Invertible Fusion Categories

A tensor category $\mathcal{C}$ over a field $\mathbb{K}$ is said to be invertible if there's a tensor category $\mathcal{D}$ such that $\mathcal{C}\boxtimes\mathcal{D}$ is Morita equivalent to $\mathrm{Vec}_{\mathbb{K}}$. When $\mathbb{K}$ is algebraically closed, it is well-known that the only invertible fusion category is $\mathrm{Vec}_{\mathbb{K}}$, and any invertible multi-fusion category is Morita equivalent to $\mathrm{Vec}_{\mathbb{K}}$. By contrast, we show that for general $\mathbb{K}$ the invertible multi-fusion categories over a field $\mathbb{K}$ are classified (up to Morita equivalence) by $H^3(\mathbb{K};\mathbb{G}_m)$, the third Galois cohomology of the absolute Galois group of $\mathbb{K}$. We explicitly construct a representative of each class that is fusion (but not split fusion) in the sense that the unit object is simple (but not split simple). One consequence of our results is that fusion categories with braided equivalent Drinfeld centers need not be Morita equivalent when this cohomology group is nontrivial.