Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory infinity integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 22nd 2012
    • (edited Apr 22nd 2012)

    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)(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.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 22nd 2012

    added an Examples-section on Equivalences of 2-groups.

    Almost done, but am being interrupted now..

    • CommentRowNumber3.
    • CommentAuthorTim_Porter
    • CommentTimeApr 23rd 2012

    I fixed some typos. I could not see what was going wrong so replaced something by Grp\infty Grp, which does come up with the right characters.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 23rd 2012

    Thanks, Tim!

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 31st 2016

    At 2-group where it says

    That should be

    w 2:BSOB 2 2? w_2 : \mathbf{B}SO \to \mathbf{B}^2 \mathbb{Z}_2 ?
    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 31st 2016
    • (edited Jun 1st 2016)

    Yep. And BSpin is the homotopy fibre of it.

    EDIT: now fixed.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 1st 2016

    Thanks for catching that. Weird.

    • CommentRowNumber8.
    • CommentAuthorJanPulmann
    • CommentTimeSep 25th 2019
    • (edited Sep 25th 2019)
    should the definition of the weak 2-group also demand that all morphisms are invertible? The definition in HDA V does that. Thanks!
    edit: to be clear, I mean the definition starting at
    • CommentRowNumber9.
    • CommentAuthorUlrik
    • CommentTimeSep 26th 2019

    Fix the definitions of weak 2-groups

    diff, v40, current

    • CommentRowNumber10.
    • CommentAuthorUlrik
    • CommentTimeSep 26th 2019

    You’re right, so I just fixed these definitions, but maybe my phrasing is suboptimal. Feel free to edit, of course.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 10th 2021
    • (edited Jul 10th 2021)

    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):

    diff, v41, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2023

    added pointer to:

    diff, v43, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2023
    • (edited Jun 4th 2023)

    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.

    diff, v46, current

    • CommentRowNumber14.
    • CommentAuthorJohn Baez
    • CommentTimeJul 4th 2023

    Added references to Hoàng Xuân Sính’s papers on 2-groups.

    diff, v47, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeAug 3rd 2023
    • (edited Aug 3rd 2023)

    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.

    • CommentRowNumber16.
    • CommentAuthorperezl.alonso
    • CommentTimeAug 3rd 2023

    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.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeAug 4th 2023

    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.)

    • CommentRowNumber18.
    • CommentAuthorvarkor
    • CommentTimeAug 4th 2023
    • (edited Aug 4th 2023)

    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.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeAug 4th 2023
    • (edited Aug 4th 2023)

    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

    In its top right corner, this scan carries the following somewhat cryptic dating:




    ? \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?

    diff, v49, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeAug 4th 2023

    added also pointer to Solian 1980

    diff, v49, current