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

which with every vertex of $D$ associates a functor and with every arrow associates a natural transformation between corresponding vertex functors. So $i : A --> B$ is mapped to the natural transformation $dia(i) : dia(A) --> dia(B)$ .

With these notations, we can define a semiotic of forms as follows:

Definition 1 A semiotic of forms is a set map

@ @ sem : F ORM S --> T ypes × M onoM od × Dia(N ames/M od )

defined on a subset $F ORM S < Names$ with the following properties (i) to (iv). To ease language, we use the following notations and terminology:

An element $F (- FORM S$ is called a form name, and the pair $(F,sem)$ a form (if $sem$ is clear, the form is identified with its name)
$pr1 .sem(F ) = t(F )$ (=type of $F$ )
$pr2 .sem(F ) = id(F )$ (= identifier of $F$ )
$domain(id(F)) = fun(F )$ (= functor or »space« of $F$ )
$codomain(id(F )) = frame(F )$ (= frame or »frame space« of $F$ )
$pr3 .sem(F ) = coord(F )$ (= coordinator of $F$ )

Then these properties are required:

The empty word $Ø$ is not a member of $FORM S$
Within the coordinator of $F$ , if $t(F ) /= Simple$ , the vertices of the diagram are form names, i.e. elements of $FORM S$
For any vertex $X$ of the coordinator diagram $coord(F)$ , we have $coord(F )(X) = fun(X)$
If the type is given, we have the following for the corresponding frames:
- For $Syn$ and $Power$ , the coordinator has exactly one vertex $G$ and no arrows, i.e. $coord(F ) : G --> fun(G)$ , what means that in theses cases, the coordinator is determined by a form name $G$ .
  Further, for $Syn$ , we have $frame(F) = fun(G)$ , and for $Power$ , we have $f rame(F ) = _O_fun(G)$ .
- For $Limit$ and $Colimit$ , the coordinator is any diagram $coord(F )$ . For $Limit$ , we have the frame $frame(F ) = lim(coord(F))$ , and for $Colimit$ , we have the frame $frame(F ) = colim(coord(F ))$ .
- For type $Simple$ , the coordinator has the unique vertex $Ø$ , and a value $coord(F) :Ø --> @M$ for a module $M$ , or, in a more sloppy notation: $coord(F) = M$ .