Definition:Exponential (Category Theory)

Definition
Let $\mathbf C$ be a metacategory with binary products.

Let $B$ and $C$ be objects of $\mathbf C$.

An exponential of $C$ by $B$ consists of an object $C^B$ of $\mathbf C$ and a morphism:


 * $\epsilon: C^B \times B \to C$

subject to the following UMP:


 * For all objects $A$ and morphisms $f: A \times B \to C$, there exists a unique morphism:


 * $\tilde f: A \to C^B$


 * such that:


 * $\begin{xy}\xymatrix@+1em@L+3px{

C^B \times B \ar[r]^*+{\epsilon} & C

\\ A \times B \ar[u]_*+{\tilde f \times \operatorname{id}_B \hskip{2cm}} \ar[ur]_*+{f} }\end{xy}$


 * is a commutative diagram, i.e. $\epsilon \circ \left({\tilde f \times \operatorname{id}_B}\right) = f$.

Evaluation
The morphism $\epsilon: C^B \times B \to C$ is called evaluation.

Exponential Transpose
The morphism $\tilde f: A \to C^B$ is called the exponential transpose of $f$.

Given $g: A \to C^B$, the morphism $\bar g: A \times B \to C$ defined by $\bar g = \epsilon \circ \left({g \times \operatorname{id}_B}\right)$ is also called the exponential transpose of $g$.

Category with Exponentials
Suppose $\mathbf C$ has an exponential $C^B$ for all objects $B$ and $C$ of $\mathbf C$.

Then $\mathbf C$ is called a category with exponentials.

Also see

 * Evaluation Mapping