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

The morphisms between local functorial compositions can be defined in an alternative way that, although less elegant, makes us relate them to the morphisms between objective local compositions and helps us realize proofs. An equivalent definition of morphisms between functorial local compositions runs as follows: Consider the following fiber product diagram (»pullback«):

La --> @A × G |, |, @a×1G L --> @B × G

Then the natural transformations $f : K --> La$ and $h : F --> G$ make the following diagram commute:

Kf --> @A@1××Fh |, |, A La --> @A × G

In other words, we have the following universal situation:

K \, f La --> @A × G |, |, @a×1 L --> @B × G

Theorem 1 If $g|a : K --> L$ is a morphism of objective local compositions, then $g|a : K --> L$ is a morphism of functorial local compositions induced by the same $h$ and $a$ as in $g|a$ .

Proof. We will evaluate an element in $X@K$ to see if, by $X@a × X@h$ , we arrive at an element in $X@L$ . If this is the case, by point evaluation of functors in the presheaf $@ Mod$ , we can affirm that there exists $g| a : K --> L$ with the same $a$ and $h$ as the morphism of objective local compositions $g| a : K --> L$ . Afterwards we will check that $g| a$ is well defined, that is, that it depends only on $g| a$ and not on $h$ .

Let $(b,kb) (- X@K = {b (- X@A, kb (- X@F : kb (- (b@F )(K)}$ $<$ $(@A × F)(X)$ . Then we apply ( $X@a × X@h)(b, kb) = (ab,(X@h)(kb)$ (note that $X@a$ is covariant). We know that $ab (- X@B$ . We must show that $(X@h)(kb) (- X@L$ . However, on the one hand:

X@L = {ab (- X@B, lab (- X@G : lab (- (ab@G)(L)} < @B × G

and, on the other:

(X@h)(kb) = (A@h)(k)(b)(with k (- K < A@F ),