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

and the formula $si = f o si$ is immediate.

To define global »free« objects among the $A$ -addressed objective compositions with finite charts, we consider the natural weight function $n : n(GI) --> N$ with $n(S) = card( /~\ S) - 1$ , where we set $/~\ S = /~\ s (- Ss$ . The pair $n*(GI) = (n(GI),n)$ is an object in the category of naturally weighted simplicial complexes the morphims of which are the simplicial maps which commute with the weight functions. We shall represent the naturally weighted nerve $n*(GI)$ by an isomorphic standard representative $n*$ induced by a covering of the natural interval $[0,m] = {0,1,2,3,...m = card(G) - 1}$ of natural numbers.

For $* n$ , we define the global standard composition $ADn*$ at address $A$ by the interpretation of the local standard composition $ADm$ which is given by the present covering of $[0,m]$ . We are also given a standard atlas of $ADn*$ . In fact, for any subset $q = {t0,...tc}< [0,m]$ of $c+ 1$ elements, we have the canonical injection $iq : ADc --> ADm$ via $sj '--> stj$ . This defines the standard atlas.

The universal property of this global standard composition reads as follows. Take the category $Covens$ of coverings of sets¹⁰

¹⁰: Its objects are coverings of sets $XI$ , its morphisms $XI --> YJ$ are compatible pairs of set maps $f :X --> Y,f:I-- > J$ , i.e., $f(x)< f(x)$ .

, and consider the covariant functor

finite I * I ACovn*: ObGlobA --> Sets : G '--> HomCovens(n ,(G, n0(G )))

(6)

where the covering $n*$ denotes the naturally weighted simplicial complex after forgetting about its weight. Then we have this straightforward result:

Proposition 3 The functor $ACovn*$ is representable by the standard global composition $ADn*$ , i.e., we have a bijection

* I ~ I HomCovens(n ,(G, n0(G ))) --> HomObGlobfAinite(ADn*,G )

which is functorial in the $A$ -addressed composition $GI$ .

In particular, if we take the standard covering $* * I n = n (G )$ of the nerve of $I G$ and then the corresponding >identity< morphism $* ~ I Id : n --> (G,no(G ))$ , we obtain a corresponding bijective morphism

I resGI : DGI --> G

(7)

with the notation $DGI = ADn*(GI)$ , this object and the morphism $resGI$ being called the resolution of $GI$ . Clearly, the associated simplicial morphism $n(resGI) : n*(DGI )-- > n*(GI)$ is an isomorphism, but $resGI$ is not, in general, an isomorphism!

In particular, due to the universal propertiy of the global standard compositions, every morphism $fi : GI-- > HJ$ can uniquely be lifted to a corresponding morphism $resfi$ of resolutions to make the diagram

D I -r-es-f-i--> D J G H resGI |, |, resHJ fi GI ------> HJ