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2007 Classification of braids which give rise to interchange
Stefan Forcey, Felita Humes
Algebr. Geom. Topol. 7(3): 1233-1274 (2007). DOI: 10.2140/agt.2007.7.1233


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.


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Stefan Forcey. Felita Humes. "Classification of braids which give rise to interchange." Algebr. Geom. Topol. 7 (3) 1233 - 1274, 2007.


Received: 17 January 2007; Revised: 12 July 2007; Published: 2007
First available in Project Euclid: 20 December 2017

zbMATH: 1140.57011
MathSciNet: MR2350281
Digital Object Identifier: 10.2140/agt.2007.7.1233

Primary: 57M99

Rights: Copyright © 2007 Mathematical Sciences Publishers


Vol.7 • No. 3 • 2007
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