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 YX, together with a morphism App: YX × X → Y, and a collection of morphisms ∧X(f): Z → YX, one for each morphism f: Z × X → Y, where Z is an arbitrary object of C, satisfying the equations:
An object X is exponentiable in C if and only if it has an exponential YX for every object Y of C.
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