Closed category explained

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

l{C}

with a so-called internal Hom functor

\left[- -\right]:l{C}op x l{C}\tol{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

l{C}

with a natural isomorphism

iA:A\cong\left[IA\right]

and a dinatural transformation

jA:I\to\left[AA\right]

,

all satisfying certain coherence conditions.

Examples

I

is the monoidal unit.

References