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) |
Closed monoidal category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Closed_monoidal_category&oldid=42386