Classification of braids which give rise to interchange

Abstract

It is well known that the existence of a braiding in a monoidal category V allows many structures to be built upon that foundation. These include a monoidal 2-category V-Cat of enriched categories and functors over V, a monoidal bicategory V-Mod of enriched categories and modules, a category of operads in V and a 2-fold monoidal category structure on V. We will begin by focusing our exposition on the first and last in this list due to their ability to shed light on a new question. We ask, given a braiding on V, what non-equal structures of a given kind in the list exist which are based upon the braiding. For instance, what non-equal monoidal structures are available on V-Cat, or what non-equal operad structures are available which base their associative structure on the braiding in V. We demonstrate alternative underlying braids that result in an infinite family of associative structures. The external and internal associativity diagrams in the axioms of a 2-fold monoidal category will provide us with several obstructions that can prevent a braid from underlying an associative structure.