Tortile Yang-Baxter operators for crossed group-categories (Q1599807)

\textit{M. C. Shum} [``Tortile tensor categories'', J. Pure Appl. Algebra 93, No.~1, 57-110 (1994; Zbl 0803.18004)] proved that the category \(T^\sim\) of tangles on ribbons is a free tortile monoidal category generated by a single object. \textit{A. Joyal} and \textit{R. Street} [``Tortile Yang- Baxter operators in tensor categories'', J. Pure Appl. Algebra 71, No.~1, 43-51 (1991; Zbl 0726.18004)] give a purely algebraic proof that this category is also the free monoidal category containing an object equipped with a tortile Yang-Baxter operator. This means that, in order to define a monoidal functor \(F:T^\sim \rightarrow V\), it is sufficient to get a tortile Yang-Baxter operator in \( V\). The starting point of the paper is the set-up of crossed G-categories in the sense of \textit{V. G. Turaev} [``Quantum invariants of knots and 3-manifolds'', de Gruyter Stud. Math. 18. (1994; Zbl 0812.57003)], introduced in the context of his programme of homotopy quantum field theory. Independently and motivated by applications to algebraic topology, \textit{P. Carrasco} and \textit{J. Martínez-Moreno} [``Categorical \(G\)-crossed modules and 2-fold extensions'', J. Pure Appl. Algebra 163, No.~3, 235-257 (2001; Zbl 0996.18003)] defined a notion of categorical G-crossed module. The latter are simply crossed G-categories which are categorical groups, i.e., monoidal groupoids with invertible objects. However, the G-action considered by Turaev is strict and the action in the concept of categorical G-crossed module need not to be strict. Categorical G-crossed modules cover the usual notions of Whitehead's crossed modules as well as Conduché's 2-crossed modules. The present paper is devoted to a study of the notion of a tortile Yang-Baxter operator in crossed G-categories. A Yang-Baxter operator on an object \(X\) is a pair of invertible arrows \(y\colon X\otimes X\rightarrow ~^XX\otimes X\) and \(z:X\rightarrow ~^XX\) satisfying the hexagonal condition \((1\otimes y)(y'\otimes 1) (1\otimes y)=(^Xy\otimes 1) (1\otimes y) (y\otimes 1)\), where \(y'=(^Xz\otimes 1)y(1\otimes z^{-1})\). The main result of the paper asserts that the crossed \(Z\)-category freely generated by an object \(X\), its left dual \(X^*\) and a tortile Yang-Baxter operator \((y,z)\) on \(X\) admits a unique braiding \(c\) and a twist \(\theta\) such that \(c_{X,X}=y\), \(\theta_X=z\) and \(\theta_{X^*}=^{X^*}(\theta_X)^*\).