\(\pi\)-twist and orientability in braided categories. (Q1428094)

Given a braided monoidal category \({\mathcal C}\), the reviewer [\textit{R. H. Street}, J. Pure Appl. Algebra 132, 195--206 (1998; Zbl 0934.16037)] showed that the category of objects in \({\mathcal C}\) equipped with an automorphism is balanced monoidal; then, for any balanced monoidal category, a tortile monoidal category was constructed. The present paper constructs a tortile (ribbon) monoidal category of orientable objects in \({\mathcal C}\) that seems tantalizingly similar to that of the reviewer. A cylinder is defined on any orientable object of \({\mathcal C}\) while a Möbius band requires stability of the object.