It there some notion of ribbon 2-category, which would allow for, say, talking about the Seifert surface of links (which is a 1-morphism in some ribbon category) as a 2-morphism in the category?


Thank you! (I'm sorry this question is so vague.)



Is there a notion of "ribbon 2-category"?
A ribbon 1-category is a 3-category which has only one 0-morphism, has only one 1-morphism, and is (strict) "3-pivotal", where "$n$-pivotal" is the property that would be called "pivotal" if $n=2$. (I don't know if there is a standard term for this. Candidates are "n-pivotal", "has strong duality", "disk-like", "is an $[S]O(n)$ homotopy fixed point of ...".)


So one can analogously define a ribbon 2-category to be a 4-pivotal 4-category with only one 0-morphism and one 1-morphism. I think everyone would agree with some version of this statement.


The above begs the question of how best to define an $n$-pivotal $n$-category. (Again, this is not standard terminology, and so far as I know no standard term for this notion has been established.) I'm aware of three approaches.


(1) Imitate the approach traditionally used for $n\le 3$; i.e. explicitly write out all the coherence diagrams. As $n$ grows large (e.g. $n\ge 4$), this quickly becomes unwieldy.


(2) Jacob Lurie's approach. First define $n$-categories with a weaker notion of duality. This weak duality allows one to define a homotopy $O(n)$ action. Now define an $n$-pivotal $n$-category to be a homotopy fixed point of this action. This is the approach described in André's answer. (I'm not an expert on this approach, so please let me know if I've misdescribed it.)


(3) "Disk-like" $n$-category approach (Section 6 of [this paper](https://arxiv.org/abs/1009.5025)). Define an $n$-category to be collection of functors on $k$-balls and homeomorphisms, for $k\le n$. The pivotal structure comes from the actions of Homeo($B^k$).


The advantage of approach #3 is that it is easy to verify for examples which are topological in origin (like bordism $n$-categories, $n$-categories built out of mapping spaces, $n$-categories built out of embedded cell complexes, mod relations). The disadvantage is that it doesn't specify any generators or relations. If you have some algebraic or combinatorial gadgets (like representations of a quantum group) and you are wondering whether they generate an $n$-pivotal $n$-category, you would like a (finite) list of relations to check. Approach #3 does not give you this, but approach #2 (or #1, if it exists) does. More specifically, if you write down in detail just what "$O(n)$ homotopy fixed point" means, you will end up with a finite list of generators and relations. (That's in theory; I haven't seen it carried out in practice.)


Quibbles with and corrections to the above are welcome.


