Examples
In the category of sets, the exponential object is the set of all functions from to . The map is just the evaluation map which sends the pair (f, y) to f(y). For any map the map is the curried form of :
In the category of topological spaces, the exponential object ZY exists provided that Y is a locally compact Hausdorff space. In that case, the space ZY is the set of all continuous functions from Y to Z together with the compact-open topology. The evaluation map is the same as in the category of sets. If Y is not locally compact Hausdorff, the exponential object may not exist (the space ZY 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 be cartesian closed. However, the category of locally compact topological spaces is not cartesian closed either, since ZY need not be locally compact for locally compact spaces Z and Y.
Read more about this topic: Exponential Object
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