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In [[category theory]], a branch of [[mathematics]], a '''closed category''' is a [[category (mathematics)|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 homomorphism]]s between two [[abelian group]]s 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]], <math>C</math>, the internal Hom can be obtained by adding additional "structure" to the ordinary Hom set, so that the [[forgetful functor]] <math>\mathcal C\to\mathbf{Set}</math> takes the internal Hom onto the external Hom. However, this need not be the case&mdash;sometimes the internal Hom functor takes on quite a different form than the external Hom functor. What remains true is that every closed category <math>\mathcal C</math> comes equipped with a (not necessarily [[faithful functor|faithful]]) [[functor]] <math>V\colon \mathcal C\to\mathbf{Set}</math> 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 category|closed monoidal categories]], where the internal Hom functor forms an [[adjoint functor|adjoint]] to the monoidal product <math>\otimes</math>. 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==
A '''closed category''' is a [[category (mathematics)|category]] <math>\mathcal{C}</math> together with the following data:
* a [[bifunctor]] <math>[-, -]\colon \mathcal{C}^{op} \times \mathcal{C} \to \mathcal{C}</math> called the '''internal Hom functor''',
* an object <math>I</math> of <math>\mathcal{C}</math> called the '''unit object''',
* a [[natural isomorphism]] <math>i_X\colon X \to [I, X]</math>,
* an [[extranatural transformation]] <math>j_X\colon I \to [X, X]</math>,
* a transformation <math>L^X_{YZ}\colon [Y,Z]\to [[X,Y],[X,Z]]</math> natural in <math>Y</math> and <math>Z</math>, and extranatural in <math>X</math>.
which satisfy the following five axioms:
#the map <math>\gamma_{XY}\colon\operatorname{Hom}(X, Y)\to \operatorname{Hom}(I, [X,Y])</math> given by <math>\gamma_{XY}(f) = [X, f]\circ j_X = [f, Y]\circ j_Y</math> is a bijection.
#<math>j_{[X,Y]} = L^X_{YY}\circ j_Y</math>,
#<math>i_{[X,Y]} = \left[j_X, [X,Y]\right]\circ L^X_{XY}</math>,
#<math>\left[X,i_Y\right] = \left[i_X,[I,Y]\right]\circ L^I_{XY}</math>,
#<math>\left[[Y,U],L^X_{YV}\right]\circ L^Y_{UV} = \left[L^X_{YU},\left[[X,Y],[X,V]\right]\right]\circ L^{[X,Y]}_{[X,U][X,V]}\circ L^X_{UV}</math>,

The first axiom says that the functor <math>V\colon\mathcal C\to\mathbf{Set}</math> given by
:<math>V = \operatorname{Hom}(I,-)</math>
maps the internal Hom object <math>[X, Y]</math> onto a set isomorphic to the external Hom set <math>\operatorname{Hom}(X,Y)</math>:
:<math>V([X,Y])\cong\operatorname{Hom}(X,Y)</math>
In other words, the bifunctor <math>V[-,-]</math> is naturally isomorphic to the external Hom functor <math>\operatorname{Hom}(-,-)</math>.

==Examples==

*[[Cartesian closed category|Cartesian closed categories]] are closed categories. In particular, any [[topos]] is closed. The canonical example is the [[category of sets]].
*[[Compact closed category|Compact closed categories]] are closed categories. The canonical example is the [[category (mathematics)|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 <math>I</math> is the monoidal unit.

==Properties==

In any closed category the following statements hold:
*<math>i_{[I,X]} = [I,i_X] : [I, X]\to\left[I,[I,X]\right]</math>. This follows purely from the fact that <math>i\colon 1_{\mathcal C}\to [I,-]</math> is a natural isomorphism.
*<math>\gamma_{XX}(\operatorname{id}_X) = j_X</math>
*<math>i_I = j_I : I\to[I,I]</math>
*<math>\gamma_{IX} = \operatorname{Hom}(I,i_X) : \operatorname{Hom}(I, X)\to\operatorname{Hom}(I,[I,X])</math>
*the [[endomorphism monoid]] of <math>I</math> is commutative

== References ==

* {{cite conference |last1=Eilenberg |first1=S. |last2=Kelly |first2=G.M. |title=Closed categories |book-title=Proceedings of the Conference on Categorical Algebra |pages=421&ndash;562 |location=La Jolla |year=1965 |publisher=Springer |publication-date=1966}}
* {{cite arXiv|eprint=0904.3137|last1=Manzyuk|first1=Oleksandr|title=Closed categories vs. Closed multicategories|class=math.CT|year=2009}}
* {{nlab|id=closed+category|title=Closed category}}

{{Portal|Category theory}}


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