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

because $Xb kbb@ (- F (b@F )(K) < X@F | , |^ A k (- K < A@F$ and $KA@ (- h K < A@F |, (A@h)(k) (- La < A@G |, (A@h)(k)(b) (- Lab < X@G$ Therefore, we see that $A@h(k) (- La$ and ( $A@h)(k)(b)$ is an element of $Lab$ , that is, $lab (- (ab@G)(L)$ . Thus, $(X@a × X@h)(b,kb) = (ab,lab) (- X@L$ , and we can affirm that $g|a : K --> L$ is induced by $g|a : K --> L$ .

Now we will check to see that $g|a$ is well defined, that is, that it depends only on $g |a$ and not on $h$ . Remember that $1 @K A$ is defined as the objective trace $\/ K$ of $K$ . Let $1A@K --> a@L$ , the evaluation of $g$ in $1A$ , that is:

1 @K < A@A × A@F |, A a@L < A@B × A@G

with $K = K < A@F$ $\text{[math]}$ and $La < A@G$ . Then we have $g : K --> La$ as the second coordinate. Also, we have the commutative diagram that guarantees that $g$ depends only on $g :$

1 @K p1 b@K < X@A × X@F Ag ---> ^g |, |, a@L p--->2 ab@L < X@B × X@G

To see that it is commutative, observe the surjections: $g|a(p1(1A,k)) = g|a(b,kb) = (ab,lab) and p2(a, la) = (ab, lab)$ .

Now that we have defined morphisms between local compositions, we are ready to define the category (or categories) of local compositions. First we will define the category of objective local compositions, $ObLoc$ .

Definition 10 The category $ObLoc$ consists of the set $0ObLoc$ of all objective local compositions and the set $1ObLoc$ of morphisms, which is just the disjoint union of the sets $ObLoc(Li, Lj)$ of morphisms $f|a : Li-- > Lj$ . (To check the details of the identity morphism and the composition of morphisms (see Mazzola, 2002).

Definition 11 The category $Loc$ consists of the set $Loc 0$ of all functorial local compositions and the set $Loc 1$ of morphisms, which is the disjoint union of the sets $Loc(L ,L ) i j$ of morphisms $f : L -- > L |a i j$ . We also have the identity morphism and the composition of morphisms, see Mazzola (2002).

7 Examples of Forms, Denotators and Local Compositions

At this point, we wish to give examples of forms and denotators. We will also include the example of the denotator Träumerei and its form Pianoscore, developed