nLab Ehresmann coring

Given a coalgebra-Galois extension $U\hookrightarrow E$ of a $k$-algebra $U$, which is the appropriate generalization of a Hopf-Galois extension, where $E$ is faithfully flat over the base $U$ as a left $E$-module, one constructs a coring, Ehresmann coring, out of these data. Its role is somewhat analogous to the gauge groupoid (see Atiyah Lie groupoid), and in Hopf-Galois case it is an intermediate stage in constructing another analogue, (Ehresmann-)Schauenburg bialgebroid, see there.

• T. Brzeziński, R. Wisbauer, Corings and comodules, London Math. Soc. Lec. Note Series 309, Cambridge 2003.