[[Category theory MOC]]
# Closed category

A **closed category** is a category with objects resembling hom-sets. #m/def/cat 
Explicitly, a closed category $\cat C$ is equipped with[^1966][^1977]

1. a [[multifunctor]] $[-,-] : \op{\cat C} \times \cat C \to \cat C$ called the **internal hom-functor**;
2. an object $1$ called the **unit**;
3. a [[Natural isomorphism|natural isomorphism]] with components $\epsilon_{X} : X \to [1,X]$ in [[Endofunctor category|$\cat C^{\cat C}$]], which may be thought of as enabling generalized elements;
4. an [[(Extra)natural transformation|extranatural transformation]] with components $\iota_{X} : 1 \to [X,X]$, which may be thought of as the generalized element for the identity;
5. an [[(extra)natural transformation]] with components $L^X_{Y,Z} : [Y,Z]\to [[X,Y],[X,Z]]$,
   which may be thought of as encoding composition

such that

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commute for any objects $X,Y,Z,U,V \in \cat C$,
and the map defined by
$$
\begin{align*}
\gamma : \cat C(X,Y) &\to \cat C(1, [X,Y]) \\
f &\mapsto [\id_{X},f](\iota_{X})
\end{align*}
$$
is a [[Surjectivity, injectivity, and bijectivity|bijection]].

  [^1966]: 1966\. [[Sources/@eilenbergClosedCategories1966|Closed categories]], §I.2, pp. 428–430. Note the refined definition uses only CC1–4
  [^1977]: 1977\. [[Sources/@laplazaEmbeddingClosedCategories1977|Embedding of Closed Categories Into Monoidal Closed Categories]], §1, p. 86. Refines the original definition with CC5, which guarantees the bijection $\gamma$


> [!tip]- 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
> $$
> \begin{align*}
> \epsilon_{X} : X &\to (1 \to X) \\
> x &\mapsto (1 \mapsto x)
> \end{align*}
> $$
> and
> $$
> \begin{align*}
> \iota_{X} : 1 &\to (X \to X) \\
> 1 &\mapsto 1_{X}
> \end{align*}
> $$
> and
> $$
> \begin{align*}
> L^X_{Y,Z} : (Y \to Z) &\to ((X \to Y) \to (X \to Z)) \\
> f &\mapsto (g \mapsto  f \circ g)
> \end{align*}
> $$

A [[Monoidal closed category]] is a category which is also monoidal in a compatible way.

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