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

turned out to include circular constructions, a completely new situation which also mathematically has some serious implications: this is the subject of the conceptual Galois theory we want to sketch below. Therefore, the following general framework was created, a framework which is rooted in the modern topos theory rather than in classical algebra and geometry.

2.2 Forms

Forms are the structure type which mimic a generic space concept³

³: Fro further motivations, see Mazzola (2000).

. They are based on the category Mod of (left) modules over associative, rings⁴

⁴: The empty module (!) is included in this category to guarantee universal constructions.

with identity. The morphisms of this category are the diaffine morphisms. This means that if $M, N$ are modules over rings $R,S$ , respectively, a diaffine morphism $f : M --> N$ is the composition $f = enof0$ of a dilinear morphism $f0$ with respect to a ring homomorphism $r : R --> S$ and a translation $en$ on the codomain $N$ . The morphism set from $M$ to $N$ is denoted by $M @N$ . The category of presheaves over Mod is denoted by $@ M od$ ; in particular, the representable presheaf of a module $M$ is denoted by $@M$ . More generally, for any presheaf $F$ in $@ M od$ , its value at module $M$ will be denoted by $M @F$ . In the context of $@ M od$ , we shall call a module an address, a terminology which stresses the Yoneda philosophy, stating that the isomorphism class of a module is determined by the system $@M$ of all the >perspectives< it takes when >observed< from all possible addresses. Recall Mac Lane and Moerdijk (1994) that $@ M od$ is a topos the subobject classifier $_O_$ of which evaluates to $M @_O_ = {S |S = sieve in M }$ . Its exponential $F _O_$ for a presheaf $F$ evaluates to $F M @_O_ = {S| S = subfunctor of @M × F}$ , and for a representable $F = @N$ , we have $@N ~ M @_O_ --> (M × N )@_O_$ , the set of sieves in $M × N$ . For a subfunctor $S < @M × F$ , an address $B$ , and a morphism $f : B --> M$ , we write $f@S = {(f,s)|(f,s) (- B@S}$ , i.e., $|_| B@S = f (- B@M f@S$ .

To construct the formal setup of forms, we consider the set $M onoM od@$ of monomorphisms in $M od@$ . We further consider the set

T ypes = {Simple,Syn, Limit,Colimit,Power}

of form types. We then need the free monoid $N ames = <UN ICODE >$ over the $UN ICODE$ alpabet ⁵

⁵: This is the current extension of the $ASCII$ alphabet code to non-European letters

. We next need the set $Dia(N ames)$ of all diagram schemes with vertices in $N ames$ . More precisely, a diagram scheme over $N ames$ is a finite directed multigraph the vertices of which are elements of $N ames$ , and the arrows $i : A --> B$ of which are triples $(i,A, B)$ , with $i = 1,...$ natural numbers to identify arrows for given vertices.

Next, consider the set $@ Dia(N ames/M od )$ of diagrams on $Dia(N ames)$ with values in $@ M od$ . Such a diagram is a map

@ dia : D --> M od