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\begin{document}

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\section*{cotensor product}

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

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_tensor_product}{Relation to tensor product}\dotfill \pageref*{relation_to_tensor_product} \linebreak
\noindent\hyperlink{ForComodulesOverCommutativeHopfCoalgebroids}{For comodules over commutative Hopf coalgebroids}\dotfill \pageref*{ForComodulesOverCommutativeHopfCoalgebroids} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Given a [[monoidal category]] $\mathcal{M}$ and a [[coalgebra]] $C$ in $\mathcal{M}$ denote by $\mathcal{M}^{C}$ ($resp. {}^{C}\mathcal{M}$) the category of right (resp. left) ${C}$-[[comodules]]; similarly for an algebra $E$, denote by ${}_E\mathcal{M}$ (resp. $\mathcal{M}_E$) the [[category of modules|category of left E-modules]] (right $E$-modules). If the monoidal category is [[symmetric monoidal category|symmetric]] or there is instead an appropriate [[distributive law]], then there are extensions of this notation to [[bimodules]], [[bicomodules]], relative [[Hopf modules]], entwined modules etc. e.g. Write ${}_E\mathcal{M}^B$ for left-right relative $(E,B)$-Hopf modules where $E$ is a $B$-comodule algebra over a bialgebra $B$.

Let $k$ be a commutative unital [[ring]], and let $\mathcal{M}$ be $k$-linear (in particular it has [[zero morphisms]]).

\begin{defn}
\label{}\hypertarget{}{}
Given a [[coalgebra]] $C$ in $\mathcal{M}$, a left $C$-[[comodule]] $(N,\rho_N \colon N\to N\otimes C)$, a right $C$-comodule $(M,\rho_M \colon M\to C\otimes M)$, their \textbf{cotensor product} is an object in $\mathcal{M}$ given by the [[kernel]]

\begin{displaymath}
N \Box M  
  \coloneqq
  \mathrm{ker} (\rho_N \otimes \mathrm{id}_M - \mathrm{id}_N \otimes \rho_M ).
\end{displaymath}
\end{defn}
If [[equalizers]] exist in $\mathcal{M}$, this formula extends to a [[bifunctor]]

\begin{displaymath}
{}\Box = \Box^{C}  
  \colon
\mathcal{M}^{C} \times {}^{C}\mathcal{M} \rightarrow \mathcal{M}
  \,.
\end{displaymath}
If $B$ is a [[bialgebra]] in $\mathcal{M}$ and $E$ is a right $B$-comodule algebra then the same formula defines a bifunctor

\begin{displaymath}
\Box 
    \colon 
  {}_{E}\mathcal{M}^{B} \times {}^{B}\mathcal{M} \rightarrow
{}_{E}\mathcal{M}
  \,.
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_tensor_product}{}\subsubsection*{{Relation to tensor product}}\label{relation_to_tensor_product}

Let now $\mathcal{M}=({}_k\mathrm{Mod},\otimes_k)$ be the symmetric monoidal category of $k$-modules.

Let $D$ be another $k$-coalgebra, with coproduct $\Delta_C$. If $D$ is [[flat module|flat]] as a $k$-module (e.g. $k$ is a field), and $N$ a left $D$- right $C$-bicomodule, then the cotensor product $N \Box M$ is a $D$-subcomodule of the [[tensor product]] $N \otimes_k M$. In particular, under the flatness assumption, if $\pi \colon D \rightarrow C$ is a surjection of coalgebras then $D$ is a left $D$- right $C$-bicomodule via $\Delta_D$ and $(\id \otimes \pi) \circ \Delta_D$ respectively, hence $\mathrm{Ind}^D_C \coloneqq D \Box^C -$ is a functor from left $C$- to left $D$-comodules called the [[induced comodule|induction]] functor for left comodules from $C$ to $D$.

