Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

Therefore, $(u,v) (- B@S$ and $<> S < S$ . Let $S$ be objective, and $S = Y$ , with $Y < A@F$ . Then $S = Y$ and $S <> = Y = S$ . Conversely, if $S = S <>$ then $S = S$ is the functorial version of the local objective composition $S = S$ .

It is very important to emphasize the difference between functorial and objective local compositions because, before the development of denotator theory, MMT used exclusively objective local compositions, which have certain restrictions that limit theoretical and practical applications.

Let $S --> @A × F$ be a local composition, $A (- Mod, F$ the functor of $CF$ (the ambient space), and $u : B --> A$ a diaffine morphism in $Mod$ . Fix $u@S = {t (- B@S : t = (u,v)} < B@S < B@A × B@F$ . Then we have

|_| B@S = u@S. u:B-->A

6 Morphisms of Local Compositions

First we will define the morphisms for objective local compositions, and then for the functorial local compositions; once this is done, we will show the relationship that exists between them.

Definition 8 Let $K --> A@F$ and $L --> B@G$ be two local objective compositions, with $A, B (- Mod$ and $F,G$ their respective ambient spaces (that is, $F = F un(CF ),$ $G = F un(CG))$ . A morphism from $K$ to $L$ is a pair $(f,a)$ , in which $a : A --> B$ is a morphism (diaffine transformation) of modules, and $f : K --> La$ a set function. The set $La$ is the image of $L$ ( $im(L))$ under $a@G : B@G --> A@G$ such that there exists a natural transformation $h : F --> G$ with $f = A@h |K :$

Kf=A@h|K --> A@AF@h |, |, La --> A@G

This morphism is denoted as $f| : K --> L a$ . Any natural transformation $h$ will be called an underlying symmetry of this morphism.

Now let us se how morphisms between functorial local compositions behave.

Definition 9 Let $A, B$ be two modules and let $K --> @A × F, L --> @B × G$ be two local compositions. A morphism from $K$ to $L$ is a pair ( $f,a)$ where $a : A --> B$ is a diaffine transformation of modules and $f|a : K --> L$ is a natural transformation, such that there exists an underlying symmetry which makes the following diagram commute.

K --> @A × F f| a |, |, @a ×h L --> @B × G