Category theoretic aspects of chain-valued frames: Part II: Applications to lattice-valued topology

This paper is Part II of a two-part series dealing with category theoretic aspects of chain-valued frames. Using the categorical properties established for L-Frm in Part I for L a complete chain, this paper constructs “upper” free functor L and “lower” free functor R. The functor L is used to create a class of non-generated L-frames, factor the LPT spectrum functor through the Σ spectrum functor, resolve the relationship between L-sobriety and ιL-sobriety, and give insight into the construction of “universal” L-topological spaces; and the functor R is used to create a class of non-generated L-frames, factor the ωL functor through a new spectrum functor Σ*, and give insight into the construction of “co-universal” L-topological spaces. These facts—universal spaces are anti-stratified, co-universal spaces are stratified, L-Frm produces both kinds of spaces via L and R, L-Frm (via its dual L-Loc) is adjunctive with L-Top—comprise a coherent argument that L-Top must accommodate both kinds of spaces, resolving the philosophical debate over the place of the constant maps condition with respect to the axioms of L-topologies.