Braidings can also be defined by observing that a braided monoidal category is a tricategory with (one object and) one arrow, from which it is formally distinguished by looping and delooping operations 2 and 2 respectively (to wit, one deloops a braided monoidal category to a one-arrow tricategory). Invoking the coherence theorem for tricategories, a braiding on C is essentially 0-composition in a Gray-category 2 (C). Because this composition is ``governed by'' Gray's tensor product of 2-categories, and noticing that 2 (C) (*,*) = (C), this gives braiding as a (2-)functor
R: (C) G (C) -> (C) .
The axioms on a braiding are now the familiar axioms for composition: functoriality in each variable, and associativity ((pseudo-)naturality of R is already contained in it being a functor).Now this definition is easily generalized to higher dimensions: calling the structures to which weak n-categories are equivalent n-dimensional teisi - these are hypothetical above dimension 5, and the equivalence is hypothetical above dimension 3 - one defines a braided nD tas as an (n+2)D tas with one arrow, and assuming that there is a tensor product of teisi ``governing'' composition, the braiding is - again - a functor (C) (C) -> (C), which is (natural,) functorial, and associative.
See the paper ``On braidings, syllepses, and symmetries''.