PlanetMath: exponential object

(more info)

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exponential object

(Definition)

Let $\mathcal{C}$ be a category with finite products and be objects in $\mathcal{C}$ . An object in $\mathcal{C}$ is called an exponential object from to if it satisfies the following conditions:

there is a morphism $f:E\times A\to B$ , called an evaluation morphism
for any morphism $g:C\times A\to B$ , there is a unique morphism $h:C\to E$ such that $f\circ (h\times 1_A)=g$ , where $h\times 1_A:C\times A\to E\times A$ is the product morphism of and the identity morphism on .

The two conditions can be summarized by the following commutative diagram:

$\xymatrix@R-=20pt{ E\times A\ar[dr]^f\ &B\ C\times A\ar[ur]_g\ar[uu]^{h\times 1_A} }$

where

is uniquely determined by

. It is easy to see that any two exponential objects from

are isomorphic, hence the existence of an exponential objects between two objects is a universal property. We may write $B^A (\cong E$ above) the exponential object from

For example, in the category of sets, $\textbf{Set}$ , where products clearly exist (empty set is a set!) between pairs of objects (sets), the exponential from to is the set , which is defined as the set of all functions from to . The evaluation morphism is the function $ev: B^A\times A\to B$ given by , where $f\in B^A$ and $a\in A$ . If $g:C\times A\to B$ is any function, then we define $h:C\to B^A$ by . Then $ev\circ (h\times 1_A)(c,a)=ev(h(c),a)=h(c)(a)=g(c,a)$ , and is universal (in the sense of the second condition above).

Since each is uniquely determined by in the above definition, and conversely every determines a by the formula $g=f\circ (h\times 1_A)$ , we have a bijection

$\displaystyle \hom(C\times A,B)\cong \hom(C,B^A).$

If an exponential object exists between every pair of objects in category with finite products, then we say that has exponentials. According to the bijection above, we see that the functor $\cdot\times A:\mathcal{C}\to \mathcal{C}$ has a right adjoint, namely $\cdot ^A:\mathcal{C}\to\mathcal{C}$ , called the exponential functor.

It can be seen that a category with finite products has exponentials iff the covariant function $\cdot\times A:\mathcal{C}\to\mathcal{C}$ has a right adjoint for every object in $\mathcal{C}$ .

"exponential object" is owned by CWoo.

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