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• 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 $(2,1)$-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 $\infty Grp$, 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

• 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

$w_2 : \mathbf{B}SO \to \mathbf{B}^2 \mathbb{Z}_2 ?$
• CommentRowNumber6.
• CommentAuthorDavidRoberts
• CommentTimeMay 31st 2016
• (edited May 31st 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)
Hi,
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 https://ncatlab.org/nlab/show/2-group#weak_groups
• CommentRowNumber9.
• CommentAuthorUlrik
• CommentTimeSep 26th 2019

Fix the definitions of weak 2-groups

• 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)