**5.5 A bit of category theory II**

**5.5.1 Exponential objects**

Given any two objects *X*, *Y* in a category **C** with binary products, an *exponential* object, if it exists, is an object *Y ^{X}*, together with a morphism App:

An object *X* is *exponentiable* in **C** if and only if it has an exponential *Y ^{X}* for every object

The constructions of Section 5.3 show that the exponentiable objects of **Top** are exactly the core-compact spaces, and that the space [*X* → *Y*] is an exponential object, with App(*h*, *x*) = *h* (*x*

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