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.
l{C}
\left[- -\right]:l{C}op x l{C}\tol{C}
with left Yoneda arrows
L:\left[B C\right]\to\left[\left[A B\right]\left[A C\right]\right]
natural in
B
C
A
I
l{C}
iA:A\cong\left[I A\right]
and a dinatural transformation
jA:I\to\left[A A\right]
all satisfying certain coherence conditions.
I