TITLE: Braided Monoidal 2-categories with duals
QUESTION [7 upvotes]: 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.

REPLY [5 votes]: 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 $A'$.