Heavily separable cowreaths

Abstract

Motivated by an example related to the tensor algebra, a stronger version of the notion of separable functor, called heavily separable (h-separable for short), was introduced and investigated in [1]. Here we study h-coseparable coalgebras in monoidal categories with special concern with the monoidal category TA♯ of right transfer morphisms through an algebra A in a monoidal category. We characterize the h-separability of the forgetful functor from the category of entwined modules associated to a cowreath to the base category using suitable Casimir morphisms. Even if there are non trivial examples of h-coseparable coalgebras over a field [1, Theorem 4.4], here we provide non trivial examples of h-coseparable coalgebras in the monoidal category TA⊗Hop# where H=H4 is the Sweedler 4-dimensional Hopf algebra over a field k and A=Cl(α,β,γ) the Clifford algebra.