Category theory MOC

Closed category

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

1. a multifunctor [ −, −] :𝖢𝐨𝐩 ×𝖢 →𝖢 called the internal hom-functor;
2. an object 1 called the unit;
3. a natural isomorphism with components 𝜖𝑋 :𝑋 →[1,𝑋] in [[Endofunctor category|𝖢𝖢]], which may be thought of as enabling generalized elements;
4. an extranatural transformation with components 𝜄𝑋 :1 →[𝑋,𝑋], which may be thought of as the generalized element for the identity;
5. an (extra)natural transformation with components 𝐿𝑋𝑌,𝑍 :[𝑌,𝑍] →[[𝑋,𝑌],[𝑋,𝑍]], which may be thought of as encoding composition

such that

commute for any objects 𝑋,𝑌,𝑍,𝑈,𝑉 ∈𝖢, and the map defined by

𝛾:𝖢(𝑋,𝑌)→𝖢(1,[𝑋,𝑌])𝑓↦[id𝑋,𝑓](𝜄𝑋)

is a bijection.

Archetypal example: Category of sets

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

\begin{align*} (- \to -) = [-,-] = \Set : \op\Set \times \Set\to \Set \end{align*}

and the unit 1 is any singleton. Then the (extra)natural transformations are given by

𝜖𝑋:𝑋→(1→𝑋)𝑥↦(1↦𝑥)

and

𝜄𝑋:1→(𝑋→𝑋)1↦1𝑋

and

𝐿𝑋𝑌,𝑍:(𝑌→𝑍)→((𝑋→𝑌)→(𝑋→𝑍))𝑓↦(𝑔↦𝑓∘𝑔)

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 𝛾 ↩