A survey of fuzzifications of frames, the Papert-Papert-Isbell adjunction and sobriety

This paper gives a survey of fuzzifications of frames, the Papert-Papert-Isbell adjunction between the categories of topological spaces and locales as well as the related concept of sobriety. It then establishes categorical isomorphisms between Yao-L-frames, Zhang-Liu-L-frames and L-algebras. Lastly, it studies the relationships between L-sobriety, modified L-sobriety and ιL-sobriety. The obtained results suggest that modified L-sobriety is the most fruitful notion between the above concepts of lattice-valued sobriety.