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It’s been getting a little lonely here, recently. Somehow I suspect it’s not unrelated to a flood of physics-related posts that I am making, to which maybe none of the regulars here can – or probably wants to – connect? – except notably David Corfield, that is.
There is hope, though: Before long I’ll be posting a lot about stable homotopy theory, when our group seminar starts again. I bet that will trigger more reactions around here… at least if anyone is still left by then.
Here is the actual question that I have, concerning the reception of physics-related posts among the $n$-crowd:
a while back Davic C. had published an interview that he did with me in this issue.pdf of “The Reasoner”. In that interview, I believe, I had tried to say something about the relation of physics to topics more manifestly in the $n$-arena. That need not mean much, but in my memory what was unusal was that something I said back then managed to transmit some spark, and right afterwards I was contacted by some more pure mathematicians expressing interest in this kind of connection.
Notably, I seem remember Todd Trimble here (hi Todd! :-) to come back from reading that interview and expressing the desire to dive into cohesive homotopy type theory due to some – if I remember correctly – “excitement” that something which I said in the interview did induce. Am I remembering that correctly? Sorry if I am hallucinating. I am not sure how to search for the thread where this happened.
I think back then I had the bad taste to react to that by being busy with some task and promising to join into the project of expanding the entry on cohesive homotopy type theory later. That was probably stupid of me and a lost opportunity. Not sure what happened.
In any case, what I am wondering is: does anyone here – in addition to David C. that is – see a chance to be re-ignited by that spark, if it was one, find some interest in the application of higher cohesive topos theory to quantum physics and by jointly exploring this make the nForum here a more lively and more rewarding place… as it maybe was a while back when we were all throwing around more pure category theory?
Hi, Urs. Since you called me out specifically, I’ll respond as best I can.
I think you should keep doing whatever is in your natural path. If it happens to be a lot of physics these days, then so be it. Although I can understand your general feeling of loneliness; it has indeed been relatively quiet here. Partly this must be due to various life circumstances; for example, I can guess one very strong and compelling reason why Mike has been quiet here as of late. :-)
My own relative quietude here has largely to do with mathematical humps I’m trying to get over and have chosen to keep to myself for now. It doesn’t have much to do with the fact that I can’t really follow the discussions between you and David (for example). [Fact is, living as I do in a small Connecticut hamlet, not being around people and not being in a university setting, and generally not seriously keeping up to date, this was bound to happen in my case anyway.]
The only public discussion of this “excitement” I found (having anything to do with me) was around here – I said that I thought I detected real excitement coming from you in that interview. I’ll have a look again at the email I referred to and see about possibly re-igniting the spark in my brain that came from reading it.
On general principle, I would welcome very much a discussion of cohesive higher topos theory that I can participate in. It would be very nice. I can’t make promises myself – life (not just mathematics) seems to pull in hundreds of directions, and it’s hard to say how much time I can make for such a project. But of course I would love to learn more from you and Mike and others!
Perhaps we need some new personnel. Relying on a handful of people, many of whom are not university based mathematicians and so have other things to do, seems unlikely to engender the liveliest discussions. But I’m not sure this venue is designed to help either, despite its convenient proximity to the nLab.
To take your example of cohesive homotopy type theory, the most active part of the work took place at the Cafe, where it piqued the interest of Guillaume Brunerie. Just the kind of person you want to attract, but is he likely to come here? For the casual visitor, there’s not enough of a demarcation between a note recording a small change and the initiation of a serious collaborative investigation.
I should think online discussion of what you’re doing in stable homotopy theory would attract a lot of interest, but it would need to be presented in the right kind of way.
Thanks for the reactions.
Enganging more people is maybe another question, here I am being concerned about (not) dis-enganging those people who use to be engaged. I was getting worried that me filling the nForum with physics-related stuff drives some contributors away.
I should also say that I am not meaning to push people to look specifically into the cohesive homotopy type theory pet. That is a fun entry point to some extent, but of course that’s just where the story starts, and not an end in itself. Also, there is probably only so much one can say here.
What I’d be hoping for is that generally behind all the physics application and examples, the general abstract matrix remains visible and allows $n$-contributors to connect. For instance, much of my additions here in the last weeks concern working out examples of nothing but: families of correspondences in higher slice toposes. Disregarding all the physics jargon that goes with these examples, it would seem that this pure topic “systems of correspondences in slice topos” would be of the kind that some time back would have been the seed of a good bit of back and forth activity here. Maybe I am somehow obstructing this from happening?
Or maybe, as Todd suggests, times have just changed and the momentum we had just dissipated.
I should think the majority of any reduction in momentum is just down to Mike’s turn to parenthood, rather than anyone being put off by the physics. It’s pretty clear that you’re reaching for the “general abstract matrix”.
