In category theory, a branch of mathematics, a closed category is a special kind of category.

In a locally small category, the external hom (x, y) maps a pair of objects to a set of morphisms. So in the category of sets, this is an object of the category itself. In the same vein, in a closed category, the (object of) morphisms from one object to another can be seen as lying inside the category. This is the internal hom [x, y].

Every closed category has a forgetful functor to the category of sets, which in particular takes the internal hom to the external hom.

Definition

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A closed category can be defined as a category   with a so-called internal Hom functor

 

with left Yoneda arrows

 

natural in   and   and dinatural in  , and a fixed object   of   with a natural isomorphism

 

and a dinatural transformation

 ,

all satisfying certain coherence conditions.

Examples

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References

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  • Eilenberg, S.; Kelly, G.M. (2012) [1966]. "Closed categories". Proceedings of the Conference on Categorical Algebra. (La Jolla, 1965. Springer. pp. 421–562. doi:10.1007/978-3-642-99902-4_22. ISBN 978-3-642-99902-4.
  • Closed category at the nLab