Closed monoidal category

A category is monoidal if it consists of the following data:

1) a category ;

2) a bifunctor ;

3) an object ; and

4) three natural isomorphisms , , such that

A1) : is natural for all and the diagram

commutes for all ;

A2) and are natural and : , : for all objects and the diagram

commutes for all ;

A3) : .

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 to be the (chosen) product of the objects and , with the terminal object; , and 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 .

Closed categories.

A monoidal category is said to be symmetric if it comes with isomorphisms : natural on such that the following diagrams all commute:

, : :

A closed category is a symmetric monoidal category in which each functor has a specified right-adjoint .

Some examples of closed monoidal categories are:

E3) the category of relations, whose objects are sets and in which an arrow is a subset ; the object is the Cartesian product of the two sets, which is not the product in this category;

E4) the subsets of a monoid (a poset, hence a category); if , are two subsets of , then is while is .

References

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