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Are there non-regular strong monics in the category of locales?

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No. The category of locales is dual to the category of frames, which is monadic (in fact, equationally presented) over Set. Any such category is Barr-exact, and in particular a regular category, which implies that every strong epic of frames is regular—hence every strong monic of locales is regular.

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Yep. Just found it myself in Sketches of an Elephant A.1.3.4. Thank you, though! –  John Iskra Jan 4 '10 at 23:01
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