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I'm reading about the Stone-Čech compactification for locales, which (I think) states that for every locale $L$ there exists a compact regular locale $\kappa L$ and a morphism $r : L \to \kappa L$ such that every locale morphism $f : L \to R$ to a compact regular locale $R$ extends uniquely to a locale morphism $\hat{f} : \kappa L \to R$ satisfying $f = \hat{f} \circ r$.

Question: is $r$ a monomorphism?

I guess it should be, in analogy with the topological manifestation of the Stone-Čech compactification. I'm getting a bit lost in Banaschewski's papers, so either an answer or a precise reference would be much appreciated.

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Even for spatial locales this is not true, since there are (sober) topological spaces whose maps to their Stone-Čech compactifications are not injective. An example is the line with two origins.

I don't know a snappy characterization of those locales for which the map $r$ is a monomorphism, and I'm not sure you'll be able to find one, because monomorphisms are not viewed as the "right" notion in the category of locales. Much more attention gets paid to regular monomorphisms (e.g. a sublocale of $L$ is defined to be an isomorphism class of regular monomorphisms into $L$).

And the theorem is that $r$ is a a regular monomorphism if and only if $L$ is a completely regular locale. Similarly, the map from a topological space to its Stone-Čech compactification is a topological embedding if and only if the space is completely regular.

The best reference for this (and all related topics) is probably Johnstone's book Stone Spaces.

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