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 ℂ^t. 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.

Key Words:

• Dualization
• Monoidal bi-closed categories
• R-rings and -corings

2010 Mathematics Subject Classification:

• Primary: 16T15
• Secondary: 18D10

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 λ_C and ρ_C 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.