Closed category

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

A closed category can be defined as a category ${\displaystyle {\mathcal {C}}}$ with a so-called internal Hom functor

${\displaystyle \left[-\ -\right]:{\mathcal {C}}^{op}\times {\mathcal {C}}\to {\mathcal {C}}}$ ,

with left Yoneda arrows natural in ${\displaystyle B}$ and ${\displaystyle C}$ and dinatural in ${\displaystyle A}$

${\displaystyle L:\left[B\ C\right]\to \left[\left[A\ B\right]\left[A\ C\right]\right]}$

and with a fixed object ${\displaystyle I}$ of ${\displaystyle {\mathcal {C}}}$ such that there is a natural isomorphism

${\displaystyle i_{A}:A\cong \left[I\ A\right]}$
${\displaystyle j_{A}:I\to \left[A\ A\right].\,}$

References

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