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\begin{document}

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\section*{Sweedler coring}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory}

[[!include (infinity,1)-topos - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_components}{In components}\dotfill \pageref*{in_components} \linebreak
\noindent\hyperlink{GeomInterpretationOfCoring}{Geometric interpretation}\dotfill \pageref*{GeomInterpretationOfCoring} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_ring_extensions}{Relation to ring extensions}\dotfill \pageref*{relation_to_ring_extensions} \linebreak
\noindent\hyperlink{DescentIntermsOfCoringModules}{Descent in terms of coring comodules}\dotfill \pageref*{DescentIntermsOfCoringModules} \linebreak
\noindent\hyperlink{ComonadicDescent}{In terms of (co)monadic descent}\dotfill \pageref*{ComonadicDescent} \linebreak
\noindent\hyperlink{GeomInterpretationOfDescent}{Geometric interpretation}\dotfill \pageref*{GeomInterpretationOfDescent} \linebreak
\noindent\hyperlink{RelationToGeneralizedCohomology}{Relation to generalized cohomology and Adams spectral sequence}\dotfill \pageref*{RelationToGeneralizedCohomology} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A Sweedler [[coring]] is an algebraic structure that is roughly the [[duality|formal dual]] of the [[?ech nerve]] of a [[cover]]: it is used to describe [[descent]] in algebraic contexts.

See also [[monadic descent]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{in_components}{}\subsubsection*{{In components}}\label{in_components}

Let $f : R \hookrightarrow S$ be the extension of associative unital $k$-[[associative algebra|algebra]]s (where $k$ is a commutative unital [[ring]]).

The corresponding \textbf{canonical coring} or \textbf{Sweedler coring} is the $S$-[[coring]]

\begin{displaymath}
C = S\otimes_R S
\end{displaymath}
with coproduct

\begin{displaymath}
\Delta : C\to C\otimes_S C \cong S\otimes_R S\otimes_R S
\end{displaymath}
given by

\begin{displaymath}
\Delta: s_1\otimes s_2 \mapsto s_1\otimes 1 \otimes s_2
\end{displaymath}
and counit

\begin{displaymath}
\epsilon : C\to S
\end{displaymath}
given by

\begin{displaymath}
\epsilon: s_1 \otimes s_2 \mapsto s_1 s_2
 \,.
\end{displaymath}
The element $1\otimes 1$ is a [[grouplike element]] in the Sweedler's coring.

\hypertarget{GeomInterpretationOfCoring}{}\subsubsection*{{Geometric interpretation}}\label{GeomInterpretationOfCoring}

We give a dual geometric interpretation of the Sweedler coring.

Suppose a context of spaces and function algebras on spaces that satisfies the basic axioms of [[geometric function theory]], in that the algebra of functions $C(Y_1 \times_X Y_2)$ on a [[fiber product]]

\begin{displaymath}
\itexarray{
    Y_1 \times_X Y_2 &\to& Y_2
    \\
    \downarrow && \downarrow
    \\
    Y_1 &\to& X
  }
\end{displaymath}
is the [[tensor product]] of the functions on the factors:

\begin{displaymath}
C(Y_1 \times_X Y_2) = C(Y_1) \otimes_{C(X)} C(Y_2)
  \,.
\end{displaymath}
Then let $\pi : Y \to X$ be a morphism of spaces and set

\begin{displaymath}
R := C(X)
\end{displaymath}
and

\begin{displaymath}
S = C(Y)
\end{displaymath}
and

\begin{displaymath}
(R \hookrightarrow S) := \pi^* : C(X) \to C(Y)
  \,.
\end{displaymath}
The morphism $\pi$ induces its augmented [[?ech nerve]]

\begin{displaymath}
\left(
     \cdots
     \stackrel{\to}{\stackrel{\to}{\stackrel{\to}{\to}}}
     Y \times_X Y \times_X Y
     \stackrel{\to}{\stackrel{\to}{\to}}
     Y \times_X Y \stackrel{\to}{\to}
     Y
     \stackrel{\pi}{\to}
     X
  \right)
  \,.
\end{displaymath}
Taking function algebras of this yields, by the above,

