closed monoidal category

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

•

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

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

$\mathcal{C}$ 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:

•

Any cartesian closed category is symmetric monoidal closed.
•

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

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 come…

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