In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. Categories (such as subcategories of Top) without adjoined products may still have an exponential law.^[1][2]

Definition

Let ${\displaystyle \mathbf {C} }$ be a category, let ${\displaystyle Z}$ and ${\displaystyle Y}$ be objects of ${\displaystyle \mathbf {C} }$, and let ${\displaystyle \mathbf {C} }$ have all binary products with ${\displaystyle Y}$. An object ${\textstyle Z^{Y}}$ together with a morphism ${\textstyle \mathrm {eval} \colon (Z^{Y}\times Y)\to Z}$ is an exponential object if for any object ${\displaystyle X}$ and morphism ${\textstyle g\colon X\times Y\to Z}$ there is a unique morphism ${\textstyle \lambda g\colon X\to Z^{Y}}$ (called the transpose of ${\displaystyle g}$) such that the following diagram commutes:

This assignment of a unique ${\displaystyle \lambda g}$ to each ${\displaystyle g}$ establishes an isomorphism (bijection) of hom-sets, ${\textstyle \mathrm {Hom} (X\times Y,Z)\cong \mathrm {Hom} (X,Z^{Y}).}$

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

Equational definition

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 ${\displaystyle \forall g\colon X\times Y\to Z,\ \mathrm {eval} \circ (\lambda g\times \mathrm {id} _{Y})=g}$.
• Uniqueness of ${\displaystyle \lambda g}$ is guaranteed by the equality ${\displaystyle \forall h\colon X\to Z^{Y},\ \lambda (\mathrm {eval} \circ (h\times \mathrm {id} _{Y}))=h}$.

Universal property

The exponential ${\displaystyle Z^{Y}}$ is given by a universal 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 ${\textstyle \mathrm {eval} \colon (Z^{Y}\times Y)\to Z}$.

Examples

In the category of sets, an exponential object ${\displaystyle Z^{Y}}$ is the set of all functions ${\displaystyle Y\to Z}$.^[4] The map ${\displaystyle \mathrm {eval} \colon (Z^{Y}\times Y)\to Z}$ is just the evaluation map, which sends the pair ${\displaystyle (f,y)}$ to ${\displaystyle f(y)}$. For any map ${\displaystyle g\colon X\times Y\to Z}$ the map ${\displaystyle \lambda g\colon X\to Z^{Y}}$ is the curried form of ${\displaystyle g}$:

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

A Heyting algebra ${\displaystyle H}$ is just a bounded lattice 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 (${\displaystyle \Rightarrow :H\times H\to H}$) being right adjoint to meet (${\displaystyle \wedge <!--\; \wedge \; -->:H\times <!--\; \times \; -->H\to <!--\; \to \; -->H}$Zdroj:https://en.wikipedia.org?pojem=Exponential_object
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