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

Definition 3 Let $S < A@RU$ be a local composition in the $R$ -module $U$ . For a submodule $L < G(S)$ of affine functions on $S$ , the evaluation map $˙: S --> A@RL*$ into the $A$ -valued points of the dual module $L*= HomR(L, R)$ of $L$ is defined by $˙s(a)(l) = l(s)(a)$ .

The problem is that the evaluation is not a morphism of local compositions in general. But in the special case which is of interest, we have this guarantee: Let $~n || S = ADn < A@RA$ . Then, the dotted points $* ˙si : A --> L$ define the universal map $|~| n * HL : A --> L$ , and we have interpreted $* ˙: ADn --> A@RL$ as a morphism of local compositions.

Next, suppose we are given two local compositiosns $S < A@RU, T < A@RV$ and a morphism

f : S --> T,

together with a module $LT < G(T )$ the retract $LT|f$ of which is included in a module $LS < G(S)$ . We then have a commutative diagram

˙ * S ------ > A@RL S f |, |, A@|f* T ---˙--- > A@RL* T

where $* |f$ is the $R$ -dual of the canonical linear map $|f : LT --> LS$ .

This construction yields a morphism $˙ ˙ ˙ f :S --> T$ of local compositions in the ambient spaces $* * LS,L T$ , respectively. With this technique we may associate a global composition with a module complex $N < nG(ADn*)$ of affine functions in the standard composition $ADn*$ of a standard covering $* n$ .

Assumption 2 In the following discussion of classification, we shall tacitly assume that all module complexes of affine functions have surjective transition morphisms. (We know that this is the case for retracted function modules from resolution morphisms!)

If we apply the construction from diagram (4.3) to the situation where $S = D < A@A~n || (s) A n(s)$ , and $T = D < A@A~ || n(t) A n(t)$ for simplexes $t < s$ of $D * A n$ , and with $L = N (s),L = N (t) S T$ , then we have injective vertical arrows in the corresponding commutative diagram

˙ * ADn(s) ------ > A@RN (s) inclusion |, |, A@res* ADn(t) ---˙--- > A@RN (t)*