I come from a physics background, but I work a bit with (higher-) categories, especially fusion ones. Whenever I talk to mathematicians about a fusion $n$-category someone is bound to bug, asking me "Have fusion $n$-categories even been defined?". E.g. a comment on a previous question of mine.
As far as I'm aware of, mathematicians are generally happy with Douglas' and Reutter's definition of a fusion 2-category [DR18]. What about $(n>2)$-categories? In the physics literature [KWZ15], I have found this definition of a unitary fusion $n$-category:
Definition 2.4. A unitary fusion $n$-category for $n\geqslant 0$ is a unitary $(n+1)$-category with a unique simple object $∗$. We also identify it with the unitary $n$-category $\mathrm{hom}(∗,∗)$. We define a unitary multi-fusion $n$-category to be the $n$-category $\mathrm{hom}(x,x)$ for an (not necessarily simple) object $x$ in a unitary $(n+1)$-category.
Is this definition satisfactory from a mathematical perspective? [I would expect not, since it predates [DR18], but anyway :) ]
- If yes, would the obvious modification of simply removing the adjective unitary define a fusion $n$-category?
- If not, what is wrong with this definition?
References:
[DR18] Douglas, C. L. and David Reutter, Fusion 2-categories and a state-sum invariant for 4-manifolds, arXiv: 1812.11933.
[KWZ15] Liang Kong, Xiao-Gang Wen, and Hao Zheng Boundary-bulk relation for topological orders as the functor mapping higher categories to their centers, arXiv: 1502.01690.