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[edit]
A closed category can be defined as a category with a so-called internal Hom functor
![{\displaystyle \left[-\ -\right]:{\mathcal {C}}^{op}\times {\mathcal {C}}\to {\mathcal {C}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d14d6760aed8289f5403c6d92f3572e3983fe394)
with left Yoneda arrows
![{\displaystyle L:\left[B\ C\right]\to \left[\left[A\ B\right]\left[A\ C\right]\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e917b70cc236fd807788d1b9f624318a4791a64)
natural in and and dinatural in , and a fixed object of with a natural isomorphism
![{\displaystyle i_{A}:A\cong \left[I\ A\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33b3873f8c3fa732d6c1b0a5bca2e187d24bde83)
and a dinatural transformation
- ,
![{\displaystyle j_{A}:I\to \left[A\ A\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48727e6e752f7194747a1a533c0678efc4671071)
all satisfying certain coherence conditions.
Examples[edit]
- 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 is the monoidal unit.
References[edit]
- 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
|  | |||||||||||||||||
| ||||||||||||||||||
