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.
2010 Mathematics Subject Classification:
Notes
We here use Sweedler's original notation for these maps: Thus, and
here should not be mistaken for the left and right unit constraints λC and ρC in a monoidal category, \bi.e., the canonical isomorphisms mentioned above.
Here R and are considered as an R-R-bimodule and an
-
-bimodule, respectively.
Note that Ψ and ψ are invertible and, thus, provide the respective morphisms in by considering their inverses.
We use the somewhat clumsy notation I⊗R in order to stress the fact that this functor depends on the monoidal structure.