On the structure of braided crossed G-categories (Appendix to book by Turaev)

The main aim of this appendix is to discuss, for any finite group G, a close connection between braided crossed G-categories and ribbon categories containing the representation cat-egory of G as a full braided subcategory. In fact, in the context of finite semisimple categories, this will turn out to be a bijection (modulo suitable equivalences). As an application we prove that every braided G-crossed fusion category is equivalent to a strict monoidal category with strict G-action, justifying the strictness assumption made in Chapters VI and VII of this book. In the last section we touch upon the open problem of obtaining braided crossed G-categories as ‘crossed products ’ of braided categories by finite group actions. The existence of such crossed products is shown to be equivalent to the conjectured existence of embeddings of braided fusion categories into modular categories of minimal size. The original references on which this appendix is based are [4] by Bruguières, [20, 21] by Kirillov Jr. and [24, 27, 28] by the author. For a more extensive treatment of some of the matters discussed in this appendix, cf. [12], in particular Section 4, “Equivariantization and de-equivariantization”