The Dual Rings of an R-Coring Revisited

Abstract

It is shown that for every monoidal bi-closed category ℂ left and right dualization by means of the unit object not only defines a pair of adjoint functors, but that these functors are monoidal as functors from , the dual monoidal category of ℂ into the transposed monoidal category ℂt. We thus generalize the case of a symmetric monoidal category, where this kind of dualization is a special instance of convolution. We apply this construction to the monoidal category of bimodules over a not necessarily commutative ring R and so obtain various contravariant dual ring functors defined on the category of R-corings. It becomes evident that previous, hitherto apparently unrelated, constructions of this kind are all special instances of our construction and, hence, coincide. Finally, we show that Sweedler's Dual Coring Theorem is a simple consequence of our approach and that these dual ring constructions are compatible with the processes of (co)freely adjoining (co)units.