# closed monoidal category

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

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$\mathcal{C}$ is closed, or left closed, if the functor $A\otimes-$ on $\mathcal{C}$ has a right adjoint $[A,-]_{l}$

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$\mathcal{C}$ is right closed if the functor $-\otimes B$ on $\mathcal{C}$ has a right adjoint $[B,-]_{r}$

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$\mathcal{C}$ 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, $A\otimes B\cong B\otimes A$, so $[A,B]_{l}\cong[A,B]_{r}$. In this case, we denote the right adjoint by $[A,B]$.

Some examples:

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Any cartesian closed category is symmetric monoidal closed.

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In particular, as a category with finite products is symmetric monoidal, it is biclosed iff it is cartesian closed.

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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, $[A,B]_{l}\neq[A,B]_{r}$ in general.

more to comeβ¦

 Title closed monoidal category Canonical name ClosedMonoidalCategory Date of creation 2013-03-22 18:30:25 Last modified on 2013-03-22 18:30:25 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 6 Author CWoo (3771) Entry type Definition Classification msc 81-00 Classification msc 18-00 Classification msc 18D10 Related topic IndexOfCategories Defines left closed Defines right closed Defines biclosed Defines symmetric monoidal closed