Abstract The article is based on Mazzola (2002), and develops the aspect of Mathematical Music Theory (MMT) known as Local Theory, in which category and topos theory are the fundamental tools. Constructions used in Mathematical Music Theory, such as “form”and “denotator”are defined. Morphisms of Local Compositions, a special type of denotator, are also defined, giving way to the category of Local Compositions.Some results are established. There are examples of applications to MMT. The denotator Träumerei is presented; its form Pianoscore, is an example with all typologies: Limits, Colimits, Synonymy, etc. This shows that overall comparative analysis of musical works transcends the theory of Local Compositions, and that category theory for general denotators is an open question.

1 Introduction

Before we begin with our topic at hand, we want to emphasize that the source and inspiration of this article is the monumental work »The Topos of Music« (Mazzola, 2002), abbreviated by ToM from here on, and in which we have the honor of being a contributor. In the prologue of their book »Sheaves in Geometry and Logic« (Mac Lane and Moerdijk, 1992), the authors say that »a startling aspect of topos theory is that it unifies two seemingly wholly distinct mathematical subjects: on the one hand, topology and algebraic geometry, and on the other hand, logic and set theory«. Echoing these authors, we could say that an amazing aspect of Mathematical Music Theory (MMT) is that it unifies topos theory (together with topology, algebraic geometry, group and module theory, etc.) and computer science, with music. Music is a multifacetical phenomenon, and while it is true that it is related to emotional and psychological manifestations, it also consists of physical and structural aspects. These last two manifestations are implicit in the whole development of this article.

The notation is identical to that used in ToM and, granted, one needs to »translate« it to the usual notation of category and topos theory; however, it does have its particular mystique, at least for those of us who have adopted its use! It is important to understand the use of the symbol »«. For example, is the category of contravariant functors, (based on the MacLane