nLab
braided object

A braided object in a monoidal category (C,,I)(C,\otimes,I) is an object VV equipped with an invertible morphism σ:VVVV\sigma : V\otimes V \to V\otimes V satisfying the Yang-Baxter equation

(σV)(Vσ)(σV)=(Vσ)(σV)(Vσ)(\sigma \otimes V) \circ (V \otimes \sigma) \circ (\sigma \otimes V) = (V \otimes \sigma) \circ (\sigma \otimes V) \circ (V \otimes \sigma)

More generally, if σ\sigma is not necessarily invertible but still satisfies the Yang-Baxter equation, then (V,σ)(V,\sigma) is called a pre-braided object.

References

  • Victoria Lebed, Objets tressés: une étude unificatrice de structures algébriques et une catégorification des tresses virtuelles, Thèse, Université Paris Diderot, 2012. (pdf) Note that the title is in French (“Braided objects: a unifying study of algebraic structures and a categorification of virtual braids”) but the main text of the thesis is in English.

  • Victoria Lebed, Categorical Aspects of Virtuality and Self-Distributivity, Journal of Knot Theory and its Ramifications, 22 (2013), no. 9, 1350045, 32 pp. (doi) According to the author, arXiv:1206.3916 is “an extended version of the above JKTR publication, containing in particular a chapter on free virtual shelves and quandles”.

Created on December 22, 2016 at 10:00:59. See the history of this page for a list of all contributions to it.