Closed category

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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.


[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



[a1] S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7 Zbl 0232.18001
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Closed category. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article