TL;DR: compact-open topology for Homs of locales?


Let $\mathcal{L}$ be a full subcategory of the category $\mathcal{Loc}$ of locales.


For two locales, $A$ and $B$, is there a nice way to make an internal (to $\mathcal{L}$ or to $\mathcal{L}$) Hom out of $Hom\_{\mathcal{L}}(A, B)$? That is, an exponential object $B^A$ in $\mathcal{L}$ or at least $\mathcal{Loc}$. Will it work to take the compact-open topology on the mapping space $Hom\_{\mathcal{Top}}(X, Y)$ for any pair of topological spaces such that the frame of opens on $X$ is $A$ and that of $Y$ is $B$?



Exponential locales and a pointless version of the compact-open topology?
For short, the exponential $(X,Y)$, characterized by the usual universal properties: morphisms from any locale $Z$ to $(X,Y)$ are functions from $X \times Z$ to $Y$, exists for all $Y$ if and only if $X$ is locally compact.


The reference for this is M.Hyland's paper [Function spaces in the category of locales](http://link.springer.com/chapter/10.1007/BFb0089910) 


There is also a chapter (C4) about it in Johnstone's Elephant (Hyland paper is a bit more complet about the case of locale because it gives a description of the geometric theory of morphism that is classified by the exponential, but Johnstone also treat the case of exponential of non localic toposes.)


