Abstract. El Kaoutit and Gómez-Torrecillas introduced comatrix corings, generalizing Sweedler's canonical coring, and proved a new version of the Faithfully Flat Descent Theorem. They also introduced Galois corings as corings isomorphic to a comatrix coring. In this paper, we further investigate this theory. We prove a new version of the Joyal-Tierney Descent Theorem, and generalize the Galois Coring Structure Theorem. We associate a Morita context to a coring with a fixed comodule, and relate it to Galois-type properties of the coring. An affineness criterion is proved in the situation where the coring is coseparable. Further properties of the Morita context are studied in the situation where the coring is (co)Frobenius.
We introduce Hopf categories enriched over braided monoidal categories. The notion is linked to several recently developed notions in Hopf algebra theory, such as Hopf group (co)algebras, weak Hopf algebras and duoidal categories. We generalize the fundamental theorem for Hopf modules and some of its applications to Hopf categories.2010 Mathematics Subject Classification. 16T05.
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