Galois extensions of braided tensor categories and braided crossed G-categories (Q1879644)

For a group \(G\), the monoidal category of \(G\)-sets over \(G_c\) (the monoid \(G\) with conjugation as \(G\)-action) was shown by \textit{P. J. Freyd} and \textit{D. N. Yetter} [Adv. Math. 77, 156--182 (1989; Zbl 0679.57003)] to possess an interesting braiding. Likewise, there is a braided monoidal 2-category \(\mathcal{C}r\mathcal{C}at(G)\) of categories over \(G_c\) on which \(G\) acts. It is therefore possible to consider braided pseudomonoids (or braided monoidal objects) in \(\mathcal{C}r\mathcal{C}at(G)\) in the sense of \textit{B. Day} and the reviewer [Adv. Math. 129, 99--157 (1997; Zbl 0910.18004)]. These were used by \textit{V. G. Turaev} [``Quantum invariants of knots and 3-manifolds'', Stud. Math. 18 (1994; Zbl 0812.57003)] to obtain invariants for 3-manifolds equipped with a principal \(G\)-bundle. The present paper shows that the Galois extensions of braided monoidal categories previously discussed by the author [\textit{M. Müger}, Adv. Math. 150, 151--201 (2000; Zbl 0945.18006)] are examples of such braided pseudomonoids, and deepens our understanding of the \(G\)-equivariant version of Turaev's passage from rational chiral conformal field theory, through a modular category, to a 3-manifold invariant.