nLab braided object

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

$$(\sigma \otimes V) \circ (V \otimes \sigma) \circ (\sigma \otimes V) = (V \otimes \sigma) \circ (\sigma \otimes V) \circ (V \otimes \sigma)$$

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

Related concepts

• group object
• monoid object
• virtual knot theory

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”.