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 symmetric 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 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

[a1]	M. Barr,C. Wells, "Category theory for computing science", CRM (1990) Zbl 0714.18001
[a2]	S. MacLane, "Categories for the working mathematician", Springer (1971)