Yetter-Drinfeld modules over weak braided Hopf monoids and center categories (Q2452404)

A Yetter-Drinfeld category is a special type of braided monoidal category. It consists of modules over a Hopf algebra which satisfy some additional axioms. Weak Hopf algebras are generalizations of Hopf algebras in the category of vector spaces. They can be extended to the context of monoidal category where the algebras are replaced by monoids. Roughly speaking, a weak braided Hopf monoid in a strict monoidal category is a monoid-comonoid with a weak Yang-Baxter operator, satisfying some compatibility conditions. A classical result states that for a Hopf monoid \(H\), the center of the category of left \(H\)-modules is equivalent to the braided monoidal category. Moreover if the Hopf monoid is finite then the braided monoidal category is equivalent to the category of modules over the Drinfeld double of \(H\). The result can be extended to the case of weak Hopf monoids. The paper under review considers a weak braided Hopf monoid \(D\) in a strict monoidal category \(C\) with split idempotents and studies a suitable analogue to the above equivalence result.