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In
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, cat ...
, a category is Cartesian closed if, roughly speaking, any
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, 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 Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal s ...
and the theory of programming, in that their internal language is the simply typed lambda calculus. They are generalized by closed monoidal categories, whose internal language, linear type systems, are suitable for both
quantum In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizat ...
and classical computation.

# Etymology

Named after (1596–1650), French philosopher, mathematician, and scientist, whose formulation of analytic geometry gave rise to the concept of
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ ...
, which was later generalized to the notion of categorical product.

# Definition

The category ''C'' is called Cartesian closed
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
it satisfies the following three properties: * 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 Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above * Exponential decay, decrease at a rate proportional to value * E ...
''Z''''Y'' in ''C''. The first two conditions can be combined to the single requirement that any finite (possibly empty) family of objects of ''C'' admit a product in ''C'', because of the natural
associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacemen ...
of the categorical product and because the
empty product In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question ...
in a category is the terminal object of that category. The third condition is equivalent to the requirement that the functor – ×''Y'' (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 In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
, usually denoted –''Y'', for all objects ''Y'' in ''C''. For locally small categories, this can be expressed by the existence of a bijection between the
hom-set 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, morphism ...
s :$\mathrm\left(X\times Y,Z\right) \cong \mathrm\left(X,Z^Y\right)$ which is natural in both ''X'' and ''Z''. Take care to note that a Cartesian closed category need not have finite limits; only finite products are guaranteed. If a category has the property that all its slice categories are Cartesian closed, then it is called ''locally cartesian closed''. Note that if ''C'' is locally Cartesian closed, it need not actually be Cartesian closed; that happens if and only if ''C'' has a terminal object.

# Basic constructions

## Evaluation

For each object ''Y'', the counit of the exponential adjunction is a natural transformation :$\mathrm_ : Z^Y \times Y \to Z$ called the (internal) evaluation map. More generally, we can construct the partial application map as the composite :$\mathrm_ : Z^ \times X \cong \left(Z^Y\right)^ \times X \xrightarrow Z^Y.$ In the particular case of the category Set, these reduce to the ordinary operations: :$\mathrm_\left(f,y\right) = f\left(y\right).$

## Composition

Evaluating the exponential in one argument at a morphism ''p'' : ''X'' → ''Y'' gives morphisms :$p^Z : X^Z \to Y^Z,$ :$Z^p : Z^Y \to Z^X,$ corresponding to the operation of composition with ''p''. Alternate notations for the operation ''p''''Z'' include ''p''* and ''p∘-''. Alternate notations for the operation ''Z''''p'' include ''p''* and ''-∘p''. Evaluation maps can be chained as :$Z^Y \times Y^X \times X \xrightarrow Z^Y \times Y \xrightarrow Z$ the corresponding arrow under the exponential adjunction :$c_ : Z^Y \times Y^X \to Z^X$ is called the (internal) composition map. In the particular case of the category Set, this is the ordinary composition operation: :$c_\left(g,f\right) = g \circ f.$

## Sections

For a morphism ''p'':''X'' → ''Y'', suppose the following pullback square exists, which defines the subobject of ''X''''Y'' corresponding to maps whose composite with ''p'' is the identity: :$\begin \Gamma_Y\left(p\right) &\to& X^Y \\ \downarrow & & \downarrow \\ 1 &\to& Y^Y \end$ where the arrow on the right is ''p''''Y'' and the arrow on the bottom corresponds to the identity on ''Y''. Then Γ''Y''(''p'') is called the object of sections of ''p''. It is often abbreviated as Γ''Y''(''X''). If Γ''Y''(''p'') exists for every morphism ''p'' with codomain ''Y'', then it can be assembled into a functor Γ''Y'' : ''C''/''Y'' → ''C'' on the slice category, which is right adjoint to a variant of the product functor: : $\hom_\left(X \times Y \xrightarrow Y, Z \xrightarrow Y\right) \cong \hom_C\left(X, \Gamma_Y\left(p\right)\right).$ The exponential by ''Y'' can be expressed in terms of sections: : $Z^Y \cong \Gamma_Y\left(Z \times Y \xrightarrow Y\right).$

