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

Theorem 1 Mazzola (2001b) The categories $Loc$ and $RLoc$ are finitely complete.

If we admit general address changes, the subcategory of objective local compositions is not finitely complete, there are examples (Mazzola, 2001b) of musically meaningful diagrams $E --> D <-- G$ of objective local compositions the fiber product $E × D G$ of which is not objective⁸

⁸: The right adjointness of the objective trace functor for fixed addresses only guarantees preservation of limits for fixed addresses.

. Therefore address change--which is the portal to the full Yoneda point of view--enforces functorial local compositions if one insists on finite completeness. This latter requirement is however crucial if, for example, Grothendieck topologies must be defined (see below).

The dual situation is less simple: There are no general colimits in $Loc$ . This is the reason why global compositions, i.e., >manifolds< defined by (finite) atlases the charts of which are local compositions, have been introduced to Mathematical Music Theory (Mazzola, 1981, 1991).

3.3 Categories of Global Compositions

More precisely, given an address $A$ , an objective global composition $I G$ is a set $G$ which is covered by a finite atlas $I$ of subsets $Gi$ which are in bijection to $A$ -addressed objective local compositions $Hi < A@Fi$ with transition isomorphisms $fi,j/IdA$ on the inverse images of the intersections $Gi /~\ Gj$ . A functorial global composition at this address is a presheaf in $@ M od$ , together with a finite covering by subsheaves $Gi$ which are isomorphic to functorial local compositions $Hi < @A × Fi$ with transition isomorphisms $fi,j/IdA$ on the inverse images of the intersections $Gi /~\ Gj$ . Suppose we are given two global objective (functorial) compositions $GI$ at address $A$ , with atlas $(Gi)I$ , and $UJ$ at address $B$ , with atlas $(Uj)J$ . A morphism $(fi/a) : GI --> UJ$ is is a morphism $f$ of the underlying sets (presheaves), together with an address-change $a : A --> B$ , and a map $i : I-- > J$ such that

$f(G ) < U i i(i)$ , all $i (- I$ ,
the induced morphisms on the charts are morphisms of objective (functorial) local compositions unter the address-change $a$ .

This defines the category $ObGlob$ ( $Glob$ ) of objective (functorial) global compositions. The functorialization process described for local compositions works also globally to yield an injection

?@ : ObGlob >-> Glob

The significant difference of this concept from mathematical manifolds is that the covering $(Gi)I$ is part of the global composition, i.e., no passage to the limit of atlas refinements is admitted. For music this is a semiotically important information since the covering of a musical composition is a significant part of its understanding (Mazzola, 1990a). In fact, a typical construction of global compositions starts with a local composition and then covers its functor by a familiy of subfunctors, together with the induced atlas of the canonical restrictions, the result is called an