On closed and quotient maps of locales

Abstract

The category Loc of locales and continuous maps is dual to the category Frm of frames and frame homomorphisms. Regular subobjects of a locale A are elements of the form Aj = fj : A ! A j j(a) = ag: The subobjects of this form are called sublocales of A. They arise from the lattice OX of open sets of a topological space X in a natural way. The right adjoint of a frame homomorphism maps closed (dually, open) sublocales to closed (dually, open) sublocales. Simple coverings and separated frames are studied and conditions under which they are closed (or open) are those that are related to coequalizers are shown. Under suitable conditions, simple coverings are regular epimorphisms. Extremal epimorphisms and strong epimorphisms in the setting of locales are studied and it is shown that strong epimorphisms compose. In the category Loc of locales and continuous maps, closed surjections are regular epimorphisms at least for those surjections with subfit domains.