We define a family of quantum invariants of closed oriented $3$-manifolds using spherical multi-fusion categories. The state sum nature of this invariant leads directly to $(2+1)$-dimensional topological quantum field theories ($text{TQFT}$s), which generalize the Turaev-Viro-Barrett-Westbury ($text{TVBW}$) $text{TQFT}$s from spherical fusion categories. The invariant is given as a state sum over labeled triangulations, which is mostly parallel to, but richer than the $text{TVBW}$ approach in that here the labels live not only on $1$-simplices but also on $0$-simplices. It is shown that a multi-fusion category in general cannot be a spherical fusion category in the usual sense. Thus we introduce the concept of a spherical multi-fusion category by imposing a weakened version of sphericity. Besides containing the $text{TVBW}$ theory, our construction also includes the recent higher gauge theory $(2+1)$-$text{TQFT}$s given by Kapustin and Thorngren, which was not known to have a categorical origin before.
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