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.)
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.)
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). 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.
Here's a cryptic answer.
[and I hope that I now got it right -- if someone sees some more mistakes, please edit my answer]
By work of Lurie, there's an $O(n+1)$ action on the collection of all $n$-categories with duals.
A ribbon category is 3-category with one object and one 1-morphism, that is equipped with the extra structure of an $SO(3)$-homotopy fixed point.
A ribbon 2-category is 4-category with one object and one 1-morphism, that is equipped with the extra structure of an $SO(4)$-homotopy fixed point.
I'm currently trying to work out the details. Here is how far I got:
In total, this means that you have a graphical calculus where you can embed a directed graph into $\mathbb{R}^2$, label it in a way that type checks, and you don't need to worry about what's up and down.
The spherical axiom says that you are allowed to put your diagram onto $S^2 \cong \mathbb{R}^2 \cup \{\infty\}$, and you're allowed isotopies on the sphere, not just the plane. This gives you one additional move, namely isotoping across $\infty$. This implies that left and right traces are equal.
(The subtlety here is that this move may validate additional axioms in the category that don't come from isotopies on the sphere. Namely, you can "pull out" an endomorphism of the monoidal unit from within a closed diagram. This means that the graphical calculus is not complete in a naïve sense. Monoidal composition of diagrams in pivotal categories is by glueing two copies of $\mathbb{R}^2$ in an essentially unique way. But when glueing two $S^2$'s together, you have to pick a point on each to perform the connected sum, and not depending on this ambiguity validates the extra axiom.)
A ribbon category is (up to issues with too big categories such as non-fusion categories) a braided, spherical category. You can think of it as a (weak) 3-category with one 1-morphism and the appropriate duality data, or you think of it as a spherical category with extra structure, the braiding.
The subtlety we had in spherical categories now vanishes into the third dimension.
There exist graphical calculi in $\mathbb{R}^3$ for 3-categories. Higher notions of pivotality can be defined explicitly (https://arxiv.org/abs/1211.0529). A 2-spherical category should be a monoidal 2-category (basically a 3-category with one object) that satisfies a graphical calculus on $S^3$, i.e. it satisfies the additional move of isotoping a brane (corresponding to objects in the monoidal 2-category) around the point at infinity. (No such axiom needs to exist for morphisms or 2-morphisms.)
There exist explicit notions of braided 2-categories. We could define now that ribbon 2-category is a spherical 2-category with a braiding.