#### Volume 7, issue 3 (2007)

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Classification of braids which give rise to interchange

### Stefan Forcey and Felita Humes

Algebraic & Geometric Topology 7 (2007) 1233–1274
 arXiv: math/0512165
##### Abstract

It is well known that the existence of a braiding in a monoidal category $\mathsc{V}$ allows many higher structures to be built upon that foundation. These include a monoidal 2–category $\mathsc{V}$–Cat of enriched categories and functors over $\mathsc{V}$, a monoidal bicategory $\mathsc{V}$–Mod of enriched categories and modules, a category of operads in $\mathsc{V}$ and a 2–fold monoidal category structure on $\mathsc{V}$. 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 $\mathsc{V}$, 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 $\mathsc{V}$–Cat, or what non-equal operad structures are available which base their associative structure on the braiding in $\mathsc{V}$. 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.

##### Keywords
iterated monoidal categories, enriched categories, braided categories
Primary: 57M99