\begin{displaymath}
\left(
     \cdots
     \stackrel{\leftarrow}{\stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}}}
     S \otimes_R S \otimes_R S
     \stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}}
     S \otimes_R S 
     \stackrel{\leftarrow}{\leftarrow}
     S
     \stackrel{\pi^*}{\leftarrow}
     R
  \right)
  \,.
\end{displaymath}
Writing again $C = S \otimes_R S$ for the Sweedler coring, this is

\begin{displaymath}
\left(
     \cdots
     \stackrel{\leftarrow}{\stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}}}
     C \otimes_S C
     \stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}}
     C 
     \stackrel{\leftarrow}{\leftarrow}
     S
     \stackrel{\pi^*}{\leftarrow}
     R
  \right)
  \,.
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_ring_extensions}{}\subsubsection*{{Relation to ring extensions}}\label{relation_to_ring_extensions}

Various properties of canonical coring correspond to adequate properties of the ring extension. For example, [[coseparable coring|coseparable]] Sweedler corings correspond to [[split extension]]s (the $k$-algebra extension $R\to S$ is split if there is an $R$-bimodule map $h: S\to R$ with $h(1_S) = 1_R$).

\hypertarget{DescentIntermsOfCoringModules}{}\subsubsection*{{Descent in terms of coring comodules}}\label{DescentIntermsOfCoringModules}

Given a morphism $f : R \to S$ with corresponding Sweedler coring $(C = S \otimes_R S,\Delta,\epsilon)$ as above, the category of [[descent]] data $\mathrm{Desc}(S/R)$ for the categories of right modules along $k$-algebra extension $R\to S$ is precisely the category of right $C$-[[comodule]]s.

In other words, the objects of $\mathrm{Desc}(S/R)$ are the pairs $(N,\alpha)$ where $N$ is a right $S$-[[module]], and $\alpha: N\to N\otimes_R S$ is a right $S$-module morphism and if we write $\alpha(m) = \sum_i m_i \otimes s_i$ then

\begin{itemize}%
\item $\sum_i \alpha(m_i)\otimes s_i = \sum_i m_i\otimes 1\otimes s_i$,


\item $\sum_i m_i s_i = m$.



\end{itemize}
\hypertarget{ComonadicDescent}{}\paragraph*{{In terms of (co)monadic descent}}\label{ComonadicDescent}

This [[coring]]-formulation of [[descent]] may be understood as special case of [[comonadic descent]] (see also the discussion at [[Bénabou?Roubaud theorem]]). See e.g. (\hyperlink{Hess10}{Hess 10, section 2}) for a review. We spell this out in a bit more detail:

The [[bifibration]] in question is

$p :$ [[Mod]] $\to$ [[CRing]]

that sends an object in the category [[Mod]] of [[module]]s to the ring that it is a module over.

A descent datum for a morphism $f : R \to S$ with respect to this bifibration is a (co)algebra object over the co[[monad]] $f_* f^*$ induced by this morphism. We have that

\begin{itemize}%
\item the morphism $f_*$ sends an $R$-module $N$ to the $S$-module $N \otimes_R S$;


\item the morphism $f^*$ sends an $S$-module $P$ to the $R$-module $P \otimes_S S_R$, where $S_R$ is $S$ regarded as a left $S$- and a right $R$-module. So $P \otimes_S S_R$ is just the $S$-module $P$ with only the right $R$-action remembered.



\end{itemize}
Accordingly, the comonad with underlying functor $f_* f^*$ sends an $S$-module $P$ to the $S$-module $P \otimes_S S \otimes_R S = P \otimes_S C$.

A (co)algebra object for this comonad is hence a co-action morphism

\begin{displaymath}
P \to P \otimes_S C
\end{displaymath}
compatible with the monad action. This is precisely a comodule over the Sweedler coring, as defined above.

