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) .

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''.