Interweaving algebra and topology: Lattice-valued topological systems

This paper is primarily dedicated to understanding the natural role that topological systems and lattice-valued topological systems play in understanding the relationship between algebra and topology, a relationship expressed by an “interweaving” of embeddings of (essentially) algebraic and topological categories. This natural role is revealed by how various categories for systems relate to locales, topological spaces, and various categories for lattice-valued topology.