Inverse limits in the category of locales
The authors give an explicit construction of filtered colimits in the category of frames -- equivalently, of (cofiltered) inverse limits in the category of locales. Using their construction, they give direct proofs that the limit of an inverse system of locales whose bonding maps are epimorphisms projects epimorphically onto each term of the system (this result was first conjectured by \textit{M. Barr}, and proved by \textit{A. Joyal} and \textit{M. Tierney} by exploiting the relations between frames and sup-lattices); and that an arbitrary inverse limit of compact locales is compact (which was proved, under the additional hypothesis that the bonding maps are epimorphic, by the reviewer and \textit{S. J. Vickers} exploiting the relations between frames and preframes). A number of topological applications are given, and their relationship to the Boolean prime ideal theorem is discussed.
scientific article; zbMATH DE number 1187434