There is an embedding theorem on which we have put Cayley's name: if M is a monoid in a closed category then the structural coretraction M --> [M,M] into the endohom is a nice monoid map. A bicategorical version of this gives a nice module (distributor) A --|--> A^{op} #A for any (pro)monoidal V-category A. This leads to a monoidal embedding of any such A into the category of A-bimodules. (E.g. see Section 4 of Pastro-St: http://www.tac.mta.ca/tac/volumes/21/4/21-04.pdf however Brian Day also knew about these things.) So the abstract case is not so much more abstract. I think Peter Johnstone says somewhere that one view of the Abelian Category Embedding Theorem is not so much that it means we should use module-proofs to work in abelian categories but rather, when working in categories of modules, we might as well work in an abelian category. I think the same applies here for monoidal categories. The coring people I have spoken to seem quite comfortable with this development. Luckily we all have our own sources of motivation. Ross On 15/09/2008, at 10:57 PM, Joost Vercruysse wrote:
cocategory), corings provide examples of these internal cocategories, but they (usually) refer to a much more concrete situation: a coring is a co-monoid in the monoidal category of bimodules over a given (possibly non-commutative) ring, this dualizes usual ring extensions.