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, $[A,B]_l \ne [A,B]_r$ in general.

more to come...