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

where $res$ is the restriction map. We write $/~\ s = D ˙ A n(s)$ , and therefore get a surjective morphism $( D -- > /~\ s) * A n(s) n$ of diagrams of local compositions over the nerve $n*$ . Setting $D */N = colim * /~\ s A n n$ , we have a commutative diagram of sets

˙ ADn(s) ------ > /~\ s |, |, D * -/N-=-c-o-l-im--->˙ D */N A n A n

induced by the dot morphisms of local compositions. In order to complete the construction and to obtain a concise theorem, we shall now suppose the following:

Assumption 3 We henceforth suppose that the global composition $I G$ has a finitely generated projective atlas, i.e., an atlas the charts of which have finitely generated projective $R$ -modules. We also suppose that $I G$ has projective affine functions, i.e., that the affine function modules on the zero-simplexes (the charts) are projective.

It is easily seen that under this assumption, the colimit diagram (4.3) has bijective horizontal arrows, and the images $/~\ s$ are injected into the limit $ADn*/N$ . So these images cover the limit and the images of the zero-dimensional simplexes build a canonical atlas of a global $A$ -addressed composition, i.e., the diagram (4.3) becomes a bijective morphism of $A$ -addressed global compositions.

So, we have constructed a canonical global composition and a bijective morphism from the free object to a global composition which is defined by the functions of $N$ .

Definition 4 We call this composition $ADn*/N$ the $N$ -quotient of $ADn*$ . The morphism $ADn* --> ADn*/N$ from diagram (4.3) is denoted by $/N$ .

In particular, if $resGI : DGI --> GI$ is the resolution of the composition $GI$ , we have the resolution complex $DnG(GI)$ . Here is the crucial theorem (Mazzola, 2001b):

Theorem 5 Under assumption 3 (for example, if $R$ is semi-simple) we have a commutative triangle of morphisms of global compositions $DGI / \ /DnG(GI)// \\ resGI / \ / I \ I DGI/DnG(G ) ----f----------G$ with an isomorphism $f$ . All morphisms are isomorphisms of covering sets.