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Concept
Closed category
Summary
In , a branch of mathematics, a closed category is a special kind of .
In a , the external hom (x, y) maps a pair of objects to a set of s. So in the , 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.
A closed category can be defined as a with a so-called internal Hom functor
with left Yoneda arrows
natural in and and dinatural in , and a fixed object of with a natural isomorphism
and a dinatural transformation
all satisfying certain coherence conditions.
are closed categories. In particular, any topos is closed. The canonical example is the .
are closed categories. The canonical example is the FdVect with finite-dimensional vector spaces as objects and linear maps as morphisms.
More generally, any is a closed category. In this case, the object is the monoidal unit.
Official source
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