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Which categorifications give explicit braided monoidal 2-categories with duals? This question is in response to Ben Webster's questions in recent history. The point is that given a braided monoidal 2-category with duals (other than the 2-category of tangle surfaces) an invariant of knotted surfaces can be constructed.

I've been told that Lurie's work gives examples, but I don't know where to look therein.

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5 Answers 5

Khovanov homology can be thought of as a braided monoidal 2-category with duals, i.e. a 4-category with duals where the 0- and 1-morphisms are trivial.

0-morphisms: an unmarked point

1-morphisms: an unmarked interval

2-morphisms: a disk with some points in it

3-morphisms: a tangle in a 3-ball

4-morphisms: Given tangles $T_1$ and $T_2$ with matching boundary conditions, we have a closed link $\overline{T_1}\cup T_2$ in the 4-sphere. Define $Mor(T_1 \to T_2)$ to be $Kh(\overline{T_1}\cup T_2)$, the Khovanov homology of this link.

Composition and duality in dimensions 0 through 3 are obvious, since we have geometrically defined morphisms in those dimensions. Once we have the well-known fact that surface bordisms act on Khovanov homology, it's relatively little additional work to define composition and duality for 4-morphisms.

For a few more details, see these slides from a talk.

Unfortunately from your point of view, Rasmussen showed that the invariants of knotted tori in $B^4$ that arise from this 4-category are always trivial (or rather always equal to 2, the same as an unknotted torus). On the other hand, I think the TQFT corresponding to this 4-category will provide interesting invariants of 4-manifolds (work in progress).

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Kevin, Thanks. I did not recognize you at first at UCR, and did not really get a chance to say hello. Both Ramussen and Tanako showed the 2-knot invariant from KhoHo was trivial; Tanaka did so after Shin Satoh, Masahico Saito and I did so for a large number of cases. Perhaps the question is in regard to other such constructions. Oszvath-Szabo, or the sl(3) invariant, for example. –  Scott Carter Nov 13 '09 at 17:33
    
@Scott, the fact that the invariants of 2-knots are trivial isn't that interesting. In particular, Khovanov homology gives interesting invariants 2-tangles. –  Scott Morrison Nov 20 '09 at 2:39

Just a fun think to check out if you don't know, Crane and Yetter used a braided monoidal 2 category with duals to build their state-sum model for quantum gravity. I am not at a place now to explain how or why this is relevant, but I just wanted to make you aware. Consult their papers on Arxiv for the details.

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Is that a very old paper? Does it have duals? –  Scott Carter Nov 13 '09 at 22:45
    
In some sense yes the papers are fairly old. The do have duals, and they construct the categories in question. arxiv.org/PS_cache/hep-th/pdf/9409/9409167v1.pdf and arxiv.org/abs/hep-th/9301062 are the ones I was speaking about specifically. Louis likes to say that the braided two category allows us to build the state-sum model "on the cheap". :) Hope this helps. –  B. Bischof Nov 16 '09 at 3:02
    
@B.Bischop, could you point to the page that you are thinking of? Thanks. –  Urs Schreiber May 14 '13 at 20:08
    
That's a braided monoidal category, and not a 2-category. Also, it can be shown that the Crane-Yetter invariant is a combination of Euler characteristic and signature, so it's uninteresting for either physics or mathematics. –  Turion Feb 11 at 16:16
    
There is this idea of "Hopf categories" which are supposed to be categorified Hopf algebras. Maybe these have braided monoidal 2-categories as representation categories? –  Turion Feb 11 at 16:17

One simple way of producing symmetric monoidal $(\infty,n)$-categories with all duals is to form $n$-fold spans/correspondences, hence an (∞,n)-category of spans $Span_n(\mathbf{H})$ in some ambient $\infty$-topos $\mathbf{H}$.

This is discussed around section 3.2 in Jacob Lurie's "On the classification of TFTs".

In fact in $Span_n(\mathbf{H})$ every object is fully self-dual even. For low $n$ this is spelled out a bit at the beginning of these notes here

For $X \in \mathbf{H} \hookrightarrow Span_n(\mathbf{H})$ any object, the corresponding invariant assigned to a closed framed $n$-manifold $\Sigma$ is $X^{\Pi(\Sigma)}$, where $\Pi(\Sigma) \in \infty Grpd \simeq L_{whe} sSet$ is the homotopy type of $\Sigma$ and the exponential notation denotes the powering of $\mathbf{H}$ over $\infty Grpd$.

