Higher Dijkgraaf-Witten Theory

Question

I am trying to understand higher-form symmetries in TQFT. In particular the higher-form version of Dijkgraaf-Witten Theory.

It is known that for a 0-form symmetry we can specify the principal G-bundle through homotopy classes of the classifying map $$ M \rightarrow BG = K(G,1). $$ This is known from Homotopy Theory and Eilenberg-MacLan spaces. Indeed the homotopy classes of these maps are in bijection with the first cohomology group $H^1(M,G)$ that for a finite group is isomorphic to $\operatorname{Hom}(\pi_1(M),G)$ and fit the usual gauge theory: $$ [M,K(G,1)] \simeq H^1(M,G) \simeq \operatorname{Hom}(\pi_1(M),G) $$

I cannot find any reference for a higher version of this. Should I expect a naive generalization? This is motivated by the fact that for a 1-form symmetry $H^2(M,G)$ works as a straightforward generalization to the previous case. But does homotopy theory tell me something about the classification of gerbes via classifying maps?

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There is a follow-up question to this, when the symmetry structure is an honest 2-group.

Answer 1

Higher-form symmetries are abelian so, with $G$ a discrete abelian group and $p\in\mathbb{Z}_{\geq 0}$ (or $G$ a discrete group, not necessarily abelian, if $p=0$): $$ [M, K(G,p+1)] \cong \mathrm{H}^{p+1}(M;G) \cong \operatorname{Hom}\left(\pi_{p+1}(M),G\right) $$ and everything works as it should. See e.g. the Wikipedia page for Eilenberg-McLane spaces or the paper From gauge to higher gauge models of topological phases by Delcamp and Tiwari$^{(*)}$.

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$^{(*)}$In that paper they do not explicitly mention $K(G,p+1)$, but they define it descriptively as a higher-classifying space $B^{p+1}G$.