Exponentiable locales I: every exponentiable locale is continuous

The exponentiable objects of Top are exactly the core-compact spaces. Through Stone duality, the core-compact spaces are related to the continuous frames. So here is a wild guess: would the exponentiable locales be exactly the continuous frames? That is indeed true, as was proved by Martin Hyland in 1979 (published in 1981). I will concentrate on one half of the this result for this time, and I will explain why every exponentiable locale must be a continuous dcpo. The proof is very close to the similar result in Top, but, as usual, locales are so much more abstract that similar arguments tend to be harder to understand in Loc; I will do my best. Read the full post.