View source for Closed monoidal category

A [[category]] $\mathcal{C}$ is monoidal if it consists of the following data:

1) a category $\mathcal{C}$;

2) a [[bifunctor]] $\otimes : \mathcal{C}\times\mathcal{C}\rightarrow\mathcal{C}$;

3) an object $e\in\mathcal{C}$; and

4) three natural isomorphisms $\alpha,\lambda,\rho$ such that

A1) $\alpha_{a,b,c} : a \otimes (b \otimes c) \cong (a \otimes b) \otimes c$ is natural for all $a,b,c \in \mathcal{C}$ and the diagram
$$
\begin{array}{ccccc}
a \otimes (b \otimes (c \otimes d)) & \stackrel{\alpha}{\rightarrow} & (a \otimes b) \otimes (c \otimes d) &  \stackrel{\alpha}{\rightarrow} & ((a \otimes b) \otimes c) \otimes d \\
          \downarrow\mathrm{id}\otimes\alpha & & & & \uparrow \alpha\otimes\mathrm{id} \\
a \otimes ((b \otimes c) \otimes d) &  & \stackrel{\alpha}{\rightarrow} &  & (a \otimes (b \otimes c)) \otimes d
\end{array}
$$
commutes for all $a,b,c,d \in \mathcal{C}$;

A2) $\lambda$ and $\rho$ are natural and $\lambda : e \otimes a \cong a$, $\rho : a \otimes e \cong a$ for all objects $a \in \mathcal{C}$ and the diagram
$$
\begin{array}{ccc}
a \otimes (e \otimes c) & \stackrel{\alpha}{\rightarrow} & (a \otimes e) \otimes c \\
\downarrow\mathrm{id}\otimes\lambda & & \downarrow\rho\otimes\mathrm{id} \\
a \otimes c & = & a \otimes c
\end{array}
$$
commutes for all $a.c \in \mathcal{C}$;

A3) $\lambda_e = \rho_e : e \otimes e \rightarrow e$.

These axioms imply that all such diagrams commute.

Some examples of monoidal categories are:

E1) any category with finite products is monoidal if one takes $a\otimes b$ to be the (chosen) product of the objects $a$ and $b$, with $e$ the [[terminal object]]; $\alpha,\lambda,\rho$ are the unique isomorphisms that commute with the appropriate projections;

E2) the usual  "tensor products"  give monoidal categories — whence the notation. Note that one cannot identify all isomorphic objects in $\mathcal{C}$.

==Closed categories.==
A monoidal category $\mathcal{C}$ is said to be ''symmetri''c if it comes with isomorphisms $\gamma_{a.b} : a \otimes b \cong b \otimes a$, natural on $a,b \in \mathcal{C}$ such that the following diagrams all commute:
$$
\gamma_{a,b} \circ \gamma_{b,a} = \mathrm{id}\,;
$$
$$
\rho_b = \lambda_b \circ \gamma_{b,e} : b\otimes e \cong b\,;
$$
$$
\begin{array}{ccccc}
a \otimes (b \otimes c) & \stackrel{\alpha}{\rightarrow} & (a \otimes b) \otimes c & \stackrel{\gamma}{\rightarrow} & c \otimes (a \otimes b) \\
\downarrow_{\mathrm{id}\otimes\gamma} & & & & \downarrow_\alpha \\
a \otimes (c \otimes b) & \stackrel{\alpha}{\rightarrow} & (a \otimes c) \otimes b & \stackrel{\gamma\otimes\mathrm{id}}{\rightarrow} & (c \otimes a) \otimes b
\end{array}
$$

A '''closed category''' $\mathcal{V}$ is a symmetric monoidal category in which each functor ${-}\otimes b: \mathcal{V} \rightarrow \mathcal{V}$ has a specified [[Adjoint functor|right-adjoint]] $({-})^b : \mathcal{V} \rightarrow \mathcal{V}$.

Some examples of closed monoidal categories are:

E3) the category $\mathsf{Rel}$ of relations, whose objects are sets $a,b,c,\ldots$ and in which an arrow $\sigma:a\rightarrow b$ is a subset $\sigma \subseteq a \otimes b$, the object $a \otimes b$ being the [[Cartesian product]] of the two sets (which is not the product in this category);

E4) the subsets of a monoid $M$ (partially ordered by inclusion, hence a category); if $A$, $B$ are two subsets of $M$, then $A \otimes B$ is $\{ab : a \in A,\,b \in B\}$ while $C^A$ is $\{b \in M : ab\in C\ \text{for all}\ a \in A\}$.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Barr,C. Wells, "Category theory for computing science", CRM (1990) {{ZBL|0714.18001}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> S. MacLane, "Categories for the working mathematician", Springer (1971)</TD></TR>
</table>

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