TITLE: Is there a notion of "ribbon 2-category"?
QUESTION [7 upvotes]: 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.)

REPLY [3 votes]: I'm currently trying to work out the details. Here is how far I got:
Pivotal categories

You can do graphical calculus on $\mathbb{R}^2$ with monoidal categories.
Boxes represent morphisms, lines are objects. You're allowed isotopies that don't upset the order of composition.
If you have duals, you can change inputs to outputs and vice versa.
In the graphical calculus, you still have to be careful not to "rotate" your boxes, though.
If you have a pivotal category, you only have to differentiate between inputs and outputs of a box up to cyclic order, that is, you can arbitrarily rotate boxes. You're allowed all isotopies now.

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.
Spherical categories
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.)
Ribbon categories
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.
Spherical 2-categories
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.)
Ribbon 2-categories
There exist explicit notions of braided 2-categories. We could define now that ribbon 2-category is a spherical 2-category with a braiding.