Strong connections and invertible weak entwining structures. (Q1757969)

Let \(\mathcal C\) be a strict monoidal category. A right-right weak entwining structure in \(\mathcal C\) is a triple \((A,C,\psi_R)\), where \(A\) is an algebra, \(C\) is a coalgebra and \(\psi_R\colon C\otimes A\to A\otimes C\) is a morphism satisfying some compatibility conditions. The category whose objects are \(M\) with structure of right \(A\)-module and right \(C\)-comodule and a compatibility condition is denoted by \(\mathcal M_A^C(\psi_R)\), the category of right-right weak entwined modules. If \((A,C,\psi_R)\) is a right-right weak entwining structure and there exists a coaction \(\rho_A\) such that \((A,\mu_A\rho_A)\in\mathcal M_A^C(\psi_R)\), it is possible to obtain an extension \(A^R_C\to A\) and a canonical morphism. If this morphism is an isomorphism, it is called a right \(C\)-Galois extension. The authors introduce the notion of strong connection for an invertible weak entwining structure. The main result of the paper (Theorem 3.12) establishes that the existence of a strong connection form is equivalent, under certain conditions, to the bijectivity of the canonical morphism of a weak Galois extension associated to a weak invertible entwining structure. An explicit formula of a strong connection for an invertible weak entwining structure with coseparable coalgebra is obtained, generalizing the formula by \textit{E. J. Beggs} and \textit{T. Brzeziński} [Appl. Categ. Struct. 16, No. 1-2, 57-63 (2008; Zbl 1195.16030)].