In this paper we examine on a pair of adjoint functors (ϕ^* ,ϕ_*)for subcategory the category crossed modules over commutative algebras where ϕ_*: XMod/P → XMod/Q, induced, and ϕ^*:XMod/Q XMod/P, pullback (co-induced), which enables us to move from Q-modules P-modules by an algebra morphism ϕ : P Q. We show that functor (ϕ^*,ϕ_*) makes p∶ XMod k-Alg into bi- fibred k-Alg, algebras, p is given p...

A braided monoidal category may be considered a $3$-category with one object and $1$-morphism. In this paper, we show that, more generally, $3$-categories $1$-morphisms given by elements of group $G$ correspond to $G$-crossed categories, certain mathematical structures which have emerged as important invariants low-dimensional quantum field theories. More precisely, that the 4-category $\mathca...

Double-bosonisation associates to a braided group in the category of modules of a quantum group, a new quantum group. We announce the semiclassical version of this inductive construction.

We compute the representation-theoretic rank of a finite dimensional quasi-Hopf algebra H and of its quantum double D(H), within the rigid braided category of finite dimensional left D(H)-modules.