I am reading about generalised global symmetries or higher-form symmetries (for example, in Generalized Global Symmetries, Gaiotto, Kapustin, Seiberg, and Willett), and came across this question. One of the paragraphs in the answer is given below,
I would say that 1-form group symmetries are in some ways simpler than 0-form symmetries, because they are related to the 2nd homotopy group, which is always abelian, while the 1st homotopy group can be nonabelian. The real nonbelian higher symmetries are as complicated as TQFTs though.
What is the relation between higher homotopy groups and higher-form symmetries? Also, are there examples of non-abelian higher symmetries? How are these different from higher-form symmetries? I mean, what are they?
Finally, in the above cited paper, the authors say that a $(q+1)$-form symmetry relates an operator $U_g(M^{d-q-1})$, corresponding to the group element $g$, to a codimension-$(q+1)$ manifold $M^{d-q-1}$. They then say that for $q=0$, which corresponds to ordinary symmetries, due to time-ordering, the operators need not commute. However, for $q>0$, they say that there is no equivalent of time-ordering and hence the operators commute. How is this related to the higher homotopy group perspective?