\hypertarget{GeomInterpretationOfDescent}{}\paragraph*{{Geometric interpretation}}\label{GeomInterpretationOfDescent}

Descent for Sweedler corings is a special case of [[monadic descent|comonadic descent]]. We describe this in detail and relate it by duality to the geometrically more intuitive \href{http://ncatlab.org/nlab/show/monadic+descent#ForCodomainFibs}{monadic descent for codomain fibrations}.

Assuming again a suitable geometric context as above, we may identify a [[module]] over $R = C(X)$ with (the collection of [[section]]s of) a [[vector bundle]] (or rather a suitable generalization of that: a [[coherent sheaf]]) over $X$. Similarly for $Y$. So we write

\begin{displaymath}
Vec(X) := R Mod
\end{displaymath}
and

\begin{displaymath}
Vec(Y) := S Mod
\end{displaymath}
for the corresponding [[category|categories]] of [[module]]s. The assignment of such categories to spaces

\begin{displaymath}
Vec : Z \mapsto Vec(Z)
\end{displaymath}
extends to a contravariant [[pseudofunctor]]

\begin{displaymath}
Vec : Spaces^{op} \to Cat
\end{displaymath}
by assigning to a morphism $f : Y \to X$ of spaces the corresponding [[functor]]

\begin{displaymath}
Vec(X) \simeq C(X) Mod \stackrel{- \otimes_{f} C(Y)}{\to} C(Y) Mod \simeq Vec(Y)
  \,.
\end{displaymath}
This way $Vec$ becomes a [[stack|prestack]] of categories on our category of spaces.

If this prestack satisfies [[descent]] along suitable [[cover]]s, it is a [[stack]].

Geometrically this is the case if for each morphism $\pi : Y\to X$ that is regarded as a [[cover]], the category $Desc(Y,Vec)$ whose objects are tuples consisting of

\begin{itemize}%
\item an [[object]] $a \in Vec(Y)$


\item an [[isomorphism]] $g : \pi_1^* a \to \pi_2^* a$


\item such that

\begin{displaymath}
\itexarray{
     && \pi_2^* a
     \\
     & {}^{\mathllap{\pi_{12}^* g}}\nearrow && \searrow^{\mathrlap{\pi_{23}^* g}}
     \\
     \pi_1^* a &&\stackrel{\pi_{13}^*}{\to}&& \pi_3^*
  }
\end{displaymath}
commutes.



\end{itemize}
Morphism are defined similarly (see [[stack]] and [[descent]] for details).

To get the geometric pucture that underlies, by duality, the above comodule definition of descent, we need to reformulate this just a little bit more:

every ordinary [[vector bundle]] $E \to X$ (of finite rank) is the [[associated bundle]] $E \simeq P \times_{O(n)} V$ of an [[orthogonal group|O(n)]]-[[principal bundle]] $P \to X$, and as such its [[section]]s may be identified with $O(n)$-equivariant functions $P \to V \simeq \mathbb{R}^n$ on the total space of $P$.

Using this we may think of the $C(X)$-module of sections of $E$ as a submodule of the $C(X)$-module of all functions on $P$

\begin{displaymath}
\Gamma(E) \subset C(P)
  \,.
\end{displaymath}
We now reformulate the geometric descent for [[vector bundle]]s in terms of geometric descent for their underlying [[principal bundle]]s, and then take functions on everything in sight to obtain the comodule definition of descent that we want to describe:

A descent datum (transition function) for a [[principal bundle]] $Q \to Y$ may be thought of as the the morphism $g$ in the double [[pullback]] diagram

\begin{displaymath}
\itexarray{
    &&Y\times_X Y \times_Y Q &\to& Q
    \\
    &&\downarrow && \downarrow
    \\
    &{}^{\mathllap{g}}\swarrow&Y \times_X Y &\to& Y
    \\
    &&\downarrow && \downarrow^{\mathrlap{\pi}}
    \\
    Q &\to& Y &\stackrel{\pi}{\to}& X
  }
  \,.
\end{displaymath}
Because here $Y \times_X Y \times_Y Q$ is the space whose points consist of a point in a double overlap of the cover and a point in the fiber of $Q$ over that with respect to one patch, and the morphism identifies this with a point in the fiber of $Q$ regarded as sitting over the other patch. Analogously, there is a cocycle condition on $g$ on triple overlaps.