\hypertarget{ForComodulesOverCommutativeHopfCoalgebroids}{}\subsubsection*{{For comodules over commutative Hopf coalgebroids}}\label{ForComodulesOverCommutativeHopfCoalgebroids}

\begin{prop}
\label{LeftComodulesToRightComodules}\hypertarget{LeftComodulesToRightComodules}{}
Consider a [[commutative Hopf algebroid]] $\Gamma$ over $A$ (\href{commutative+Hopf+algebroid#CommutativeHopfAlgebroidDefinitionInExplicitComponents}{def.}). Any left comodule $N$ over $\Gamma$ (\href{commutative+Hopf+algebroid#CommutativeHopfAlgebroidComodule}{def.}) becomes a right comodule via the coaction

\begin{displaymath}
N
    \overset{\Psi}{\longrightarrow}
  \Gamma \otimes_A N
    \overset{\simeq}{\longrightarrow}
  N \otimes_A \Gamma
    \overset{id \otimes_A c}{\longrightarrow}
  N \otimes_A \Gamma
  \,,
\end{displaymath}
where the isomorphism in the middle the is [[braiding]] in $A Mod$ and where $c$ is the conjugation map of $\Gamma$.

Dually, a right comodule $N$ becoomes a left comodule with the coaction

\begin{displaymath}
N
    \overset{\Psi}{\longrightarrow}
  N \otimes_A \Gamma
    \overset{\simeq}{\longrightarrow}
  \Gamma \otimes_A N
    \overset{c \otimes_A id}{\longrightarrow}
  \Gamma \otimes_A N
  \,.
\end{displaymath}
\end{prop}
\begin{defn}
\label{CotensorProductOfComodules}\hypertarget{CotensorProductOfComodules}{}
Given a [[commutative Hopf algebroid]] $\Gamma$ over $A$, (\href{commutative+Hopf+algebroid#CommutativeHopfAlgebroidDefinitionInExplicitComponents}{def.}), and given $N_1$ a right $\Gamma$-comodule and $N_2$ a left comodule (\href{commutative+Hopf+algebroid#CommutativeHopfAlgebroidComodule}{def.}), then their \textbf{cotensor product} $N_1 \Box_\Gamma N_2$ is the [[kernel]] of the difference of the two coaction morphisms:

\begin{displaymath}
N_1 \Box_\Gamma N_2
    \;\coloneqq\;
  ker
  \left(
    N_1 \otimes_A N_2
    \overset{\Psi_{N_1}\otimes_{A} id - id \otimes_A \Psi_{N_2} }{\longrightarrow}
  \right)
  \,.
\end{displaymath}
If both $N_1$ and $N_2$ are left comodules, then their cotensor product is the cotensor product of $N_2$ with $N_1$ regarded as a right comodule via prop. \ref{LeftComodulesToRightComodules}.

\end{defn}
e.g. (\hyperlink{Ravenel86}{Ravenel 86, def. A1.1.4}).

\begin{example}
\label{PrimitiveElementInAComodule}\hypertarget{PrimitiveElementInAComodule}{}
Given a [[commutative Hopf algebroid]] $\Gamma$ over $A$, (\href{commutative+Hopf+algebroid#CommutativeHopfAlgebroidDefinitionInExplicitComponents}{def.}), and given $N$ a left $\Gamma$-comodule (\href{commutative+Hopf+algebroid#CommutativeHopfAlgebroidComodule}{def.}). Regard $A$ itself canonically as a right $\Gamma$-comodule Then the cotensor product

\begin{displaymath}
Prim(N) \coloneqq A \Box_\Gamma N
\end{displaymath}
is called the \textbf{[[primitive elements]]} of $N$:

\begin{displaymath}
Prim(N) = \{ n \in N \;\vert\; \Psi_N(n) = 1 \otimes n \}
  \,.
\end{displaymath}
\end{example}
\begin{prop}
\label{}\hypertarget{}{}
Given a [[commutative Hopf algebroid]] $\Gamma$ over $A$, and given $N_1, N_2$ two left $\Gamma$-comodules , then their [[cotensor product]] (def. \ref{CotensorProductOfComodules}) is commutative, in that there is an [[isomorphism]]

\begin{displaymath}
N_1 \Box N_2 \;\simeq\; N_2 \Box N_1
  \,.
\end{displaymath}
\end{prop}
(e.g. \hyperlink{Ravenel86}{Ravenel 86, prop. A1.1.5})

