Classification of braids which give rise to interchange

Abstract

It is well known that the existence of a braiding in a monoidal category $\mathcal{V}$ allows many higher structures to be built upon that foundation. These include a monoidal 2–category $\mathcal{V}$–Cat of enriched categories and functors over $\mathcal{V}$, a monoidal bicategory $\mathcal{V}$–Mod of enriched categories and modules, a category of operads in $\mathcal{V}$ and a 2–fold monoidal category structure on $\mathcal{V}$. These all rely on the braiding to provide the existence of an interchange morphism $\eta$ necessary for either their structure or its properties. We ask, given a braiding on $\mathcal{V}$, 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 $\mathcal{V}$–Cat, or what non-equal operad structures are available which base their associative structure on the braiding in $\mathcal{V}$. 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.