It is well known that the existence of a braiding in a monoidal category
allows many
higher structures to be built upon that foundation. These include a monoidal 2–category
–Cat of enriched categories
and functors over , a
monoidal bicategory –Mod
of enriched categories and modules, a category of operads in
and a 2–fold monoidal
category structure on .
These all rely on the braiding to provide the existence of an interchange morphism
necessary for either their structure or its properties. We ask, given a braiding on
, what
non-equal structures of a given kind from this list exist which are based upon the
braiding. For example, what non-equal monoidal structures are available on
–Cat, or
what non-equal operad structures are available which base their associative structure on the
braiding in .
The basic question is the same as asking what non-equal 2–fold monoidal structures
exist on a given braided category. The main results are that the possible 2–fold
monoidal structures are classified by a particular set of four strand braids
which we completely characterize, and that these 2–fold monoidal categories
are divided into two equivalence classes by the relation of 2–fold monoidal
equivalence.
