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{C} with a so-called internal Hom functor
\left[-\ -\right] : \mathcal{C}^{op} \times \mathcal{C} \to \mathcal{C}
with left Yoneda arrows
L : \left[B\ C\right] \to \left[\left[A\ B\right] \left[A\ C\right]\right]
natural in B and C and dinatural in A, and a fixed object I of \mathcal{C} with a natural isomorphism
i_A : A \cong \left[I\ A\right]
and a dinatural transformation
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
Iis 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.