# 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 |

[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) |

[3] | E.J. Dubuc, "Kan extensions in enriched category theory" , Springer (1970) |

#### Comments

#### References

[a1] | S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7 |

**How to Cite This Entry:**

Closed category.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Closed_category&oldid=16972