I am trying to understand higher form symmetries in TQFT. In particular the higher form version of Dijkgraaf Witten Theory.
It is know that for a 0-form symmetry we can specify the principal G-bundle through homotopy classes of the classyfing map $$ M \rightarrow BG = K(G,1). $$ This is known from Homotopy Theory and Eilenberg-MacLan spaces. Indeed the homotopy classes of this maps are in bijection with the first cohomology group $H^1(M,G)$ that for a finite group is isomorphic to $Hom(\pi_1(M),G)$ and fit the usual gauge theory: $$ [M,K(G,1)] \simeq H^1(M,G) \simeq Hom(\pi_1(M),G) $$
I cannot find any reference about higher version of this. Should I expect a naive generalization? This is motivated from the fact that for a 1-form symmetry $H^2(M,G)$ works as a straightforward generalization to the previous case. But does homotpy theory tell me something about classification of gerbes via classyfing maps?
Moreover the 2-Group approach works in a similar way, we introduce $K(G,2)$ and stuff.
But how should I formulate this for a single 1-form symmetry? I hope I was clear, any help is appreciated.