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

of $M @N$ through the Identifier $IF$ ; in this case, the coordinates are identified with a diaffine transformation of modules, $Mod(M, N )$ . It’s interesting and of practical importance to note that if we have the »zero address«, $M = 0Z$ , then $Mod(0, N ) ~= N$ .

(b) In the case of Syn, the $CD$ correspond to an element of $M @F un(C(FD))$ , that is, the functor of the Coordinator of the form of the denotator, evaluated at the address $M$ . From here on, we will use $F$ to designate $FD$ , the form of the denotator.

(c) The coordinates $CD$ that correspond to $TF$ Power are identified with a subfunctor of $@M × Fun(CF )$ because of the isomorphism

M @_O_F un(CF ) ~= Subfunctors(@M × F un(CF ))

We will examine this in more detail. In a topos of presheaves like $Mod@$ there exists a bijection between the set of sieves $_O_(M )$ and the subfunctors $Mod(x, M ),$ and the subobject classifier in this topos is $True : 1 --> _O_$ . The Yoneda functor, $@ : M --> M @$ which is full and faithful, plus the existence of power objects in any topos, permits the following construction. Remember that $F un(CF ) (- Mod@$ and $N @M = Mod(N, M )$ . Furthermore, the power set $2M@F u$ in $Sets$ induces a functor $2Fu (- Mod@$ (for all $Fu (- Mod@)$ defined as $M @2Fu = 2M@F u$ . In fact, $2Fu$ is the composition of $F u : Modop --> Sets$ and the covariant »power set« functor $- 2 : Sets --> Sets$ .

We have: $Fun(CF) Fun(CF) M @_O_ ~= N at(@M, _O_ ) ~= N at(@M × F un(CF ),_O_)$ . Then, if $Sub(@M × Fun(CF ))$ denotes the subfunctors of $@M × Fun(CF )$ , there is an isomorphism

F un(CF) Sub(@M × F un(CF )) ~= M @_O_ ,

which can be illustrated by the following fiber product diagram for such a subfunctor $T$ :

T >-> @M ×F un(CF ) |, |, 1 >-> _O_

On the other hand, let $@ Fun(F ) (- Mod$ and $S (- M @2F un(F)$ . Then $S (- 2M@F un(F)$ and, of course, $S < M @F un(F )$ .

Definition 3 To each $S (- M @2Fun(F)$ we associate a subfunctor $S --> @M × F un(F)$ defined as follows: For any module $N$ , $N@S = {u (- N @M, v (- N @F un(F) : v (- (u@F un(F))(S)}$ . Then, for any morphism $q : N --> L$ , we have that $q@S = q@M × q@F un(F )| S(L)$ .

(d) For the TF Limit, the corresponding denotator consists in special lists in the product of all the $M @F un(Fi) (- Sets$ . Once again we will present this construction in detail: Suppose that the diagram of forms has vertices $Fi$ , (with $Fi$ a form) and functors $Fun(Fi)$ that correspond to each $Fi$ . This means that $fi$ is an arrow in the