Candidate definitions for "1-braided 2-category"?

Question

Recall that a braided monoidal category is a category $\mathcal C$, a functor $\otimes: \mathcal C \times \mathcal C \to \mathcal C$ satisfying some properties, and a natural isomorphism $b_{V,W}: V\otimes W \to W\otimes V$ satisfying some properties. Recall also that a monoidal category (just $\mathcal C,\otimes$ and their properties) is the same as a one-element 2-category: the objects of $\mathcal C$ become the morphisms, and the monoidal structure becomes composition.

Thus, is there a natural definition of "1-braided 2-category"? I'm calling it "1-braided" because the braiding acts on 1-morphisms (as opposed to "braided monoidal 2-category", where the braiding acts on the 0-morphisms).

I realize, of course, that if $V,W$ are morphisms of a 2-category so that $V\circ W$ is defined, then generally $W\circ V$ is not defined, so a priori asking for any relationship $V\circ W \cong W\circ V$ doesn't make sense. On the other hand, consider Aaron Lauda's categorification of $U\_q(\mathfrak{sl}\_2)$. It is a 2-category, but different hom-sets can be more-or-less identified.

Answer 1

In the Baez-Dolan periodic table a braided monoidal category is just a 2-monoidal category (that is a 3-category with one object and one 1-morphism). If you just think that a braiding means that the structure is inherently 3-dimensional, then you might just want to think about a 1-monoidal 2-category (that is a 3-category with one object).

Answer 2

As you mentioned, the commutor doesn't generally make sense if your 2-category has morphisms between distinct objects. Therefore, here's a first try at a definition: a collection of braided monoidal categories.

There is another perspective: you can fold the commutor into the structure of a 3-category with one object and one 1-morphism instead of describing it as an extra datum. Here's a more refined candidate definition: a 3-category with one object.

Answer 3

You might be interested in this paper. While the notion of "bicategory with a shadow" is not exactly what you're describing, I think it might be at least related. It is, at least, a structure on a bicategory which generalizes the notion of symmetry on a monoidal category. (It doesn't generalize braidings that aren't symmetries, though.)