As explained in grouplike element, to an -coring with a grouplike element one associates a semi-free differential graded algebra , sometimes called its (generalized) Amitsur complex. The simplest notion of a connection for the coring is a connection for the corresponding Amitsur complex.
Let now be a -algebra and be an -coring with grouplike element and its Amitsur complex.
determines a flat connection on . Conversely, any flat connection determines a right -coaction by
This amounts to a bijection between -coactions and flat connections on . Regarding that coactions correspond to descent data in the context of comonadic descent, this gives the flat connection interpretation of such descent data. A first instance is probably Grothendieck’s identification of flat connections and the first order costratifications in Grothendieck’s theory of differential calculus on schemes (foundations of crystalline cohomology, see book by Berthelot and Ogus; cf. also Grothendieck connection).
One the other hand, one can consider more generally additive comonads, and define connections on comodules over them rather directly. Or dually one can work with connection on modules over additive monads.
Menini and Ştefan first define an intermediate notion of a quasi-connection for monads. Let be an additive category an additive monad in and an action on some object in . Then a quasi-connection on is a map such that
A quasi-connection is a connection if, in addition,
For every connection in Menini–Ştefan sense, one defines its curvature by the formula
As usually, we define a flat connection as a connection whose curvature vanishes.
In this setting one again has a bijection between flat connections and descent data.
P. Nuss, Noncommutative descent and non-abelian cohomology, -Theory 12 (1997), no. 1, 23–74.
T. Brzeziński, R. Wisbauer, Corings and comodules, London Math. Soc. Lec. Note Series 309, Cambridge 2003.
C. Menini, Talk at MSRI: Connections, symmetry operators and descent data for triples, 1999, link
C. Menini, D. Ştefan, Descent theory and Amitsur cohomology of triples, J. Algebra 266 (2003), no. 1, 261–304.
T. Brzeziński, Flat connections and (co)modules, [in:] New Techniques in Hopf Algebras and Graded Ring Theory, S Caenepeel and F Van Oystaeyen (eds), Universa Press, Wetteren, 2007 pp. 35-52 arxiv:math.QA/0608170