In terms of timing, it’s a shame if momentum dissipation is happening. We live in great times. It’s like Riemann, Cantor, Dedekind, Frege, Einstein and Hilbert all happening at once, where changes to the foundations of maths in general and to geometry in particular, combine to allow a new way to formulate physics. Even Hilbert never got his formalist foundational work to link in with his 6th problem.
I know that I am more rarely reporting my edits and new stubs than before: it slows me down when I have little online time like in recent couple of months and anyway there is almost no constructive feedback to my posts anyway (e.g. about the reported dual gebra entry); and I do not find making the noise that useful for the community. I mean I am contributing somewhat less but reporting even less, what makes a bit wrong impression.
@jim_stasheff
The standard answer is “both”, but I think ‘theory of homotopy types’ is closer to the truth. The fact of the matter is that $\equiv$ in homotopy type theory is not homotopy-invariant and can be used to discern finer structure than the homotopy-invariant $=$. This is actually a good thing, because it allows us to distinguish between the unit type (which is just any contractible space) and the higher inductive interval type (which has two distinct points)!
@Urs
I am interested to hear more about cohesive toposes (both ordinary and higher), but my lack of background makes it difficult to see what you are doing with them, whether in physics or in differential geometry.
It’s true that I have other things on my mind right now: not just a new baby but also a new job.
@jim and @Zhen, I think the question to which the standard answer is “both” is “should I read ’homotopy type theory’ as ’the theory of homotopy types’ or as ’the homotopy theory of type theory’”, and I think that that answer is correct. I agree with Zhen that “type theory up to homotopy” is not quite right.
We live in great times. It’s like Riemann, Cantor, Dedekind, Frege, Einstein and Hilbert all happening at once, where changes to the foundations of maths in general and to geometry in particular, combine to allow a new way to formulate physics.
Yes. And now if only this observation would percolate beyond a group of size of order $\frac{1}{2} \cdot 10^0$…
(It certainly hasn’t propagated to Dutch hiring committee’s yet ;-)
size of order $\frac{1}{2} \cdot 10^0$…
Did you mean
size of order $\frac{1}{2} \cdot 10^1$… ?
(It certainly hasn’t propagated to Dutch hiring committee’s yet ;-)
Are you having a hard time landing the next job?
With those orders of magnitude I was of course joking around, but I really did mean $10^0$. Here is how I feel about it: given something of order $10^0$ I expect “a handful”, namely 3 or 4 or 5 or 6, a little less likely 7 or 8 or 9 and if its gets 10 or 11 then we need to start thinking about calling it order $10^1$, which in turn most likely means something like 30 or 40 or 50 or 60, and so on. So something of order of magnitude $\frac{1}{2} 10^0$ is a “small handful”, likely 1 or 2 or 3, less likely 4 or 5 or 6, etc.
Concerning having a hard time: not sure yet, depends on what happens next. But one rumour I heard the other day indicated that your insight from #5 is not shared by all hiring committees ;-)
What kind of take-up has there been of your ideas? I mean is anyone else talking about cohesiveness of $\infty$-toposes? I guess Joyal is. (By the way, I asked him if he’d like to join in our locus discussion. He says he’s too busy at the moment, but may find time later.)
Uli Bunke has now been picking up the idea of formulating differential cohomology internal to cohesive $\infty$-toposes. In Pittsburgh at the GAP XI meeting he has given a three-part lecture series on this.
He is boosting it a good bit further by considering actual spectrum-valued cohesive $\infty$-stacks, hence the proper stabilization of the cohesive $\infty$-topos. That’s the context in which one finds the smooth cohomology spectra that in Nuiten’s thesis thesis are explained will give the fully internal formulation of “motivic quantization”.
I guess the first relevant article is this one here:
In the lectures in Pittsburgh at least he had a nice acknowledgement.
Re #14, does that bear on our discussion of stabilized cohesion? Does his stabilization generate a cohesive $\infty$-topos?
That article considers the stabilization of the cohesive $\infty$-topos, that means the spectrum objects inside the $\infty$-topos, that means the spectrum-valued sheaves, localized at the covers.
The article doesn’t dwell on general abstract characterizations of what this means, but just writes it down and then works with it.
12 Hm, I was taught in school that order of magnitude of $a$ is somewhere around $a$ plus minus some exponent of whatever pace is relevant. So order of magnitude of 10 to 50 may be as well 48 or 52; and order of 10 to 1 is then somewhere between say 5 and 50 (exponentially actually better between sqrt of 10 and 10 times sqrt 10) if we measure order by power of ten, what is not necessary in order of magnitude estimates. So I interpret things more like 11 then like 12.
I would take order $10^n$ to mean necessarily between $10^{n-1}$ and $10^{n+1}$ and probably between $10^{n-1/2}$ and $10^{n+1/2}$. So $11 \times 10^n$ would be right out.
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