# Higher Dijkgraaf-Witten Theory

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?

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

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