Epimorphisms of Metric Frames

This paper deals with several aspects of epimorphisms in the category MFrm of metric frames and contractive homomorphisms. In particular, it is shown that (i) the epicomplete metric frames are uniquely determined by the power-set lattices of sets, (ii) episurjective is the same as Boolean, (iii) a metric frame has an epicompletion iff it is spatial, and (iv) the subcategory of epicomplete L in MFrm is reflective. Moreover, we show that the counterpart of the latter does not hold for uniform frames. It is a well-known fact that the epimorphisms in the category of frames can be very far from surjective. The same phenomenon is encountered also in categories of more special or enriched frames such as the paracompact and the uniform ones (see [2]). In this paper we make first steps in investigating this phenomenon in the category of metric frames and contractive homomorphisms. The question whether there are metric frames L for which there exist epimorphisms L → M with arbitrarily large M remains still open (and seems to be rather difficult). However, we can prove that the metric frames L such that every epimorphism L→M is surjective (the episurjective objects) are very special, and those for which every epimorphic monomorphism L → M is an isomorphism (the epicomplete ones) are indeed very rare. Thus, unlike for mere frames, episurjectivity and epicompleteness do not coincide here. Further, we show that only spatial metric frames have epicompletions, but, on the other hand, the epicomplete objects do constitute a reflective subcategory in the whole of the category of metric frames; hence, each metric frame L has a canonical epimorphism L → M into an epicomplete one, very much in contrast, as we show, with the behaviour of epicomplete objects in the uniform case. 2000 Mathematics Subject Classification. 18A20, 54E35, 06D99.