# Examples

Examples of Cartesian closed categories include: * The category Set of all sets, with
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orien ...
s 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 curried 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 Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which ...
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 set of all functions from ''Y'' to ''Z'' with ''G'' action defined by (''g''.''F'')(''y'') = F(''g''−1.y) for all ''g'' in ''G'', ''F'':''Y'' → ''Z'' 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 In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object in t ...
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 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 particular, the category of
simplicial set In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined ...
s (which are functors ''X'' : Δop → Set) is Cartesian closed. * Even more generally, every elementary topos is Cartesian closed. * In algebraic topology, Cartesian closed categories are particularly easy to work with. Neither the category of
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
s with continuous maps nor the category of
smooth manifolds In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...
with smooth maps is Cartesian closed. Substitute categories have therefore been considered: the category of
compactly generated Hausdorff space In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space ''X'' is compactly generated if it satisfies the following condition: :A subs ...
s is Cartesian closed, as is the category of Frölicher spaces. * In order theory, complete partial orders (''cpo''s) have a natural topology, the Scott topology, whose continuous maps do form a Cartesian closed category (that is, the objects are the cpos, and the morphisms are the Scott continuous maps). Both currying and ''
apply In mathematics and computer science, apply is a function that applies a function to arguments. It is central to programming languages derived from lambda calculus, such as LISP and Scheme, and also in functional languages. It has a role in ...
'' are continuous functions in the Scott topology, and currying, together with apply, provide the adjoint. * A
Heyting algebra In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of ''i ...
is a Cartesian closed (bounded) lattice. An important example arises from topological spaces. If ''X'' is a topological space, then the
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are s ...
s in ''X'' form the objects of a category O(''X'') for which there is a unique morphism from ''U'' to ''V'' if ''U'' is a subset of ''V'' and no morphism otherwise. This poset is a Cartesian closed category: the "product" of ''U'' and ''V'' is the intersection of ''U'' and ''V'' and the exponential ''U''''V'' is the interior of . * A category with a
zero object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...
is Cartesian closed if and only if it is equivalent to a category with only one object and one identity morphism. Indeed, if 0 is an initial object and 1 is a final object and we have $0 \cong 1$, then $\mathrm\left(X, Y\right) \cong \mathrm\left(1, Y^X\right) \cong \mathrm\left(0, Y^X\right) \cong 1$ which has only one element. **In particular, any non-trivial category with a zero object, such as an
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abel ...
, is not Cartesian closed. So the category of modules over a ring is not Cartesian closed. However, the functor tensor product $-\otimes M$ with a fixed module does have a
right adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
. The tensor product is not a categorical product, so this does not contradict the above. We obtain instead that the category of modules is monoidal closed. Examples of locally Cartesian closed categories include: * Every elementary topos is locally Cartesian closed. This example includes Set, ''FinSet'', ''G''-sets for a group ''G'', as well as Set''C'' for small categories ''C''. * The category ''LH'' whose objects are topological spaces and whose morphisms are local homeomorphisms is locally Cartesian closed, since ''LH/X'' is equivalent to the category of sheaves . However, ''LH'' does not have a terminal object, and thus is not Cartesian closed. * If ''C'' has pullbacks and for every arrow ''p'' : ''X'' → ''Y'', the functor ''p''* : ''C/Y'' → ''C/X'' given by taking pullbacks has a right adjoint, then ''C'' is locally Cartesian closed. * If ''C'' is locally Cartesian closed, then all of its slice categories ''C/X'' are also locally Cartesian closed. Non-examples of locally Cartesian closed categories include: * Cat is not locally Cartesian closed.

# 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 λ''f'' : ''X'' → ''Z''''Y''). In
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...
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. The Curry–Howard–Lambek correspondence provides a deep isomorphism between intuitionistic logic, simply-typed lambda calculus and Cartesian closed categories. Certain Cartesian closed categories, the topoi, have been proposed as a general setting for mathematics, instead of traditional
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
. 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.

# Dependent sum and product

Let ''C'' be a locally Cartesian closed category. Then ''C'' has all pullbacks, because the pullback of two arrows with codomain ''Z'' is given by the product in ''C/Z''. For every arrow ''p'' : ''X'' → ''Y'', let ''P'' denote the corresponding object of ''C/Y''. Taking pullbacks along ''p'' gives a functor ''p''* : ''C/Y'' → ''C/X'' which has both a left and a right adjoint. The left adjoint $\Sigma_p : C/X \to C/Y$ is called the dependent sum and is given by composition $p \circ \left(-\right)$. The right adjoint $\Pi_p : C/X \to C/Y$ is called the dependent product. The exponential by ''P'' in ''C/Y'' can be expressed in terms of the dependent product by the formula $Q^P \cong \Pi_p\left(p^*\left(Q\right)\right)$. The reason for these names is because, when interpreting ''P'' as a dependent type $y : Y \vdash P\left(y\right) : \mathrm$, the functors $\Sigma_p$ and $\Pi_p$ correspond to the type formations $\Sigma_$ and $\Pi_$ respectively.

# 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''. One may ask 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 *''x''1 = ''x'' *(''x''×''y'')''z'' = ''x''''z''×''y''''z'' *(''x''''y'')''z'' = ''x''(''y''×''z'')

## Bicartesian closed categories

Bicartesian closed categories extend Cartesian closed categories with binary coproducts and an
initial object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...
, with products distributing over coproducts. Their equational theory is extended with the following axioms, yielding something similar to Tarski's high school axioms but with a zero: *''x'' + ''y'' = ''y'' + ''x'' *(''x'' + ''y'') + ''z'' = ''x'' + (''y'' + ''z'') *''x''×(''y'' + ''z'') = ''x''×''y'' + ''x''×''z'' *''x''(''y'' + ''z'') = ''xy×xz'' *0 + ''x'' = ''x'' *''x''×0 = 0 *''x''0 = 1 Note however that the above list is not complete; type isomorphism in the free BCCC is not finitely axiomatizable, and its decidability is still an open problem.

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