Now, blindly applying our functor that takes functions of spaces to the above diagram yields the double [[pushout]] diagram

\begin{displaymath}
\itexarray{
    &&C(Y\times_X Y \times_Y Q) &\leftarrow& C(Q)
    \\
    &&\uparrow && \uparrow
    \\
    &{}^{\mathllap{g}^*}\nearrow&C(Y \times_X Y) &\leftarrow& C(Y)
    \\
    &&\uparrow && \uparrow^{\mathrlap{\pi^*}}
    \\
    C(Q) &\leftarrow& C(Y) &\stackrel{\pi^*}{\leftarrow}& C(X)
  }
  \,.
\end{displaymath}
We may restrict to $N := \Gamma(E) \subset C(Q)$ as just discussed and switch to the notation from above to get

\begin{displaymath}
\itexarray{
    &&N \otimes_{S} C &\leftarrow& N
    \\
    &&\uparrow && \uparrow
    \\
    &{}^{\mathllap{g}^*}\nearrow& C &\leftarrow& S
    \\
    &&\uparrow && \uparrow^{\mathrlap{\pi^*}}
    \\
    N &\leftarrow& S &\stackrel{\pi^*}{\leftarrow}& R
  }
  \,.
\end{displaymath}
The morphism

\begin{displaymath}
\alpha := g^* : N \to N \otimes_S C
\end{displaymath}
obtained this way is the co-action morphism from the above algebraic definition.

The further cocycle condition on $g$ similarly translates into the condition that $\alpha$ really satisfies the [[comodule]] property.

\hypertarget{RelationToGeneralizedCohomology}{}\subsubsection*{{Relation to generalized cohomology and Adams spectral sequence}}\label{RelationToGeneralizedCohomology}

Applied to [[E-infinity rings]] the Sweedler coring construction yields the [[Hopf algebroids]] of dual [[Steenrod algebras]] and appears in the [[Adams spectral sequence]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Amitsur complex]], [[Hopf algebroid]]


\item [[descent]]

\begin{itemize}%
\item [[cover]]


\item [[cohomological descent]]


\item [[monadic descent]],

\begin{itemize}%
\item \textbf{Sweedler coring}


\item [[monadic descent]], [[higher monadic descent]]


\item [[descent in noncommutative algebraic geometry]]



\end{itemize}


\end{itemize}

\item [[sheaf]], [[(2,1)-sheaf]], [[2-sheaf]] [[(∞,1)-sheaf]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Sweedler corings are named after [[Moss Sweedler]].

A textbook account is in

\begin{itemize}%
\item [[Tomasz Brzezinski]], [[Robert Wisbauer]], section 29 of \emph{Corings and Comodules}, Cambridge University Press, London Math. Soc. LN 309 (2003), (\href{http://www.math.uni-duesseldorf.de/~wisbauer/corinerr.pdf}{errata pdf})

\end{itemize}
Section 29 there discusses the relation to the [[Amitsur complex]] and the [[descent theorem]].

\begin{itemize}%
\item MO: \href{http://mathoverflow.net/questions/150547/how-is-a-descent-datum-the-same-as-a-comodule-structure}{how-is-a-descent-datum-the-same-as-a-comodule-structure}

\end{itemize}
Discussion in the context of ([[higher monadic descent|higher]]) [[monadic descent]] is around example 2.24 of

\begin{itemize}%
\item [[Kathryn Hess]], section 6 of \emph{A general framework for homotopic descent and codescent} (\href{http://arxiv.org/abs/1001.1556}{arXiv:1001.1556})

\end{itemize}
[[!redirects Sweedler corings]] [[!redirects Sweedler's coring]] [[!redirects Sweedler's corings]]

[[!redirects canonical coring]] [[!redirects canonical corings]] [[!redirects canonical co-ring]] [[!redirects canonical co-rings]]



\end{document}