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

with the identifier $Id : space(List(I,F))-- > 2space(G) < _O_space(G)$ is the subpresheaf whose value at address $A$ is the set of subsets $L < A@space(G) ~--> I × A@space(F)$ such that the first projection $pr1 : A@space(G) --> I$ is injective on $L$ , and $pr1(L)$ is an initial interval of $I$ . Moreover, this construction yields the lexicographic linear ordering relation among lists if $F$ bears a linear ordering, see (Mazzola, 2002, 6.8) for orderings on denotators.

The subject of a circular form definition is a first example of the Galois problem in defining concepts: We are given a determined stage of a form semiotic and would like to add new forms by specific properties, as for example the above list property. Besides this existence problem, we also would like to know, how many solutions we may expect, and whether they have an influence on the denotators which will live in these new forms.

3 The Category of Form Semiotics

Evidently, the problem of successive extensions of form semiotics must be accessed by use of a tool for comparing form semiotics: We need to know when a semiotic is an extension of a given form semiotic, what is an isomorphism between form semiotics, and so forth. In other words, we have to introduce morphisms between form semiotics. We shall therefore elaborate the categorical aspect which has been sketched in (Mazzola, 2002, G.5.3.1). The type set $T$ being fixed once for all (in our context), a form semiotic $Sem$ is given by four objects: $E,R,F, D$ , and seven maps $T,D, Id,DN, FN,DF, C$ . We write

Sem = (E,R, F,D;T, D,Id,DN, FN, DF,C)

for these data. For the following definition we consider the subset $Dias(F/E)$ of $Dia*(F/E)$ consisting of $E$ and of those diagrams $D : D --> E$ such that for every vertex form $F (- D$ , we have $D(F ) = space(F)$ . The maps $D$ and $Id$ may then be given the codomain $Dias(F/E)$ instead of $Dia*(F/E)$ .

Definition 2 Given two form semiotics

Sem1 = (E1,R1,F1,D1;T1,D1,Id1,DN1, FN1, DF1,C1) Sem2 = (E2,R2,F2,D2;T2,D2,Id2,DN2, FN2,DF2, C2),

a morphism $f : Sem --> Sem 1 2$ is a triple of maps $f = (f ,f ,f ) F D E$ with the following properties:

We have two set maps $fF : F1 --> F2$ , $fD : D1 --> D2$ and a logical³

³
It preserves finite limits, exponentials, and subobject classifiers, see (Mac Lane and Moerdijk, 1994, p.170); to be clear, we also require here that it preserves colimits.
morphism of topoi $fE : E1 --> E2$ which preserves addresses.
They commute with all seven maps of the respective forms, more precisely: $t = t o f 1 2 F$ , $DN o f = f o DN 2 D D 1$ , $DF o f = f o DF 2 D F 1$ , $FN o f = f o F N 2 F D 1$ . The morphism $f$ induces a map $fs : Dias(F /E )-- > Dias(F /E ) 1 1 2 2$ , to be defined below, such that $Id o f = f s o Id 2 F 1$ and $D o f = fs o D 2 F 1$ .