiscrete. Since, of course, any field admits the indiscrete topology, but that's not very interesting.
Yeah, I think your construction exemplifies that you can in fact lift adjunctions across pairs of topological functors.
I wonder if the topology you get this way coincides with the Waterman-Bergman one.
By the way: unless I missed it, we've never yet found a topological ring with a path component that is not simply connected, nor proven that one can't exist. The trick just explained - "sewing in a path of indeterminates" - might be a strategy for create such a topological ring. Can we form the free topological ring on a loop of indeterminates , and will this loop be noncontractible?
We should certainly be able to form it, and I'd be pretty surprised if it turned out to be contractible.
But I don't know how to go about proving that it isn't.
Yes, since "freedom's just another word for nothing left to lose", as Joplin observed, it seems unlikely that the free topological ring on a loop will be contractible! Maybe I'll pose this as a question on MathOverflow, to give the experts something to chew on.
Please if you ask this on MathOverflow, which seems like a good idea, say that I started this discussion on Zulip.
Else, I will be a bit angry.
Sure, I'll say that. But by the way: when asking people for favors, it works best not to say "Else, I will be a bit angry". Saying that actually reduced the chance of me citing you, since I like doing favors but respond negatively to threats. (When people say "otherwise I'll be angry", that gets interpreted as a threat even if it's not, because anger is always an implicit threat - and to keep people from threatening us, we often resist doing what they asked.)
That’s kind of a threat indeed because I’m not asking for a favor but just to cite your sources. But thank you. Now, I’m happy. :)
(I agree that I'm not an expert in diplomacy and communication though so thank you for the advice.)
I guess I should cite you, Mike Shulman (who in this conversation pointed out the idea of constructing a free ring on a space), and maybe Kevin Carlson (who described how to get path-connected fields of characteristic ), Peter Arndt (who pointed out a paper that does that), and Waterman and Bergman (who actually wrote a paper doing that), and Caicedo and Mantilla-Soler (since I'm mentioning one of their results on path connected topological fields).
You can do that. There are papers with dozen or hundred of authors sometimes when a lot of people are involved.
Normally I wouldn't cite all these people, since I'm just posting a question to a website, not writing a paper, and I feel the question is my own. But you're asking me to cite you, I feel that to be fair I have to cite them as well.
Now I want to work on that question.
You can explain what everyone did like that I guess. I think this is perfectly fine, polite and fair.
That’s your choice. As to me, I’ve already said what I’ve said!
John Baez said:
as Joplin
following Kristofferson!
Whoops, I forgot to cite Kristofferson! Do I have to cite him too in my MathOverflow question? :face_with_spiral_eyes:
Brr but some important facts to consider are that: 1) I’m the one who started the discussion 2) the other guys in the discussion are professors, postdocs or work at the Topos institute I think while I’m just a PhD student.
If I was a tenured prof, I don’t think I would care that much.
Another time, you cited a guy on Mathsodon who participated in a conversation on Zulip without citing me who started the conversation and asked the relevant question. I think it was unfair.
(It was about multiplicatively idempotent rigs.)
In fact I predict that even if you get tenure you will be worrying that people don't cite you enough. And as you get older and older, you will become ever more resentful of how you haven't been cited enough.
Of course I can't really predict what you'll do. But this is a common pattern which I observe in many people, including many famous mathematicians but also myself. I try to fight it in myself, with mixed success.
I have a theory for why this happens. When mathematicians are young, they hope that eventually they will become famous, or at least widely respected. But as they get older, even if they are famous, they start realizing that they are going downhill, not up: they'll never become as respected as they hoped, and they'll eventually be a mere footnote in the history of mathematics. So they become resentful - unless they deal with this problem in their own minds, and transcend this pettiness.
Maybe you're right. For sure, I hope that eventually I will become famous and probably I will just be a footnote in the history of mathematics. But for now, I also know that the more people can read my name, the better it will be for me professionally, to be paid at the end of the month, stable etc... So right now, it is not only about ego. At least that's why I feel. But that's very hard for me to understand why I'm saying this or this, why I'm doing this or this, what are the real reasons, so maybe this is mostly about ego and I just try to find a justification. I don't know!
This is all fundamentally tangential to the simple point that it’s inappropriately aggressive in a professional setting to tell someone they’ll make you angry if they don’t do something you want. This remains just as true even if you have a knock-down argument that it’s ethically correct for them to do that thing. That’s the takeaway here.
Ok, I will not say anymore "I will be a bit angry." if that's inappropriate. And I'm also doing this: I kindly ask people to cite my name if they talk about a math problem on the Internet, somewhere else that on this Zulip if I started the discussion where the question emerged. Especially if I don't just started the discussion but also asked at the same time a relevant question which is very close to what they are talking about on the other website.
Here is my Mathoverflow question:
Note that I actually conjecture a construction of a topological ring that has nontrivial nth homotopy group at a certain basepoint (not or ).
Thank you, that's perfect!