category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, a branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 asso ...

, the ''external hom'' (''x'', ''y'') maps a pair of objects to a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

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, morphisms a ...

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 of m ...

, 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 sign ...

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

\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 ...

: \mathcal^ \times \mathcal \to \mathcal

with left

Yoneda arrow In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewi ...

s :

\left[A\_C\right.html" ;"title="\_B\right.html" ;"title="left[A\ B\right">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

\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 natur ...

i_A : A \cong \left \ A\right /math>

and a

dinatural transformation In category theory, a branch of mathematics, a dinatural transformation \alpha between two functors :S,T : C^\times C\to D, written :\alpha : S\ddot\to T, is a function that to every object c of C associates an arrow :\alpha_c : S(c,c)\to T(c ...

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

. * Compact closed categories are closed categories. The canonical example is the

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 can ...

as objects and

linear maps In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...

as morphisms. *More generally, any

monoidal closed category In mathematics, especially in category theory, a closed monoidal category (or a ''monoidal closed category'') is a category (mathematics), category that is both a monoidal category and a closed category in such a way that the structures are compati ...

is a closed category. In this case, the object

I

is the monoidal unit.

References

* * {{Category theory