B equals
B A, but there
is a natural isomorphism between these, satisfying certain axioms.
Braidings on monoidal 2-categories are usually motivated by replacing
axioms for braidings on monoidal categories by isomorphisms, which then
have to satisfy certain axioms themselves. (Some people call this
``categorification''.)
This list of axioms is quite long, has been put together by trial and
error, and does not give any clues for higher dimensions.
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''.