While these are not the quantum invariants that you are looking for, this are in some precise sense the PREquantum invariants of a local field with moduli sstack $X$, before quantization. An exposition of this is in the lecture notes geometry of physics in the section on prequantum field theory

A slight variant of this (also discussed there in more detail) works as follows: for $G \in Grp(\mathbf{H})$ an abelian $\infty$-group object, also the $(\infty,n)$-category $Span_n(\mathbf{H}_{/G})$ of $n$-fold spans in the slice $\infty$-topos over $G$ is symmetric monoidal with all duals. Objects are now maps $\exp(i S) : X \to G$ and their duals are now

$$ \exp(-i S) : X \to G $$

(using the inversion operation in $G$). As the notation suggests, the manifold invariant induced by that now are prequantum fields equipped with a local action functional.

These are still not the interesting quantum invariant that you are looking for, but this is now that data which upon "quantization" should give rise to them.

For discrete higher gauge theories (Dijkgraaf-Witten-type theories) this is indicated in sections 3 and 8 of Freed-Hopkins-Lurie-Teleman.

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Let $A$ be an $E_3$-algebra, so that $A$ is an $E_2$-algebra in the category of $E_1$-algebras by Dunn additivity. The functor $$ E_1-alg \to Cat $$ $$ A \mapsto A-mod $$ is symmetric monoidal, so it will send a "banana" algebra in $E_1$-alg to a "banana" algebra in categories. In particular, the category of left $E_1$-modules over $A$ is an $E_2$-category; i.e., a braided monoidal (but not symmetric monoidal) category.

If you want the target to be 2-cats, rather than Cat, you can enhance by considering an $E_4$-algebra $A$, forget it to an $E_2$ algebra $A'$ and looking at the Morita 2-category of algebras over $A'$, or at the category of $E_2$-algebras over $A'$.

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Looking at $(\infty,2)$ rather than $2$-categories, Luries paper on the cobordism hypothesis link text provides hints on how to find examples of categories with duals. Lurie sketches in Section 4.1 how to obtain examples for TQFTs using $E_n$-algebra objects in good symmetric monoidal $(\infty,1)$ categories $\mathcal{S}$. Given such a category $\mathcal{S}$, the $(\infty,n)$-category $Alg^o_{(n)}(\mathcal{S})$ of $E_n$-algebra objects has duals (4.1.14) and thus every object of it determines an $n$-dimensional TQFT by the cobordism hypothesis.

Particular examples can be obtained using topological chiral homology (4.1.18). This generalizes usual Hochschild cohomology of associative algebras.

Passing to $E_2$-algebras this should provide examples. It is also worth looking at 4.2 where the cobordism hypothesis is explored for smaller dimensions.

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I think you mean it generalizes Hochschild homology, not cohomology. Also, the statement about $E_n$-algebras is true if you want to construct a TFT in the sense not of the tangle hypothesis. That is, $E_n$ algebras give n-categories with duals, but not braided monoidal n-categories with duals. To get the structure the OP wants, you'd rather consider a pair $(A,M)$ where $A$ is an $E_4$-algebra and $M$ is an $E_2$-algebra with an action of $A$. The associated TFT constructed from top. chi. homology (aka factorization homology) will then evaluate on 2,3,4-mflds with codim 2 embedded objects. –  Hiro Lee Tanaka May 14 '13 at 23:09
    
Thanks for clarifying, you are right, I meant homology. And yes, the statement in this form gives symmetric monoidal $(\infty, 2)$-categories which wasn't asked for. But given a $E_n$-category $\mathcal{S}$ to start with, the $E_k$-algebra objects have the structure of an $E_{n-k}$-category. Starting with a $E_4$-category and looking at $E_2$-algebras in it, we should be able to obtain a $E_2$-object in the $(\infty,1)$-category of $(\infty,2)$-categories, i.e a braided monoidal $(\infty, 2)$-category. If $\mathcal{S}=A$-Mod ($A$ an $E_4$-algebra), this should give your pair $(A,M)$. –  Zahlendreher May 15 '13 at 8:25
    
At this point, all these examples are too sophisticated for me in that I don't have the background knowledge to decipher the notation. Supposing that the examples work, then is there a scheme to color $2$-tangle diagrams and get non-trivial invariants of oriented knotted surfaces? –  Scott Carter May 18 '13 at 2:55
    
@ScottCarter, I'm not sure if you mean something more elaborate by "color", but if you just mean labeling surfaces by elements of some set, yes, you can. For every color of a 2-tangle, you can for instance assign a different E_2 algebra. If you just take usual E_2 and E_4 algebras with the simplest actions on each other, you'd get invariants of knotted surfaces that are framed; i.e., with a trivialization of their normal bundle. –  Hiro Lee Tanaka Jul 2 '13 at 22:09

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