In a braided monoidal category C we consider Hopf bimodules and crossed modules over a braided Hopf algebra H. We show that both categories are equivalent. It is discussed that the category of Hopf bimodule bialgebras coincides up to isomorphism with the category of bialgebra projections over H. Using these results we generalize the Radford-Majid criterion and show that bialgebra cross products over the Hopf algebra H are precisely described by H-crossed module bialgebras. In specific braided monoidal...

Let C be a coalgebra over an arbitrary field K. We show that the study of the category C-Comod of left C-comodules reduces to the study of the category of (co)representations of a certain bicomodule, in case C is a bipartite coalgebra or a coradical square complete coalgebra, that is, C = C₁, the second term of the coradical filtration of C. If C = C₁, we associate with C a K-linear functor ${ℍ}_C:C-Comod\to H_C-Comod$ that restricts to a representation equivalence ${ℍ}_C:C-comod\to H_C-como{d}_{sp}^{•}$, where $H_C$ is a coradical square complete hereditary bipartite...

A class of stratified posets $I{*}_{\varrho }$ is investigated and their incidence algebras $KI{*}_{\varrho }$ are studied in connection with a class of non-shurian vector space categories. Under some assumptions on $I{*}_{\varrho }$ we associate with $I{*}_{\varrho }$ a bound quiver (Q, Ω) in such a way that $KI{*}_{\varrho }\simeq K(Q,\Omega )$. We show that the fundamental group of (Q, Ω) is the free group with two free generators if $I{*}_{\varrho }$ is rib-convex. In this case the universal Galois covering of (Q, Ω) is described. If in addition $I_{\varrho }$ is three-partite a fundamental domain $I^{*+\times }$ of this covering is...