Let $({\cal C},\otimes)$ be a monoidal category, $X$ an object in ${\cal C}$, and $\Psi:X \otimes X \to X \otimes X$ an isomorphism such that $\Psi$ satisfies the braid relation: $$ (\Psi \otimes \text{id}) \circ (\text{id} \otimes \Psi) \circ (\Psi \otimes \text{id}) = (\text{id} \otimes \Psi) \circ (\Psi \otimes \text{id}) \circ (\text{id} \otimes \Psi). $$ What would one call such an isomorphism? The most obvious suggestion is to call it a braiding for $X$. Might this be taken to imply that $\Psi$ comes from a braiding for the category (which I do not want to assume)?
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Assuming S. Carnahan's surmise in his comment is correct, I believe the correct term for this is "YangBaxter" operator in a monoidal category (or, you could call an object $X$ equipped with such an automorphism $R: X \otimes X \to X \otimes X$ a YangBaxter object). This terminology is given in the seminal paper on the subject, Braided Tensor Categories by Joyal and Street (Adv. Math. 102, pp. 2078, 1993). In particular, as observed by Joyal and Street, the braid category can be characterized as the free (i.e., initial in a 2categorical sense) monoidal category equipped with a YangBaxter object. Edit: Another reference for this terminology:


