Weak Corings

Entwined structures (A,C, ψ) were introduced by Brzeziński and Majid to study the interdependence of an R-algebra A and an R-coalgebra C, R a commutative ring. It turned out that this relationship can also be expressed by the fact that A⊗R C has a canonical A-coring structure. More generally weak entwined structures and their modules were studied by Caenepeel and Groot and it was suggested by Caenepeel to relate these to pre-corings. Slightly modifying this notion we introduce weak corings and develop a general theory of comodules over such corings. In particular we obtain that (A,C, ψ) is a weak entwined structure if and only if A ⊗R C is a weak A-coring (with canonical structure maps). Weak bialgebras in the sense of Böhm-Nill-Szlachányi are characterized as R-modules with an algebra and coalgebra structure (B,μ,∆) such that B ⊗R B is a weak coring for the various coring structures induced by μ, μ ◦ τ , ∆ and τ ◦∆. Moreover we will characterize weak Hopf algebras as those weak bialgebras B, which are generators for the comodules over (B ⊗R B) · 1.