Stratified LMN-convergence tower spaces

We develop a general theory of convergence for lattice-valued spaces based on the concept of s-stratified LM-filters. For different choices of the frames L and M, different kinds of filters arise, and suitable choices of the lattice N allow to view many existing fuzzy and probabilistic convergence spaces as special examples of our stratified LMN-convergence tower spaces. The stratification requires a certain relation between the frames L and M and we use a so-called stratification mapping to this end. The stratification condition is used to show that the resulting category is fiber-small and hence Cartesian closedness is equivalent to the existence of natural function spaces. We give several examples for our spaces for different choices of the lattices L, M and N with a special focus on enriched LM-fuzzy topological spaces. Finally, we study diagonal axioms and regularity.