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User:Fropuff/Drafts/Closed category

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In category theory, a branch of mathematics, a closed category is a category which possess a functor, called the internal Hom functor that behaves very much like the ordinary (external) Hom functor except that it takes values in the same category, rather than the category of sets.

For example, in the category of abelian groups, Ab, the set of all group homomorphisms between two abelian groups can be given the structure of an abelian group in a natural way, so the ordinary Hom functor can be taken to have values in Ab and not just Set. For many closed concrete categories, , the internal Hom can be obtained by adding additional "structure" to the ordinary Hom set, so that the forgetful functor takes the internal Hom onto the external Hom. However, this need not be the case—sometimes the internal Hom functor takes on quite a different form than the external Hom functor. What remains true is that every closed category comes equipped with a (not necessarily faithful) functor that maps the internal Hom onto the external Hom. For concrete closed categories this functor may, or may not, coincide with the forgetful functor.

A rich collection of examples is provided by the class of closed monoidal categories, where the internal Hom functor forms an adjoint to the monoidal product . Most examples are of this form, but sometimes it is more natural to start with the closed structure and define the monoidal product as adjoint to the internal Hom, rather than the other way around. It has been shown that every closed category can be embedded as a full subcategory of a closed monoidal category.

Definition

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A closed category is a category together with the following data:

  • a bifunctor called the internal Hom functor,
  • an object of called the unit object,
  • a natural isomorphism ,
  • an extranatural transformation ,
  • a transformation natural in and , and extranatural in .

which satisfy the following five axioms:

  1. the map given by is a bijection.
  2. ,
  3. ,
  4. ,
  5. ,

The first axiom says that the functor given by

maps the internal Hom object onto a set isomorphic to the external Hom set :

In other words, the bifunctor is naturally isomorphic to the external Hom functor .

Examples

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Properties

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In any closed category the following statements hold:

  • . This follows purely from the fact that is a natural isomorphism.
  • the endomorphism monoid of is commutative

References

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  • Eilenberg, S.; Kelly, G.M. (1965). "Closed categories". Proceedings of the Conference on Categorical Algebra. La Jolla: Springer (published 1966). pp. 421–562.
  • Manzyuk, Oleksandr (2009). "Closed categories vs. Closed multicategories". arXiv:0904.3137 [math.CT].
  • Closed category at the nLab