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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):
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