closed monoidal category

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

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

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

• $𝒞$ 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, ${[A,B]}_l\ne {[A,B]}_r$ 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