It is well know that the category of locale is not a regular category, that is the pullback of a regular epimorphism is not always a regular epimorphism: for example, the classical counterexample given on the [nlab][1] for cat/top/poset also work for locales.

Now, in all the counterexamples I know of, the pullback of a regular surjection is still an epimorphism.

My question is hence: is every regular epimorphism of locale a stable epimorphism (i.e. a map whose pullback are all epimorphisms) ?


Although this might not seems really interesting it has, if it is true, several interesting consequences for the category of locales, for example it imply that every morphism of locale factor as a regular epimorphism followed by a monomorphism and that every strong epimorphism of locales is a regular epimorphism. (so counterexample to these properties would also answer my question).


NB: by locale I mean the opposite of the category of frame so this question is about pushout of regular monomorphism of frame.


  [1]: http://ncatlab.org/nlab/show/regular+category#examples