Sobriety and spatiality in categories of lattice-valued algebras

The paper provides an analogue of the famous equivalence between the categories of sober topological spaces and spatial locales for the framework of (L,M)-fuzzy topology of Kubiak and Šostak (and partly to that of Guido). To be more general, we replace locales with localic lattice-valued algebras in the sense of Di Nola and Gerla and use the respective generalized topological setting. As a result, it appears that the shift from crisp algebras to lattice-valued algebras weakens (resp. strengthens) considerably the classical (including the point-set lattice-theoretic setting of Rodabaugh) notion of sobriety (resp. spatiality).