# Have people defined fusion $n$-categories?

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.

• As far as I know there is not a widely used standard definition of $n$-category in the mathematical literature for, say, $n \ge 4$, so from a mathematical point of view it's not clear what this definition is supposed to mean rigorously. Commented Dec 3, 2020 at 19:52
• Is there then a definition for $n=3$? Can you point me to it? Also, what do you mean it's not clear what it's supposed to mean? Basically what I'm asking is, why is this definition not enough as a mathematical definition? Commented Dec 3, 2020 at 19:58
• You can look up, for example, "tricategory": ncatlab.org/nlab/show/tricategory And I mean exactly what I said: there isn't a widely used standard definition of $n$-category in the mathematical literature, so giving a definition of some other concept in terms of an unspecified definition of $n$-category is not clear. You can see some references on the nLab: ncatlab.org/nlab/show/n-category#defn Commented Dec 3, 2020 at 20:00
• Ah, sorry. I overlooked the fact that you omitted the adjective "fusion". Things make sense again! Commented Dec 3, 2020 at 20:08