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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


It is well known that the existence of a braiding in a monoidal category V allows many higher 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. These all rely on the braiding to provide the existence of an interchange morphism η necessary for either their structure or its properties. We ask, given a braiding on 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 V–Cat, or what non-equal operad structures are available which base their associative structure on the braiding in 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.

iterated monoidal categories, enriched categories, braided categories
Mathematical Subject Classification 2000
Primary: 57M99
Received: 17 January 2007
Revised: 12 July 2007
Published: 24 September 2007
Stefan Forcey
Department of Physics and Mathematics
Tennessee State University
Nashville TN 37209
Felita Humes
Department of Mathematics
College of the Bahamas
P O Box N4912 Nassau