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I looked at the entry 2-group today and found it strongly wanting. Now I have spent a few minutes with it, trying to bring it into better shape. While I think I did imporve it a little, there is clearly still lots of further room for improvement.
Mainly what I did was add more on the intrinsic meaning and definition, more on the homotopical meaning, and more on the details of how crossed modules present the $(2,1)$-category of 2-groups – amplifying the role of weak equivalences. And brief remarks on how all this generalizes to the case of 2-groups “with structure” hence internal to other $\infty$-toposes than the terminal one.
added an Examples-section on Equivalences of 2-groups.
Almost done, but am being interrupted now..
I fixed some typos. I could not see what was going wrong so replaced something by $\infty Grp$, which does come up with the right characters.
Thanks, Tim!
At 2-group where it says
the second Stiefel-Whitney class
$w_2 : \mathbf{B}Spin \to \mathbf{B}\mathbb{Z}_2$is induced this way from the central extension $\mathbb{Z}_2 \to Spin \to SO$ of the special orthogonal group by the spin group;
That should be
$w_2 : \mathbf{B}SO \to \mathbf{B}^2 \mathbb{Z}_2 ?$
Yep. And BSpin is the homotopy fibre of it.
EDIT: now fixed.
Thanks for catching that. Weird.
You’re right, so I just fixed these definitions, but maybe my phrasing is suboptimal. Feel free to edit, of course.
Added these pointers:
Further on 2-group-extensions by the circle 2-group:
of tori (see also at T-duality 2-group):
of finite subgroups of SU(2) (to Platonic 2-groups):
added pointer to:
As you may be aware of, there has been and is an ongoing flood of articles discussing “generalized symmetries” in (high energy-) and in condensed matter physics. Alas, much of it is based on handwavy arguments about “inserting higher codimension operators” and only a small fraction of authors attempts to connect this vague idea to mathematical definitions.
So while we don’t have an entry on “generalized symmetries” (and I am torn whether we ought to have one) it’s maybe good to start listing those references in this “field” which show an inkling of understanding of higher groups.
For the moment I have added a couple such reference items here. If and when this list grows, it may be worthwhile moving it to an !include
-entry.
A propos #13:
The new field of oxymoronic symmetry theory is blossoming: Today no less than 3 announcements on hep-th
with “non-invertible symmetries” in the title.
Hi Urs,
Had missed this subthread. Perhaps it’s not a bad idea to create a new entry about “generalized symmetries”, but your call. Broadly speaking, the idea of generalized symmetries is that there are quantum symmetries that do not have inverses. But why is this apparent oxymoron meaningful? In what sense are they non-invertible? The key point here is that they should arise at the quantum level. The Idea/story is then roughly as follows. As you and your collaborators established back in the day, a prequantum n-dim’l theory is a pre-n-plectic space lifted to a circle n-bundle. Of this, we can consider its n-group of automorphisms, which we call the quantomorphism n-group. These are the prequantum higher symmetries (there are some subtleties of the kind we discussed in person regarding Lepage equivalents to make this work that I haven’t been able to sort out). Say for n=1, we get a group of symmetries of a prequantum bundle.
Now, as described in the nlab page of motivic quantization, one should map Phases to BGL(E) for E some E-oo ring. This basically performs linearization in some sense. Then the main question is, what happens to the quantomorphism n-group? For n=1, E=C, and finite group, at least the naive answer is that the symmetry group becomes the group algebra C[G]. But of course, while the generators are still invertible, any of the formal sums does not have an inverse anymore. It is in this sense that Physicists consider “non-invertible” symmetries, which is different from what the name would suggest (e.g. considering the higher monoid of prequantum endomorphisms). So for n-dimensional theories, the guess or central assumption is that the prequantum higher group becomes a quantum higher algebra. The great majority of the current literature deals with the higher categorification of this finite group setting, which is why you’ll always see them discussing fusion (n-)categories. In particular, they are interested in cases where the fusion category is not the generalization of C[G], but are things such as the Tambara-Yamagami category. It’s still not obvious to me how these categories could arise from a prequantum theory, but there are some models using lattices that argue that the corresponding quantum theory has a given fusion category as its fusion category of symmetries.
Obviously there are a lot of things that need to be established (gauging, anomalies, dual symmetries, etc). But that’s as far as I can tell the general idea. “Non-invertible symmetries” arise at the quantum level due to the tensor structure of the quantum theory, even if we stick to invertible transformations at the quantum level. This is not far from the content of e.g. Entanglement of Sections, I think.
Would be happy to discuss this further.
Hi Alonso,
yes, if you would go ahead creating an entry in the direction of “generalized symmetries” that would be great! Or give it a technically more descriptive name (maybe “higher global symmetry”), whatever makes sense to you. I am (still) busy with other things, but I’d be happy to follow such an entry develop.
My quip on the terminology “non-invertible symmetry” being “oxymoronic” seems hard to evade, though: It is de facto an oxymoron, isn’t it. I don’t think it’s wise to speak this way, since it effectively means dumping the meaning of the dear word “symmetry”, for little gain. But this is a side issue which should not to get in the way of having a new entry as above.
What you say regarding quantum phases seems to refer to the construction of (pre)quantum operators as on the bottom of p.5 here?
The case of regarding a group algebra as an algebra of quantum observables has been much on my mind lately. A profound example of this is that discussed in Fundamental weight systems are quantum states. We have been expanding on the underlying idea in section 5 “Brane lightcone quantization” of Introduction to Hypothesis H. There are some classical theorems by Gelfand, Naimark and others (here) that are relevant here in view of quantization, which seems to have remained under-appreciated. This would be fun to discuss. (Again, I don’t think it would be wise to refer to elements of a group algebra as “non-invertible symmetries”, but that linguistic issue need not get in our way.)
This page claims that Hoàng Xuân Sính introduced 2-groups in 1973. However, most references cite the year as 1975. Furthermore, there’s a 1972 paper Groupe dans une catégorie by Alexandru Solian which appears to be an earlier reference.
I have added the pointer to Solian 1972
and fixed the web-link for Sinh 1973,
adding also the link to the pdf-scan at pnp.mathematik.uni-stuttgart.de/lexmath/kuenzer/sinh_I.pdf
In its top right corner, this scan carries the following somewhat cryptic dating:
12
HOA
73
? $\to$ 1975
Since the last line is in a handwriting different from the first three lines, maybe the original dating may have been 1973 and then later somebody suggested that the proper dating should be 1975? Maybe Sinh wrote “73” when starting the manuscript but did not defend it before 1975?
added also pointer to Solian 1980
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