Closed category

Template:Short descriptionIn category theory, a branch of mathematics, a closed category is a special kind of category.

In a locally small, 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 $𝒞$ with a so-called internal Hom functor

$[--]:{𝒞}^{op}\times 𝒞\to 𝒞$

with left Yoneda arrows

$L:\left[BC\right]\to \left[\left[AB\right]\left[AC\right]\right]$

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

$i_A:A\cong \left[IA\right]$

and a dinatural transformation

$j_A:I\to \left[AA\right]$,

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

• Template:Cite book
• Template:Nlab