Weak Corings

Entwined structures (A, C, ψ) were introduced by Brzezi ski 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 de Groot and Caenepeel suggested relating 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, Δ) such that B R B is a weak coring for the various coring structures induced by μ, μ τ, Δ, and τ Δ. Moreover we will characterize weak Hopf algebras as those weak bialgebras B, which are generators for the comodules over (B R B) • 1.