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 C {\displaystyle {\mathcal {C}}} with a so-called internal Hom functor
[ − − ] : C o p × C → C {\displaystyle \left[-\ -\right]:{\mathcal {C}}^{op}\times {\mathcal {C}}\to {\mathcal {C}}}with left Yoneda arrows
L : [ B C ] → [ [ A B ] [ A C ] ] {\displaystyle L:\left[B\ C\right]\to \left[\left[A\ B\right]\left[A\ C\right]\right]}natural in B {\displaystyle B} and C {\displaystyle C} and dinatural in A {\displaystyle A} , and a fixed object I {\displaystyle I} of C {\displaystyle {\mathcal {C}}} with a natural isomorphism
i A : A ≅ [ I A ] {\displaystyle i_{A}:A\cong \left[I\ A\right]}and a dinatural transformation
j A : I → [ A A ] {\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 I {\displaystyle I} is the monoidal unit.
- 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