closed monoidal category
Let $\mathrm{\pi \x9d\x92\x9e}$ be a monoidal category, with tensor product^{} $\beta \x8a\x97$. Then we say that

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$\mathrm{\pi \x9d\x92\x9e}$ is closed, or left closed, if the functor^{} $A\beta \x8a\x97$ on $\mathrm{\pi \x9d\x92\x9e}$ has a right adjoint ${[A,]}_{l}$

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$\mathrm{\pi \x9d\x92\x9e}$ is right closed if the functor $\beta \x8a\x97B$ on $\mathrm{\pi \x9d\x92\x9e}$ has a right adjoint ${[B,]}_{r}$

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$\mathrm{\pi \x9d\x92\x9e}$ is biclosed if it is both left closed and right closed.
A biclosed symmetric monoidal category is also known as a symmetric monoidal closed category^{}. In a symmetric monoidal closed category, $A\beta \x8a\x97B\beta \x89\x85B\beta \x8a\x97A$, so ${[A,B]}_{l}\beta \x89\x85{[A,B]}_{r}$. In this case, we denote the right adjoint by $[A,B]$.
Some examples:

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Any cartesian closed category is symmetric monoidal closed.
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An example of a biclosed monoidal category that is not symmetric monoidal is the category of bimodules over a noncommutative ring. The right adjoint of $A\Gamma \x97$ 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 $\Gamma \x97A$ 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^{}, ${[A,B]}_{l}\beta \x89{[A,B]}_{r}$ in general.
more to comeβ¦
Title  closed monoidal category 
Canonical name  ClosedMonoidalCategory 
Date of creation  20130322 18:30:25 
Last modified on  20130322 18:30:25 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  6 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 8100 
Classification  msc 1800 
Classification  msc 18D10 
Related topic  IndexOfCategories 
Defines  left closed 
Defines  right closed 
Defines  biclosed 
Defines  symmetric monoidal closed 