This paper by Auke Booij gives another proof of the intermediate value theorem without the axiom of choice by using locators and locally non-constant functions.

]]>Incidentally, regarding original references on the infinite tensor product in #6:

Wikipedia here calls this notion the “incomplete-” or “Guichardet-“tensor product and references

- O. Bratteli, D. W. Robinson,
*Operator Algebras and Quantum Statistical Mechanics 1*, Springer 1987/2002

where the notion is briefly mentioned on p. 144 (and only there, it seems), without, however, calling it anything, neither “incomplete” nor “Guichardet”.

The attribution to Guichardet must refer to

- A. Guichardet,
*Tensor products of $C^\ast$-algebras*Part II.*Infinite tensor products*, Aarhus Universitet Lecture Notes Series**13**(1969) (pdf)

where the case of Hilbert spaces is considered explicitly in section 6, while “incomplete” seems to refer to

- J. von Neumann,
*On infinite direct products*, Compositio Mathematica, tome 6 (1939), p. 1-77 (numdam:CM_1939__6__1_0)

where something at least similar is considered around Lem. 4.1.2. (But is it the same?)

]]>For what it’s worth, this is appears as Ex. 6.3.11 in

- K. R. Parthasarathy,
*Introduction to Probability and Measure*, Texts and Readings in Mathematics**33**, Hindustan Book Agency 2005 (doi:10.1007/978-93-86279-27-9)

It’s a link to a pdf on the HoTT Zulip, this is the thread where it is.

]]>I still don’t know which preprint you mean, as the link in your #18 is broken! (Maybe I should have said this more explicitly in #19. Please check the link and let us know where you are pointing us to.)

]]>Urs, this preprint was only released on September 2021 as said in the pdf file itself, so isn’t actually linked in the article.

]]>A basic thought on the infinite tensor product of $L^2$-spaces:

The “grounded” infinite tensor product of Hilbert spaces, which is mentioned in

- John C. Baez, Irving Ezra Segal, Zhengfang Zhou, p. 126 of:
*Introduction to algebraic and constructive quantum field theory*, Princeton University Press 1992 (ISBN:9780691634104, pdf)

and discussed in

- Nik Weaver, Def. 2.5.1
*Mathematical Quantization*, Chapman and Hall/CRC 2001 (ISBN:9781584880011)

has, I’ll suggest, a nicely transparent incarnation in the case of the Hilbert space of square-integrable functions on some $(X,\mu)$, because here it is the dual of the limit that gives the infinite product $X^{\times_\infty}$:

$X^\infty \;\coloneqq\; \underset{ \underset{ \mathclap{ S \in FinSub(\mathbb{N}) } }{\longleftarrow} }{\lim} \; X^{S} \;\;\;\;\;\;\; \mapsto \;\;\;\;\;\; \underset{ \underset{ \mathclap{ S \in FinSub(\mathbb{N}) } }{\longrightarrow} }{\lim} \; L^2(X)^{\otimes_S} \;\;\;\; =: \; L^2(X)^{\otimes_\infty} \,.$Here the “groundedness” of the product is just the contravariant functoriality of $L^2(-)$ applied to the projections

$\array{ S' &\xhookrightarrow{ \;\;\;\; i \;\;\;\; }& S \\ X^{S'} &\xleftarrow{ \;\;\;\; X^i \;\;\;\;}& X^S \\ L^2\big( X^{S'} \big) &\xrightarrow{ \;\;\; (X^i)^\ast \;\;\; }& L^2\big( X^{S} \big) }$which implicitly regards each $L^2$-spaces as having as vacuum state the constant unit function.

This makes one want to say that $L^2(-)$ takes (these) limits to (these) colimits, so that

$L^2(X)^{\otimes_\infty} \;\; \simeq \;\; L^2\big( X^\infty \big) \,.$Is this the case? (Just firing off this question, maybe it’s evident once I start to really think about it…)

]]>No problem, thanks for chatting about it. As it goes, making some noise here served to export enough entropy out of my system that I think I have now solved the problem that made me write #2 :-)

]]>Not sure where this link was intended to point, but the entry *Introduction to Homotopy Type Theory* (which #17 is referring to) lists several links to preprint versions.

