Given a crossed module of groupoids [Formula: see text], we construct (1) natural homomorphism from the product groupoid text] to and (2) transgression map singular cohomology nerve text]. The latter turns out be identical obtained by Tu–Xu in their study equivariant text]-theory.

Abstract For a braided fusion category $\mathcal{V}$, $\mathcal{V}$-fusion is $\mathcal{C}$ equipped with monoidal functor $\mathcal{F}:\mathcal{V} \to Z(\mathcal{C})$. Given fixed $(\mathcal{C}, \mathcal{F})$ and $G$-graded extension $\mathcal{C}\subseteq \mathcal{D}$ as an ordinary category, we characterize the enrichments $\widetilde{\mathcal{F}}:\mathcal{V} Z(\mathcal{D})$ of $\mathcal{D}$ ...

In this paper, we have introduced the category of 2-quasi crossed modules for Lie algebras and constructed a pair adjoint functors between that 2-crossed algebras.

We obtain some explicit calculations of crossed Q-modules induced from a crossed module over a normal subgroup P of Q. By virtue of theorems of Brown and Higgins, this enables the computation of the homotopy 2-types and second homotopy modules of certain homotopy pushouts of maps of classifying spaces of discrete groups.

We discuss properties of Yetter-Drinfeld modules over weak bialgebras over commutative rings. The categories of left-left, left-right, right-left and right-right Yetter-Drinfeld modules over a weak Hopf algebra are isomorphic as braided monoidal categories. Yetter-Drinfeld modules can be viewed as weak Doi-Hopf modules, and, a fortiori, as weak entwined modules. If H is finitely generated and p...

We discuss properties of Yetter-Drinfeld modules over weak bialgebras over commutative rings. The categories of left-left, left-right, right-left and right-right Yetter-Drinfeld modules over a weak Hopf algebra are isomorphic as braided monoidal categories. Yetter-Drinfeld modules can be viewed as weak Doi-Hopf modules, and, a fortiori, as weak entwined modules. If H is finitely generated and p...

We introduce the notion of 'bar category' by which we mean a monoidal category equipped with additional structure induced by complex conjugation. Examples of our theory include bimodules over a *-algebra, modules over a conventional Hopf *-algebra and modules over a more general object which call a 'quasi-*-Hopf algebra' and for which examples include the standard quantum groups u q (g) at q a ...

Braided tensor products have been introduced by the author as a systematic way of making two quantum-group-covariant systems interact in a covariant way, and used in the theory of braided groups. Here we study infinite braided tensor products of the quantum plane (or other constant Zamolodchikov algebra). It turns out that such a structure precisely describes the exchange algebra in 2D quantum ...

in this work we deal with actions of vector groupoid which is a new concept in the literature‎. ‎after we give the definition of the action of a vector groupoid on a vector space‎, ‎we obtain some results related to actions of vector groupoids‎. ‎we also apply some characterizations of the category and groupoid theory to vector groupoids‎. ‎as the second part of the work‎, ‎we define the notion...