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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.)

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  • $\begingroup$ @Jakob: My guess is that it is possible to make up such a definition. Do I understand correctly that your question is whether such a definition has already been made by someone? Or would you like someone to try to come up with such a definition here? The problem with the latter is that such a definition is bound to be very long and to involve multiple coherence diagrams. So I don't expect it to fit into an MO answer... $\endgroup$ – André Henriques Jun 13 '12 at 15:43
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    $\begingroup$ If your 1-morphsims are framed tangles (e.g. links) in the 3-ball, then your 2-morphisms should be framed surfaces in the 4-ball, rather than Seifert surfaces, which live in the 3-ball. $\endgroup$ – Kevin Walker Jun 13 '12 at 18:07
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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.

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    $\begingroup$ How clear is it that ribbon is the analogue of pivotal and not of spherical? $\endgroup$ – Noah Snyder Jun 14 '12 at 16:35
  • $\begingroup$ How clear is it? Probably not very. I think pivotal is a more natural condition than spherical (pivotal is local, spherical is not). Also, for an n-category with trivial 0- and 1-morphisms, n-pivotal implies n-spherical (I think). By general position, any isotopy of diagrams which takes place in the n-sphere has support in an n-ball (up to 2nd order isotopy). $\endgroup$ – Kevin Walker Jun 14 '12 at 18:06
  • $\begingroup$ Why would $n$-pivotal imply $n$-spherical? It doesn't in the original case, i.e. a pivotal 3-category with trivial 1-morphisms is not a ribbon category, it's a balanced category. The ribbon axiom is an additional axiom it might or might not satisfy. (And ribbon implies spherical.) $\endgroup$ – Manuel Bärenz Jun 14 '18 at 10:15
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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.

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  • $\begingroup$ I'm confused about the relationship between the O(n) action on the space of all dualizable objects in a particular n-category, vs. the O(n) action on the space of all dualizable n-categories... That said, your answer doesn't seem to fit my intuition, just thinking about what topological structures Reshetikhin-Turaev depends on. $\endgroup$ – Noah Snyder Jun 13 '12 at 16:37
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    $\begingroup$ Should the last paragraph start "A ribbon 2-category is a 4-category ..."? $\endgroup$ – Bruce Westbury Jun 13 '12 at 17:05
  • $\begingroup$ Oic, maybe it's just that the dimensionality is funny. Let's think about pivotal fusion categories (which is the 2-dimensional analogue). Those are 2-categories (with one object), and are dualizable objects in a 3-category (of 2-categories, but you need to be careful that the morphisms are Morita enough). Hence you have an O(3) action. But you're not claiming that they're O(3) fixed points, rather that they're O(2) fixed points. This is almost right, they're SO(2) fixed points. So maybe you mean SO(n) everywhere instead of O(n)? $\endgroup$ – Noah Snyder Jun 13 '12 at 17:16
  • $\begingroup$ THe principal article on ribbon categories theory is (or was ?) sciencedirect.com/science/article/pii/002240499200039T $\endgroup$ – Buschi Sergio Jun 13 '12 at 18:33
  • $\begingroup$ I tried to take the comments into account in order to make my answer less false. I'm still largely guessing though... $\endgroup$ – André Henriques Jun 13 '12 at 19:55
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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.

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