Category theory MOC

Closed category

A closed category is a category with objects resembling hom-sets. cat Explicitly, a closed category is equipped with¹²

1. a multifunctor called the internal hom-functor;
2. an object called the unit;
3. a natural isomorphism with components in [[Endofunctor category|]], which may be thought of as enabling generalized elements;
4. an extranatural transformation with components , which may be thought of as the generalized element for the identity;
5. an (extra)natural transformation with components , which may be thought of as encoding composition

such that

commute for any objects , and the map defined by

is a bijection.

Archetypal example: Category of sets

In the internal hom-functor is the ordinary Hom-functor

and the unit is any singleton. Then the (extra)natural transformations are given by

and

and

A Closed monoidal category is a category which is also monoidal in a compatible way.

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Footnotes

1. 1966. Closed categories, §I.2, pp. 428–430. Note the refined definition uses only CC1–4 ↩

2. 1977. Embedding of Closed Categories Into Monoidal Closed Categories, §1, p. 86. Refines the original definition with CC5, which guarantees the bijection ↩