Closed category
A category with an additional structure, thanks to which the internal Hom-functor can be used as a right-adjoint functor to the abstract tensor product.
A category is said to be closed if a bifunctor
(see Functor) and a distinguished object
have been given on it, and if it admits natural isomorphisms
![]() |
![]() |
![]() |
![]() |
such that the following conditions are satisfied: 1) the natural isomorphisms are coherent; and 2) every functor
![]() |
where is the category of sets, is representable. The representing objects are usually denoted by
, and they can be regarded as the values of the bifunctor
(the internal Hom-functor) on objects. If the bifunctor
coincides with a product and
is a right zero (terminal object) of
, then
is called a Cartesian-closed category.
The following categories are Cartesian closed: the category of sets, the category of small categories and the category of sheaves of sets over a topological space. The following categories are closed: the category of modules over a commutative ring with an identity and the category of real (or complex) Banach spaces and linear mappings with norm not exceeding one.
References
[1] | M. Bunge, Matematika , 16 : 2 (1972) pp. 11–46 MR0360082 |
[2] | F.W. Lawvere, "Introduction" F.W. Lawvere (ed.) , Toposes, algebraic geometry and logic (Dalhousic Univ., Jan. 1971) , Lect. notes in math. , 274 , Springer (1972) MR0376798 Zbl 0249.18015 |
[3] | E.J. Dubuc, "Kan extensions in enriched category theory" , Springer (1970) MR0280560 Zbl 0228.18002 |
Comments
References
[a1] | S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7 Zbl 0232.18001 |
Closed category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Closed_category&oldid=42581