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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 $B$?

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  • $\begingroup$ Have you tried looking up exponentiable locales? $\endgroup$ – Zhen Lin Jul 18 '15 at 1:10
  • $\begingroup$ @ZhenLin indeed I have $\endgroup$ – Harrison Smith Jul 18 '15 at 1:12
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    $\begingroup$ So you know, for instance, that there is a notion of locally compact locale, and that these are precisely the exponentiable locales? $\endgroup$ – Zhen Lin Jul 18 '15 at 2:18
  • $\begingroup$ @ZhenLin exponentiaBLE not exponentiaL! That answers it. Thank you. I just misread your comment. $\endgroup$ – Harrison Smith Jul 18 '15 at 2:32
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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

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.)

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