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 mathematical logic and the theory of programming. For generalizations of this notion, see monoidal category. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... Computer programming (often simply programming) is the craft of implementing one or more interrelated abstract algorithms using a particular programming language to produce a concrete computer program. ... In mathematics, a monoidal category (or tensor category) is a category equipped with a binary tensor functor and a unit object . ...

Definition

The category C is called cartesian closed iff it satisfies the following three properties: ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P...

• it has a terminal object
• any two objects X and Y of C have a product X×Y in C
• any two objects Y and Z of C have an exponential Z^Y in C

For the first two conditions above, it’s the same to require that any finite (possibly empty) family of objects of C admit a product in C, because of the natural associativity of the categorical product and noticing that the empty product in a category is nothing but the terminal object of that category. In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there... In mathematics, specifically in category theory, an exponential object is the categorical equivalent of a function space in set theory. ... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there...

For the third condition it is equivalent to ask that the –×Y functor (i.e. the functor from C to C that maps objects X to X×Y and morphisms φ to φ×id_Y) has a right adjoint usually denoted –^Y. This in turn, is expressed by the existence of a bijection between the following hom-sets, natural in both X and Z : Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ... In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

Examples

Examples of cartesian closed categories include:

• The category Set of all sets, with functions as morphisms, is cartesian closed. The product X×Y is the cartesian product of X and Y, and Z^Y is the set of all functions from Y to Z. The adjointness is expressed by the following fact: the function f : X×Y → Z is naturally identified with the function g : X → Z^Y defined by g(x)(y) = f(x,y) for all x in X and y in Y.
• The category of finite sets, with functions as morphisms, is cartesian closed for the same reason.
• If G is a group, then the category of all G-sets is cartesian closed. If Y and Z are two G-sets, then Z^Y is the G-set of equivariant maps from Y to Z with trivial G action (an G-equivariant map from Y to Z is by definition an application f : Y → Z such that g.(f(y)) = f(g.y) for every g in G and y in Y).
• The category of finite G-sets is also cartesian closed.
• The category Cat of all small categories (with functors as morphisms) is cartesian closed; the exponential C^D is given by the functor category consisting of all functors from D to C, with natural transformations as morphisms.
• If C is a small category, then the functor category Set^C consisting of all covariant functors from C into the category of sets, with natural transformations as morphisms, is cartesian closed. If F and G are two functors from C to Set, then the exponential F^G is the functor whose value on the object X of C is given by the set of all natural transformations from (X,−) × G to F.
  • The earlier example of G-sets can be seen as a special case of functor categories: every group can be considered as a one-object category, and G-sets are nothing but functors from this category to Set
  • The category of all directed graphs is cartesian closed; this is a functor category as explained under functor category.
• In algebraic topology, cartesian closed categories are particularly easy to work with, and it is regrettable that neither the category of topological spaces with continuous maps nor the category of smooth manifolds with smooth maps is cartesian closed. Substitute categories have therefore been considered: the category of compactly generated Hausdorff spaces is cartesian closed, as is the category of Frölicher spaces.
• If X is a topological space, then the open sets in X form the objects of a category O(X) for witch there's a unique morphism from U to V if U is a subset of V and no morphism otherwise. This category is cartesian closed; the "product" of U and V is the intersection of U and V and the exponential U^V is the interior of U∪(XV).

The following categories are not cartesian closed: In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique type of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... This article is about the mathematical concept. ... In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. ... In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ... A diagram of a graph with 6 vertices and 7 edges. ... In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ... This page is about a higher mathematics topic. ... In topology, a compactly generated space is a topological space X satisfying the following condition: a subspace A is closed in X if and only if A ∩ K is closed in K for all compact subspaces K ⊆ X. Equivalently, one can replace closed with open in this definition. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In mathematics, Frölicher spaces extend the notions of calculus and smooth manifolds. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...

• The category of all vector spaces over some fixed field is not cartesian closed, neither is the category of all finite-dimensional vector spaces. While they have products (called direct sums), the product functors don't have right adjoints.
• The category of abelian groups is not cartesian closed, for the same reason.

A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the dimension of a vector space V is the cardinality (i. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...

Applications

In cartesian closed categories, a "function of two variables" (a morphism f:X×Y → Z) can always be represented as a "function of one variable" (the morphism λf:X → Z^Y). In computer science applications, this is known as currying; it has led to the realization that simply-typed lambda calculus can be interpreted in any cartesian closed category. Wikibooks Wikiversity has more about this subject: School of Computer Science Open Directory Project: Computer Science Downloadable Science and Computer Science books Collection of Computer Science Bibliographies Belief that title science in computer science is inappropriate Categories: | ... In computer science, currying is the technique of transforming a function taking multiple arguments into a function that takes a single argument (the first of the arguments to the original function) and returns a new function that takes the remainder of the arguments and returns the result. ... The lambda calculus is a formal system designed to investigate function definition, function application, and recursion. ...

Certain cartesian closed categories, the topoi, have been proposed as a general setting for mathematics, instead of traditional set theory. Sheaves were introduced into mathematics in the 1940s and, a major theme since then has been to study a space by studying sheaves on that space. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

The renowned computer scientist John Backus has advocated a variable-free notation, or Function-level programming, which in retrospect bears some similarity to the internal language of cartesian closed categories. CAML is more consciously modelled on cartesian closed categories. John Backus (born December 3, 1924) is an American computer scientist, notable as the inventor of the first high-level programming language (FORTRAN), the Backus-Naur form (BNF, the almost universally used notation to define formal language syntax), and the concept of Function-level programming. ... Function-level programming refers to one of the two contrasting programming paradigms identified by John Backus in his work on Programs as mathematical objects, the other being Value-level programming. ... CAML may mean: Categorical_Abstract_Machine_Language, a version of ML Collaborative Application Markup Language, an XML-Based markup language used with the Microsoft SharePoint collaborative portal application. ...

Equational theory

In every cartesian closed category (using exponential notation), (X^Y)^Z and (X^Z)^Y are isomorphic for all objects X, Y and Z. We write this as the "equation"

(x^y)^z = (x^z)^y

What other such equations are valid in all cartesian closed categories? It turns out that all of them follow logically from the following axioms:

• x×(y×z) = (x×y)×z
• x×y = y×x
• x×1 = x (here 1 denotes the terminal object of C)
• 1^x = 1
• x¹ = x
• (x×y)^z = x^z×y^z
• (x^y)^z = x^(y×z)