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 $U\\_q(\mathfrak{sl}\\_2)$](http://arxiv.org/abs/0803.3652). It is a 2-category, but different hom-sets can be more-or-less identified.



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


