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Non-Hausdorff Topology and Domain Theory by Jean Goubault-Larrecq

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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 × XY, and a collection of morphisms X(f): ZYX, one for each morphism f: Z × XY, where Z is an arbitrary object of C, satisfying the equations:

image

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 [XY] is an exponential object, with App(h, x) = h (x

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