81-00-ClosedMonoidalCategory.tex

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\pmtitle{closed monoidal category}
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\begin{document}
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Let $\mathcal{C}$ be a monoidal category, with tensor product $\otimes$.  Then we say that 

\begin{itemize}
\item $\mathcal{C}$ is \emph{closed}, or \emph{left closed}, if the functor $A\otimes -$ on $\mathcal{C}$ has a right adjoint $[A,-]_l$
\item $\mathcal{C}$ is \emph{right closed} if the functor $-\otimes B$ on $\mathcal{C}$ has a right adjoint $[B,-]_r$
\item $\mathcal{C}$ is \emph{biclosed} if it is both left closed and right closed.
\end{itemize}

A biclosed symmetric monoidal category is also known as a \emph{symmetric monoidal closed category}.  In a symmetric monoidal closed category, $A\otimes B\cong B\otimes A$, so $[A, B]_l \cong [A,B]_r$.  In this case, we denote the right adjoint by $[A,B]$.

Some examples:
\begin{itemize}
\item Any cartesian closed category is symmetric monoidal closed.  
\item In particular, as a category with finite products is symmetric monoidal, it is biclosed iff it is cartesian closed.  
\item An example of a biclosed monoidal category that is not symmetric monoidal is the category of bimodules over a non-commutative ring.  The right adjoint of $A\times -$ is $[A,-]_l$, where $[A,B]_l$ is the collection of all left $R$-linear bimodule homomorphisms from $A$ to $B$, while the right adjoint of $-\times A$ is $[A,-]_r$, where $[A,B]_r$ is the collection of all right $R$-linear bimodule homomorphisms from $A$ to $B$.  Unless $R$ is commutative, $[A,B]_l \ne [A,B]_r$ in general.
\end{itemize}

more to come...
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