Abstract
We consider regular epimorphisms in the category Loc of locales. Closed surjections with subfit domain are regular epimorphisms However, there exists a closed surjection which is the composite of two regular epimorphisms without being regular epic itself. This example answers both of the following questions in the negative: Y. Li’s question of whether weak quotient maps are necessarily regular epimorphisms, and P. Johnstone’s related question of whether regular epimorphisms compose. It follows that not all extremal epimorphisms in Loc are regular. The weak quotient maps of Y. Li and the equationally closed subframes of A. Pultr and A. Tozzi are shown to be dual notions. We also give a new characterization of regular epimorphisms in Loc.
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