We study Lie bialgebroid crossed modules which are pairs of algebroid in duality that canonically give rise to bialgebroids. A one-one correspondence between such and co-quadratic Manin triples [Formula: see text] is established, where a pair transverse Dirac structures text].

We extend categorical Morita equivalence to finite tensor categories gra\-ded by a group $G$. show that two such are graded equivalent if and only their equivariant Drinfeld centers as braided $G$-crossed categories.

This paper is prompted by work of Grandjee an and Ladra on homol-ogy crossed modules. We deene the homology of a crossed module in dimensions one and two via the equivalence of categories with group objects in groupoids. We show that for any perfect crossed module, its second homology crossed module occurs as the kernel of its universal central extension as deened by Norrie. Our rst homology is...