Thanks, Richard, I appreciate it.

]]>My thoughts on IUTT (and this page) as expressed here and elsewhere on the nForum remain unchanged, but I have removed the material under discussion above now as I do not currently have the time/energy to work on it further, or to defend trying to proceed in such a way.

]]>Removing the experimental sketched outline of the proof logic.

]]>I have dealt with and other recent occurrences of spam on the forum. Thanks Tim!

]]>Re. #132:

Yes, I was being brief.

That’s why I thought I’d fill in some of the details.

There were statements in that 2004 talk by Mochizuki that made me seriously pause.

Yup. That, and not Scholze-Stix, was what did it for me.

While Scholze-Stix managed to derive a contradiction, that’s only a symptom of a problem. Everyone also seemed to agree that they used a simplified version of IUT, which unsurprisingly became a major point of dispute. With the 2004 talk, we get directly from Mochizuki some of the assumptions that he was working from, and that seems to point to something deep within IUT itself that makes it hard to survive contact with number-theoretic reality.

it seems like the audience was nodding sagely along to statements

Not really. There were many folded arms and bewildered looks.

Mochizuki edited his most recent set of notes back in May to have a long section (1.10) on the legal theory of intellectual property

This was discussed on Reddit, where functor7 quoted the salient bits.

how he would like to have some kind of recourse to the reputational damage to IUT based on what he sees as a false, poor imitation by Scholze

That is why I also agree with Urs in #129 that

]]>decency demands that however little we have on the nLab regarding would-be mathematical content of IUT be essentially cleared, leaving only, if unavoidable, pointers to references and to the debate.

Dating back to revision 8 (Mike Shulman): “At that time this was generally thought to be a rather unsatisfactory and unnecessarily technical solution especially when compared to the normal forms to which the community was accustomed.”

Not that I’m doubting this, but it’s the first time I’m learning about it. Any citations?

]]>Remark 3.73, “solid ring” — not to be confused with solid abelian group or solid modules (!)

]]>Yes, I was being brief. There were statements in that 2004 talk by Mochizuki that made me seriously pause. If that was a talk in another part of the world, and the speaker not so respected, there would have been serious questions about some of the mathematical ideas. Instead it seems like the audience was nodding sagely along to statements like having to “solve” “equations” of the form $x_1\in x_2 \in \ldots \in x_n \in x_1$, and how this is impossible in ZFC. There is no way in God’s green earth that statements about arithmetic geometry rely crucially on working around the Axiom of Foundation. There are talks where the speaker is motivating my metaphor, backed up by theorems that show the metaphor is true, and then …

The event that sparked Rongmin to reminding me of this talk is the fact Mochizuki edited his most recent set of notes back in May to have a long section (1.10) on the legal theory of intellectual property, and how he would like to have some kind of recourse to the reputational damage to IUT based on what he sees as a false, poor imitation by Scholze (this is me paraphrasing, I can’t really bring myself to read it just yet). I find this section troubling for various reasons.

]]>In 3.51(i), you are asking that to drop nilpotency, one needs to include an action of the fundamental group. But surely even in the nilpotent case, there is a (no trivial, in general) action, since in 3.50(ii) the higher homotopy groups are already modules over $\pi_1$.

]]>re #8:

I see that it is this connection which the preprint

- David Ayala,
*Homological Stability among Moduli Spaces of Holomorphic Curves in Complex Projective Space*(arXiv:0811.2274)

was after, before the author discovered the mistake highlighted in v2.

Mistake or not, that’s the right question to ask. But it looks like it wasn’t followed up.

]]>added this pointer:

- Sheldon Katz,
*Enumerative Geometry and String Theory*, Student Mathematical Library**32**AMS 2006 (ISBN:978-1-4704-2143-4, spire:739788)

Re. #130:

IUT is aiming for some kind of $\mathbb{F}_1$ theory, and made an idiosyncratic use of category theory that seems to me to be really be out of kilter.

The above may seem like it’s coming out of left field for some people, so let me elaborate a bit.

There is now some sort of consensus about what a reasonable theory of the field with one element $\mathbb{F}_1$ could be, but back when things were still in flux, Mochizuki gave a talk at a 2004 Tokyo conference (conference website, Youtube video, Mochizuki’s notes) about a proposal that was motivated by the search for such a theory. This proposal later became IUT. Someone pointed this out to David on Twitter, especially the bit about “rings as 2-categories” on the first page of the notes, where Mochizuki sketched out his proposed approach to categorification.

It’s unclear to me how much this misapprehension (as David saw it) has informed the development of IUT, but it seemed plausible to me that this may be behind the many peculiarities of IUT.

]]>Thanks! Have added indication that the theorem itself finally appears there as Thm. 10.6.

]]>Added:

A detailed exposition is available in Chapter II of

- James R. Munkres,
*Elementary Differential Topology*, Annals of Mathematics Studies 54 (1966), Princeton University Press, doi.

Thanks for the pointer. I have added a warning here.

]]>This comment https://mathoverflow.net/questions/139339/voronoi-cells-and-the-dual-complexes-in-riemannian-manifolds/177199#177199 seems to give a counterexample to the 1961 proof by Cairns (not Cairn).

]]>[ this is referring to arXiv:1207.0249, web ]

Since the classical model structure on simplicial sets is right proper, a pullback diagram is a homotopy pullback already when one of the two maps is a fibration, with no further condition on the objects. This is the second item of this Prop..

]]>Following a detailed referee report (thanks to the anonymous referee!, that was useful) we have expanded out details on a number of points in the article. Version 2 is now available as pdf here.

The only request not yet reacted to is to expand the Introduction. Happy to do so, but first some vacation next week.

]]>a bare subsection with a list of references, to be `!include`

-ed into the References-section of relevant enties, such as at *triangulation* and at *triangulation theorem*, for ease of synchronizing

brief `category:people`

-entry for hyperlinking references at *triangulation theorem* and *Riemann surface*

It seems that Cairn’s and Whitehead’s proofs work already for $C^1$-differentiable manifolds, not just smooth manifolds. Is that right?

]]>brief `category:people`

-entry for hyperlinking references at *triangulation theorem*

brief `category:people`

-entry for hyperlinking references at *topological manifold*