Multi-fusion categories are a generalization of fusion categories with a non-simple unit. The direct sum of two multi-fusion categories is again a multi-fusion category. By irreducible I mean that a multi-fusion category cannot be written as a non-trivial direct sum. Are all such irreducible multi-fusion categories fusion categories?


Bonus question: Are there multi-fusion categories that are not Morita equivalent to a direct sum of fusion categories?


[I'm mainly interested in the unitary case.]



Are there irreducible multi-fusion categories that are not fusion categories?
Matrix categories, $\mathrm{End}(\mathrm{Vec}^{\oplus n})$. (The identity on each copy of $\mathrm{Vec}$ are summands of the identity.)


(You can generalize this example by putting fusion categories along the diagonal of a matrix, and Morita equivalences between them into the off-diagonal entries. Over an algebraically closed field irreducible multifusion categories examples look like this, which you can see by decomposing the multifusion category as a module over itself.)


