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 $\mathcal$ with a so-called internal Hom functor

: \left \ -\right: \mathcal^ \times \mathcal \to \mathcal

with left Yoneda arrows

: L : \left \ C\right\to \left left[A\ B\right\left[A\ C\right">\_B\right.html" ;"title="left[A\ B\right">left[A\ B\right\left[A\ C\rightright]

natural transformation, natural in $B$ and $C$ and dinatural transformation, dinatural in $A$, and a fixed object $I$ of $\mathcal$ with a natural isomorphism

: i_A : A \cong \left \ A\right/math>

and a dinatural transformation

: j_A : I \to \left \ A\right/math>,

all satisfying certain coherence conditions.

Examples

* Cartesian closed categories are closed categories. In particular, any topos is closed. The canonical example is the category of sets.
* Compact closed categories are closed categories. The canonical example is the category FdVect with finite-dimensional vector spaces as objects and linear maps as morphisms.
*More generally, any monoidal closed category is a closed category. In this case, the object $I$ is the monoidal unit.

References

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{{Category theory