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

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

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

${\displaystyle i_{A}:A\cong \left[I\ A\right]}$

and a dinatural transformation

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

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 ${\displaystyle I}$ is the monoidal unit.

References

• Eilenberg, S.; Kelly, G.M. (2012) [1966]. "Closed categories". Proceedings of the Conference on Categorical Algebra. (La Jolla, 1965. Springer. pp. 421–562. doi:10.1007/978-3-642-99902-4_22. ISBN 978-3-642-99902-4.
• Closed category at the nLab