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

diagram and $m(fi)$ a natural transformation. When there is an »address change« in $Mod$ we have:

M M @fFi un(Fi) |, |^ N N@F un(Fi)

and, in a given address, we see that:

M @F un(F ) = Limit(D) M@hi ,/ \, M@hj M @F un(Fi) M@m(fi) M @F un(Fj) ------->

In fact, when the address is fixed there is an isomorphism:

M @Limit(D) = M prod @F un(F ) = {(xi) (- M @F un(Fi) : m(fi)(xi) = xj, A fi (- D},

which looks like this, in the case of two vertices:

M @F un(F) = M @Limit(D) M@hi ,/ |, \, M@hj M @F un(Fi) pi M @F un(Fi)× M @F un(Fj) pj M @F un(Fj) <--- M@m(fi) --->

This is how we see that denotators corresponding to a form with $TF$ Limit, are related canonically with the product when they have a fixed address.

(e) Finally, we have the denotator that corresponds to TF Colimit. Here we also have a diagram of forms and, for an address $M$ , we have an equivalence relation $~$ on the coproduct generated by the binary relation $xk ~ xj <====> m(fk)(xk) = xj$ for $xk (- M @F un(Fk),xj (- M @F un(Fj)$ in the coproduct $|_| M @F un(Fi)$ . This can also be inferred by the construction of the colimit in $Sets$ . Then there is a natural isomorphism:

|_| M @CoLimit(D) = M @F un(F) = M @F un(Fi)/ ~

which, with two functors, is reproduced in the following diagram:

M @F un(F) M@hk /^ |_| |^ ^\ M@hj M @F un(Fk) ik M @F un(Fi)/ ~ ij M @F un(Fj) . ---> M@m(fi) <---

5 Local Compositions

In this section we will define the objects of the category $Loc$ ; these objects are a type of denotator known as local compositions. A local composition has an indecomposable structure that, in MMT, is called »elementary«.