Inmathematics, specifically incategory theory, anexponential object ormap object is thecategorical generalization of afunction space inset theory.Categories with allfinite products and exponential objects are calledcartesian closed categories. Categories (such assubcategories ofTop) without adjoined products may still have anexponential law.^[1][2]

Let${\displaystyle \mathbf{C}}$ be a category, let${\displaystyle Z}$ and${\displaystyle Y}$ beobjects of${\displaystyle \mathbf{C}}$, and let${\displaystyle \mathbf{C}}$ have allbinary products with${\displaystyle Y}$. An object$Z^Y$ together with amorphism$\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{l}:(Z^Y\times Y)\to Z$ is anexponential object if for any object${\displaystyle X}$ and morphism$g:X\times Y\to Z$ there is a unique morphism$\lambda g:X\to Z^Y$ (called thetranspose of${\displaystyle g}$) such that the following diagramcommutes:

This assignment of a unique${\displaystyle \lambda g}$ to each${\displaystyle g}$ establishes anisomorphism (bijection) ofhom-sets,$\mathrm{H}\mathrm{o}\mathrm{m}(X\times Y,Z)\cong \mathrm{H}\mathrm{o}\mathrm{m}(X,Z^Y).$

If$Z^Y$exists for all objects${\displaystyle Z,Y}$ in${\displaystyle \mathbf{C}}$, then thefunctor${\displaystyle (-{)}^Y:\mathbf{C}\to \mathbf{C}}$ defined on objects by${\displaystyle Z\mapsto Z^Y}$ and on arrows by${\displaystyle (f:X\to Z)\mapsto (f^Y:X^Y\to Z^Y)}$, is aright adjoint to the product functor${\displaystyle -\times Y}$. For this reason, the morphisms${\displaystyle \lambda g}$ and${\displaystyle g}$ are sometimes calledexponential adjoints of one another.^[3]

Alternatively, the exponential object may be defined through equations:

• Existence of${\displaystyle \lambda g}$ is guaranteed by existence of the operation${\displaystyle \lambda -}$.
• Commutativity of the diagrams above is guaranteed by the equality${\displaystyle \mathrm{\forall }g:X\times Y\to Z,\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{l}\circ (\lambda g\times {\mathrm{i}\mathrm{d}}_Y)=g}$.
• Uniqueness of${\displaystyle \lambda g}$ is guaranteed by the equality${\displaystyle \mathrm{\forall }h:X\to Z^Y,\lambda (\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{l}\circ (h\times {\mathrm{i}\mathrm{d}}_Y))=h}$.

The exponential${\displaystyle Z^Y}$ is given by auniversal morphism from the product functor${\displaystyle -\times Y}$ to the object${\displaystyle Z}$. This universal morphism consists of an object${\displaystyle Z^Y}$ and a morphism$\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{l}:(Z^Y\times Y)\to Z$.

In thecategory of sets, an exponential object${\displaystyle Z^Y}$ is the set of allfunctions${\displaystyle Y\to Z}$.^[4] The map${\displaystyle \mathrm{e}\mathrm{v}\mathrm{a}\mathrm{l}:(Z^Y\times Y)\to Z}$ is just theevaluation map, which sends the pair${\displaystyle (f,y)}$ to${\displaystyle f(y)}$. For any map${\displaystyle g:X\times Y\to Z}$ the map${\displaystyle \lambda g:X\to Z^Y}$ is thecurried form of${\displaystyle g}$:

${\displaystyle \lambda g(x)(y)=g(x,y).}$

AHeyting algebra${\displaystyle H}$ is just a boundedlattice that has all exponential objects. Heyting implication,${\displaystyle Y\Rightarrow Z}$, is an alternative notation for${\displaystyle Z^Y}$. The above adjunction results translate to implication (${\displaystyle \Rightarrow :H\times H\to H}$) beingright adjoint tomeet (${\displaystyle \wedge :H\times H\to H}$). This adjunction can be written as${\displaystyle (-\wedge Y)⊣(Y\Rightarrow -)}$, or more fully as:$${\displaystyle (-\wedge Y):H\stackrel{⟶}{\underset{⟵}{\mathrm{\top }}}H:(Y\Rightarrow -)}$$

In thecategory of topological spaces, the exponential object${\displaystyle Z^Y}$ exists provided that${\displaystyle Y}$ is alocally compactHausdorff space. In that case, the space${\displaystyle Z^Y}$ is the set of allcontinuous functions from${\displaystyle Y}$ to${\displaystyle Z}$ together with thecompact-open topology. The evaluation map is the same as in the category of sets; it is continuous with the above topology.^[5] If${\displaystyle Y}$ is not locally compact Hausdorff, the exponential object may not exist (the space${\displaystyle Z^Y}$ still exists, but it may fail to be an exponential object since the evaluation function need not be continuous). For this reason the category of topological spaces fails to becartesian closed.However, the category of locally compact topological spaces is not cartesian closed either, since${\displaystyle Z^Y}$ need not be locally compact for locally compact spaces${\displaystyle Z}$ and${\displaystyle Y}$. A cartesian closed category of spaces is, for example, given by thefull subcategory spanned by thecompactly generated Hausdorff spaces.

Infunctional programming languages, the morphism${\displaystyle \mathrm{eval}}$ is often called${\displaystyle \mathrm{apply}}$, and the syntax${\displaystyle \lambda g}$ is oftenwritten${\displaystyle \mathrm{curry}(g)}$. The morphism${\displaystyle \mathrm{eval}}$ should not be confused with theeval function in someprogramming languages, which evaluates quoted expressions.

• Closed monoidal category

• Adámek, Jiří;Horst Herrlich; George Strecker (2006) [1990].Abstract and Concrete Categories (The Joy of Cats). John Wiley & Sons.
• Awodey, Steve (2010). "Chapter 6: Exponentials".Category theory. Oxford New York: Oxford University Press.ISBN 978-0199237180.
• Mac Lane, Saunders (1998). "Chapter 4: Adjoints".Categories for the working mathematician. New York: Springer.ISBN 978-0387984032.

• Interactive Web page which generates examples of exponential objects and other categorical constructions. Written byJocelyn Paine.

• Objects (category theory)

• Articles with short description
• Short description matches Wikidata