The bicategories of corings. (Q2369024)

The basic example of a bicategory is Bim, with rings as \(0\)-cells, bimodules as \(1\)-cells and bimodule maps as \(2\)-cells. In the present paper, a bicategory with corings as \(0\)-cells is studied. Actually, this bicategory can be obtained starting from Bim, using a construction due to \textit{S. Lack} and \textit{R. Street} [J. Pure Appl. Algebra 175, No. 1-3, 243-265 (2002; Zbl 1019.18002)]. The explicit description of this category \(\text{REM}(\text{Bim})\) is given in Section 2. \(1\)-cells are represented by bimodules together with a certain structure maps. \(\text{fREM}(\text{Bim})\) is the category with the same \(0\)-cells as \(\text{REM}(\text{Bim})\), but where the \(1\)-cells are required to be finitely generated and projective on the right side. Left handed versions \(\text{LEM}(\text{Bim})\) and \(\text{fLEM}(\text{Bim})\) are given, and a duality result asserts that there is a duality between the hom-categories of \(\text{fLEM}(\text{Bim})\) and \(\text{fREM}(\text{Bim})\). Given a \(B\)-coring \(\mathcal D\) and a \((B,A)\)-bimodule \(\Sigma\) that is finitely generated and projective as a right \(A\)-module, we can construct a new coring \(\Sigma[{\mathcal D}]=\Sigma^*\otimes_B{\mathcal D}\otimes_B\Sigma\), called a base ring extension of \(\mathcal D\) by a coring. In Section 3, it is studied how separability and Frobenius properties are reflected by such base ring extensions. In Section 4, it is shown that the \(1\)-cells in \(\text{fREM}(\text{Bim})\) can also be described as so-called module morphisms between corings \(\mathcal D\) and \(\mathcal C\). To such a module morphism, we can associate a functor \({\pmb\Sigma}_o\colon{\mathcal M}^{\mathcal D}\to{\mathcal M}^{\mathcal C}\) (the push-out functor), and this has, under appropriate purity conditions, a right adjoint \({\pmb\Sigma}^o\), called the pull-back functor. It is investigated when the push-out functor is an equivalence, leading to a generalized descent theorem.