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

interpretation. The absence of colimits in $Loc$ can be restated in terms that there are global compositions which are not isomorphic to interpretations, see Mazzola (1991, 2001b) for criteria of interpretability in terms of flasque sheaves of affine functions.

From this general definition, several specialization are derived for more specific usage. First, algebraic applications are more feasible in the smaller categories $R ObLoc$ and $R Glob$ of objective and functorial global compositions defined on the category $ModR$ of $R$ -modules and $R$ -affine morphisms over a commutative ring $R$ instead of $Mod$ . Again, we have a completeness theorem:

Theorem 2 Mazzola (2001b) The categories $Glob$ and $RGlob$ are finitely complete.

For Grothendieck topologies, one better works with a more mathematical manifold concept of global compositions. This regards uniquely the morphism concept. Two morphisms $fi/a,gk/b : GI --> HJ$ are mathematically equivalent iff $f = g$ , so just consider set maps (natural transformations for functorial global compositions) $f$ such that there is an address change and a covering map which extends $f$ to a morphism in $ObGlob$ ( $Glob$ ) (or corresponding categories $RObGlob$ , $RGlob$ ). This equivalence defines coarser categories which we index by » $m$ « for »mathematical«:

mGlob, mObGlob, RmGlob, RmObGlob.

Theorem 3 The categories $R mGlob,mGlob$ are finitely complete.

Observe that the »mathematical« categories have the same objects as the original ones, only the morphisms are >blurred<. So the mathematical categories are half way between the original musical manifold (morphism) concept and the purely mathematical manifold (morphism) concept.

3.4 Grothendieck Topology and Cohomology

Because of theorem 3, we may define a Grothendieck (pre)topology, the finite cover topology, on $mGlob$ and on $R mGlob$ via covering families. Its covering families for a global composition $GI$ are finite collections of morphisms $(HJkk --> GI)k$ which generate the functor of $GI$ . Various ech cohomology groups (in the sense of Verdier (Grothendieck and Dieudonné, 1960-1967, exposé V)) can be associated to covering families of the finite cover Grothendieck topology (Mazzola, 2001b, chapter 19).

Given an address $A$ which is a modules over a commutative ring $R$ , one can also define the $R$ -module $G(GI)$ of global affine functions on a $A$ -addressed global composition $GI$ . It entails cohomology $R$ -modules $Hi(f.,G)$ for any covering family $f.$ in $ObLocA$ . See (Mazzola, 2001b, 19.1.1) for details.

4 Classification

Classification of global musical objects deals with the determination of isomorphism classes in adequate categories of global compositions, the type of objetcs which play