Extended TQFT arising from enriched multi-fusion categories

Abstract: We define a symmetric monoidal (4,3)-category with duals whose objects are certain enriched multi-fusion categories. For every modular tensor category $\mathcal{C}$, there is a self enriched multi-fusion category $\mathfrak{C}$ giving rise to an object of this symmetric monoidal (4,3)-category. We conjecture that the extended 3D TQFT given by the fully dualizable object $\mathfrak{C}$ extends the 1-2-3-dimensional Reshetikhin-Turaev TQFT associated to the modular tensor category $\mathcal{C}$ down to dimension zero.