Category theory MOC

Closed category

A closed category is a category with objects resembling hom-sets. #m/def/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 , 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

https://q.uiver.app/#q=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

commute for any objects , and the map defined by

is a bijection.

Archetypal example:

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.

#state/tidy | #lang/en | #SemBr

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 ↩