To the guest at 17: the textbook isn’t published yet, but Egbert Rijke did release another preprint here.

]]>Hmm. Yeah, I guess it would be pretty strange if there were a group whose characters are Gegenbauer polynomials anyway. Looks like the approach I suggested will not work. Let $X=G/H$ and suppose for each irrep $\rho$ of $K$ we can find a subgroup $J\leq G$ such that $\rho$ appears in $L^2(J\backslash X)$. Then since each $L^2(J\backslash X)\subseteq L^2(X)$ we would have shown that each $\rho$ appears in $L^2(X)$. But in the case of $G=SO(n+1)$ and $H=SO(n)$, even allowing for $J\lneq G$ other than $J=H$, this will not work by a similar argument to #9 by the subgroup structure of $G$. (Also note that choosing smaller subgroups $J'\leq J$ is counterproductive since $L^2(J'\backslash X)\subseteq L^2(J\backslash X)$.) I guess it still leaves the option of writing $X=G'/H'$ for different $G'$ and $H'$ but I kind of doubt this is the right approach, especially since the freeness of the $K$-action is not coming into play anywhere.

]]>Banks argues that the corners of M-theory that have a decent theoretical underpinning – such as the matrix models – look decidedly different from the assumptions that the bulk of the string-pheno community is running their business on.

What this means for string-pheno-done-right will only be answerable once its done right – which in turn will require formulating more of the M-theory first. It’s a long way ahead. (See also Duff 2020 around 16:36.)

]]>Thanks for the heads-up.

It looks like in the “Definition”-Section 2 it’s stated correctly, at least after the words “more precisely”. Then section 3 is lacking the capitalization.

]]>Sorry, that comment #10 didn’t quite type-check.

]]>Is that action of G via the inclusion of S(H) in S(O)?

]]>Once nLab editing is open, someone should fix the mistake that (connective) tmf is defined as the global sections of a sheaf of $E_{\infty}$-rings. That’s not true - it’s only known definition is as a connective cover of Tmf. For instance, see Behrens’ survey article in the Handbook. To quote the Hill-Lawson paper (p. 6): “Finally, the construction of the object tmf by connective cover remains wholly unsatisfactory, and this is even more true when considering level structure. In an ideal world, tmf should be a functor on a category of Weierstrass curves equipped with some form of extra structure. We await the enlightenment following discovery of what exact form this structure should take.”

]]>That said, I do think that there is much room left to analyze the IKKT matrix model in more depth, notably in more mathematical sophistication. (Back in the golden 90s one could see Alain Connes talk about IKKT (arXiv:hep-th/9711162) highlighting that its eponymous matrices are hardly those but must be understood as operators on a Hilbert space, making matrix model theory a topic in operator algebra.)

Another evident suggestion that remains underappreciated is that in holographic QCD baryons are modeled by D-branes and hence multiple baryons ought to be modeled by the respective “nuclear matrix model”.

There are several promising loose ends like this sitting around waiting to be picked up, while the ST community is lost in the Swampland (to pick up Banks’ ranting from arXiv:1910.12817).

But that’s okay, this gives us time to put up the machinery for the next serious attack…

]]>Thanks Urs! ]]>

I note that Goncalo Tabuada has moved so we should edit his nLab page when editing is available: webpage

]]>It’s still all up for grabs. But possibly best not to wish too hard for it: What has hurt communities is trying to hard-wire their psychological preconceptions of what quantum spacetime should be into their theories, instead of letting the theory tell them.

Generally, if a would-be theory of quantum gravity has either

- (a) a compelling input

or

- (b) a fascinating output,

and preferably both

then it’s worth taking note of as a hint towards humanity’s quest to solve this riddle.

But if it has (b) no discernible output while being based on (a) a crazy left turn at step one, then it is a waste of time.

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