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\begin{document}

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\section*{connection for a coring}

Given an $A$-[[coring]] or even a more general additive [[comonad]] with a [[grouplike element]] there are several related (but in general nonequivalent) notions of connections.

As explained in [[grouplike element]], to an $A$-coring $C$ with a grouplike element one associates a [[semi-free differential graded algebra]] $\Omega A = \Omega(A,C)$, sometimes called its (generalized) Amitsur complex. The simplest notion of a connection for the coring $C$ is a [[connection for a differential graded algebra|connection]] for the corresponding Amitsur complex.

Let now $A$ be a $k$-algebra and $(C,\Delta,\epsilon)$ be an $A$-coring with grouplike element $g$ and $(\Omega A,d)$ its [[Amitsur complex]].

A \textbf{connection} $\nabla:M\otimes_A\Omega^\bullet\to M\otimes_A\Omega^{\bullet+1}$ on a module $M$ over a [[semifree dga]] (in the sense of the entry [[connection for a differential graded algebra]]) is determined by its value $\nabla_M|_M$ on $M\cong M\otimes_A A$. If $\rho^M:M\to M\otimes_A C$ is a right $C$-[[coaction]] then the formula

\begin{displaymath}
\nabla|_M:m\mapsto \rho^M(m)-m\otimes g
\end{displaymath}
determines a [[flat connection]] on $M$. Conversely, any flat connection determines a right $C$-coaction by

\begin{displaymath}
\rho^M(m)=\nabla(m)+m\otimes g.
\end{displaymath}
This amounts to a \emph{bijection between $C$-coactions and flat connections} on $M$. 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]]).

\hypertarget{connections_on_comodules_directly}{}\subsection*{{Connections on comodules directly}}\label{connections_on_comodules_directly}

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 monad]]s.

Menini and tefan first define an intermediate notion of a quasi-connection for monads. Let $A$ be an [[additive category]] $(T,\mu,\eta)$ an additive monad in $A$ and $\nu:TM\to M$ an action on some object $M$ in $A$. Then a \textbf{quasi-connection} on $M$ is a map $\nabla:M\to TM$ such that

\begin{displaymath}
\nabla \circ \nu - \mu\circ T(\nabla) = id_{TM} -  \eta\circ\nu: TM\to TM.
\end{displaymath}
A quasi-connection is a \textbf{connection} if, in addition,

\begin{displaymath}
\nu\circ\nabla = 0.
\end{displaymath}
For every connection in Menini--tefan sense, one defines its \textbf{curvature} $F_\nabla : M\to T^2 M$ by the formula

\begin{displaymath}
F_\nabla := (id_{T^2 M} - \eta_{TM}\circ\mu_M)\circ T(\nabla)\circ\nabla.
\end{displaymath}
As usually, we define a \textbf{flat connection} as a connection whose curvature vanishes.

In this setting one again has a bijection between flat connections and descent data.

\hypertarget{literature_and_links}{}\subsection*{{Literature and links}}\label{literature_and_links}

\begin{itemize}%
\item P. Nuss, Noncommutative descent and non-abelian cohomology, $K$-Theory 12 (1997), no. 1, 23--74.


\item [[T. Brzeziński]], R. Wisbauer, \emph{Corings and comodules}, London Math. Soc. Lec. Note Series \textbf{309}, Cambridge 2003.


\item C. Menini, Talk at MSRI: \emph{Connections, symmetry operators and descent data for triples}, 1999, \href{http://www.msri.org/communications/ln/msri/1999/hopfalg/menini/1/index.html}{link}


\item C. Menini, D. tefan, \emph{Descent theory and Amitsur cohomology of triples}, J. Algebra \textbf{266} (2003), no. 1, 261--304.


\item [[T. Brzeziński]], \emph{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 \href{http://arxiv.org/abs/math.QA/0608170}{arxiv:math.QA/0608170}



\end{itemize}
[[!redirects connection for coring]]



\end{document}