A braided monoidal 2-category C is a 4D tas
^{2} (C)
with one object and one arrow. The braiding is a functor
R: (C)
(C)
-> (C)
(where
is the tensor product of **Gray**-categories
[tpgc]) which satisfies unit and
associativity axioms. Spelling this out, one gets arrows R_{A,B}:
A B ->
B A, and the
naturality axioms of one dimension lower become 2-arrows
R_{f,B}, R_{A,g}, which themselves again satisfy naturality.
In fact, R induces a pseudo-natural transformation
C _{G} C -> C,
not - ? =>
? -,
but (? -)^{~} =>
? -,
where (? -)^{~}
is the `reversal' of ? -,
obtained from it by swapping the coordinates *and introducing some
inverses* which is necessary to make it into a functor
C _{G} C -> C.
Spelling out functoriality of R in both variables one gets an
equation of the shape of Kapranov and Voevodsky's resultohedron N_{2,2}.
The two proofs of the Yang-Baxter equation in one dimension lower become
2-arrows, which Kapranov and Voevodsky call S^{+} and S^{-},
and spelling out associativity of R one gets an equation between
these 2-arrows.

A braided monoidal **Gray**-category C is a 5D tas
^{2} (C)
with one object and one arrow. The braiding is a functor
R: (C)
(C)
-> (C)
(where there is no monoidal structure on **5D-Teisi** but it is possible to
define a 5D tas C D
via a presentation) which satisfies unit and
associativity axioms. Spelling this out, one gets arrows R_{A,B}:
A B ->
B A, 2-arrows
R_{f,B}, R_{A,g}, and the
naturality axioms of one dimension lower become 3-arrows
R_{,B},
R_{A,},
R_{f,g}, which themselves satisfy naturality and functoriality.
However, the braiding does not induce a pseudo-natural transformation
C C -> C:
(? -)^{~} =>
? -,
because in defining this there is a dimension-raising horizontal composition.

One dimension higher ? - is not even reversible.

See the papers ``On braidings, syllepses, and symmetries'' and ``Localizations of transfors''.