4 problems

Let $\Sigma\subset S^4$ be an oriented knotted surface. Let $\mathcal{G}=(G,E,\partial,\triangleright)$ be a finite crossed module. Write $I_\mathcal{G}(S^4\setminus\nu(\Sigma))$ f…

Let $\chi$ be a crossed module, let $\Bbbk\mathcal{G}_\chi$ be the spherical $\chi$-fusion category associated with it, and let $\omega$ be a normalized $3$-cocycle for $\chi$ with…

Let $\mathcal{K}$ be a crossed module with finite fundamental group $\pi_1(\mathcal{K})$, let $\widetilde{\mathcal{K}}$ denote its associated 2-group, and let $\mathcal{X}$ be a cl…

Let $\mathcal{K}=(H\xrightarrow{\partial}G)$ be a crossed module whose fundamental group $\pi_1(\mathcal{K})$ is finite. Let $\Theta$ be a 2-representation of…