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

commute. We therefore have a resolution functor

finite finite resA : ObGlob A --> ObGlob A

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and a natural transformation

d : res -- > Id finite A A ObGlobA

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The resolution of a global composition is a representation of its weighted nerve and thereby includes invariant data of the composition. But more is needed to yield a full set of invariants. The next step deals with this completion. It is related to functions on global compositions.

4.3 Global Compositions from Coefficient Systems

A global affine function on a global composition $I G$ is a morphism $I f : G -- > A@RR$ . The set of global affine functions builds a $R$ -module $I G(G )$ under pointwise addition and scalar multiplication. Moreover, the retraction $i f .g$ of an affine function $I f : G --> A@RR$ via a morphism $i J I g : H --> G$ is an affine function, and the retracted function module is a submodule of $J G(H )$ .

To control such submodules, we give an alternative description of such function modules in terms of coefficient systems from sheaf theory (Godement, 1964). Given a global composition $I G$ , reconsider the category $I n(G )$ of its abstract nerve.

Definition 2 A ( $R$ -)module complex over $I G$ is a covariant functor (a coefficient system in Godement’s terminology)

I M : n(G )-- > RMod

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into the category $Mod R$ of $R$ -modules and $R$ -affine morphisms, with transition morphisms

M : M (s) --> M (t) s,t

of modules for the simplex inclusions (morphisms) $s < t$ .

As usual in sheaf theory, we put $GM = limn(GI)M (s)$ and call this the set of global sections.

Example 3 Since any morphism of global compositions $i f/a : GI --> HJ$ yields a natural transformation $i I J n(f/a) : n(G )-- > n(H )$ , every module complex $M$ over $HJ$ induces a module complex on $i f/a *M$ over $GI$ , with

i i f /a*M (s) = M (n(f /a)(s)).

Example 4 If $M$ is any $R$ -module, the constant module complex of $M$ is the complex with $M (s) = M$ for all simplexes and identity transition. Observe that its global sections are in bijection with the set $c M$ , if $I N(G )$ has $c$ connected components.