\begin{lemma}
\label{ComoduleHomInTermsOfCotensorProduct}\hypertarget{ComoduleHomInTermsOfCotensorProduct}{}
Given a [[commutative Hopf algebroid]] $\Gamma$ over $A$, and given $N_1, N_2$ two left $\Gamma$-comodules, such that $N_1$ is [[projective module|projective]] as an $A$-[[module]], then

\begin{enumerate}%
\item The morphism

\begin{displaymath}
Hom_A(N_1, A)
    \overset{f \mapsto (id \otimes_A f) \circ \Psi_{N_1}}{\longrightarrow}
   Hom_A(N_1, \Gamma \otimes_A A)
    \simeq
   Hom_A(N_1, \Gamma)
    \simeq
   Hom_A(N_1, A) \otimes_A \Gamma
\end{displaymath}
gives $Hom_A(N_1,A)$ the structure of a right $\Gamma$-comodule;


\item The [[cotensor product]] (def. \ref{CotensorProductOfComodules}) with respect to this right comodule structure is isomorphic to the hom of $\Gamma$-comodules:

\begin{displaymath}
Hom_A(N_1, A) \Box_\Gamma N_2
  \simeq
  Hom_\Gamma(N_1, N_2)
  \,.
\end{displaymath}
Hence in particular

\begin{displaymath}
A \Box_\Gamma N_2 
   \;\simeq\;
  Hom_\Gamma(A,N_2)
\end{displaymath}


\end{enumerate}
\end{lemma}
(e.g. \hyperlink{Ravenel86}{Ravenel 86, lemma A1.1.6})

\begin{remark}
\label{}\hypertarget{}{}
In computing the second page of $E$-[[Adams spectral sequences]], the second statement in lemma \ref{ComoduleHomInTermsOfCotensorProduct} is the key translation that makes the comodule [[Ext]]-groups on the page be equivalent to a [[Cotor]]-groups. The latter lend themselves to computation, for instance via [[Lambda-algebra]] or via the [[May spectral sequence]].

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

\begin{itemize}%
\item The [[derived functor]] of cotensoring is called \emph{[[Cotor]]}.

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

Cotensor products in [[noncommutative geometry]] appear in the role of [[space of sections]] of [[associated bundle|associated]] [[vector bundles]] of [[quantum principal bundle]]s (which in affine case correspond to [[Hopf-Galois extensions]]). See e.g.

\begin{itemize}%
\item [[Shahn Majid]], \emph{Foundations of quantum groups theory}, 2nd extended edition, paperback, Cambridge Univ. Press 2000.

\end{itemize}
For a nonaffine extension of the sections of associated quantum vector bundle, using [[localization]] theory see

\begin{itemize}%
\item [[Zoran Škoda]], \emph{Coherent states for Hopf algebras}, Lett. Math. Phys. 81 (2007), no. 1, 1--17. (\href{http://front.math.ucdavis.edu/0303.5357}{arXiv:math.QA/0303357})

\end{itemize}
In [[Hopf algebra]] theory, cotensor products appear as early as in

\begin{itemize}%
\item [[John Milnor]], [[John Moore]], On the structure of Hopf algebras. Ann. of Math. (2) 81 1965 211--264.

\end{itemize}
The authors mention that they learned the notion from Mac Lane who knew it earlier in more general contexts.

A textbook account is in

\begin{itemize}%
\item [[Doug Ravenel]], section A1.1 of \emph{[[Complex cobordism and stable homotopy groups of spheres]]}, 1986/2003

\end{itemize}
An important problem is that the cotensor product of bicomodules is in general (even for $\mathcal{M}={}_k\mathrm{Mod}$) \emph{not associative}, even up to an isomorphism.

Cotensor products play a prominent role in various treatments of [[Galois theory]] in [[noncommutative geometry]]; a particularly geometric approach is within a version of [[noncommutative algebraic geometry]] based on usage of monoidal categories, as sketched in

\begin{itemize}%
\item [[Tomasz Maszczyk]], Noncommutative geometry through monoidal categories I, \href{http://front.math.ucdavis.edu/0707.1542}{arXiv:0707.1542}

\end{itemize}
[[!redirects cotensor products]]



\end{document}