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In category theory, a branch of mathematics, a closed category is a special kind of
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
. In a
locally small category In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows as ...
, the ''external hom'' (''x'', ''y'') maps a pair of objects to a set of
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphis ...
s. So in the
category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition ...
, 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 In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given sig ...
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 Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
\mathcal with a so-called
internal Hom functor In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory ...
: \left \ -\right: \mathcal^ \times \mathcal \to \mathcal with left Yoneda arrows : L : \left \ C\right\to \left left[A\ B\right\left[A\ C\right">\_B\right.html" ;"title="left[A\ B\right">left[A\ B\right\left[A\ C\rightright] natural transformation, natural in B and C and dinatural transformation, dinatural in A, and a fixed object I of \mathcal with a
natural isomorphism In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a nat ...
: i_A : A \cong \left \ A\right/math> and a dinatural transformation : j_A : I \to \left \ A\right/math>, all satisfying certain coherence conditions.


Examples

*
Cartesian closed categories In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in ma ...
are closed categories. In particular, any
topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notio ...
is closed. The canonical example is the
category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition ...
. * Compact closed categories are closed categories. The canonical example is the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
FdVect with finite-dimensional
vector spaces In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but c ...
as objects and linear maps as morphisms. *More generally, any monoidal closed category is a closed category. In this case, the object I is the monoidal unit.


References

* * {{Category theory