Cool tools: Polysemic and non-commutative Nets, subchain decompositions and cross-projecting pre-orders, object-graphs, chain-hom-sets and chain-label-hom-sets, forgetful functors, free categories of a Net, and ghosts

Expressive tools work primarily on the problem of the clarity and style of expression of models in some underlying theory. There is a structure among mathematics, a theory of the world (e.g. music), expressive constructs with their own structure, the applied theory, and models of some real entity. Lewin’s transformational networks are expressive tools of great currency in music theory. We generalize Lewin-nets to Nets, which, unlike Lewin-nets proper, are both polysemic and non-commutative. We show how all kinds of Nets work on both simple and complex data objects, and how this is all useful in expressing musical analysis. We then show some ways that Nets connect with category theory and topology. We construct chain-hom-sets and chain-labelhom-sets in Nets that form free categories on object-graphs and transformation-graphs, and show how each chain-hom-set of paths consists of all possible musical compositions between an initial and final musical object in a chain within a particular Net. We note how the temporal partial order of any piece of music plays against the structural pre-order of its Net representation, and how the various partial and pre-orders cross-project against one another. There is a family of forgetful functors that relate the various categories Net, Object-graph, Transformation-graph, and Grph. We show how to get from both proper Lewin-nets and from Nets to an underlying pre-order such that the labelling arrows of the Net can then be construed as a pre-sheaf over the pre-order. We point out the interesting ghosts that survive all the forgetful functors, that is, the particular characterizing structures of each digraph (or transformation-graph or object graph) which then, reading upwards, constrain the possibilities for labelling its arrows and nodes in any superior entity (such as a Net) built on it.