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

Once such a system is implemented in a programming environment, such a discipline pays. Therefore, in order to place the same form on different vertexes, one has to produce a number of isomorphic copies, i.e., synonymous forms (in the above sense) with the same spaces. Or else, we may enrich the form names in order to be able to define as much forms as needed via rich name spaces. In practice, we do however often refer to “copies of a given form” without specifying these naming accents.

Remark 4 In practice, lists are very useful. There is no list type in our setup, but one may easily mimick it as follows. We are working in the presheaf topos $M od@$ and its address category $@Mod$ , where we often identify modules with their representable functors, if no confusion is likely. Suppose that we want to have a list of length $n = 0,1,2,3,... (- N$ , the denotators of which have the form $F$ . This is achieved by a form $L (F ) n$ of limit type and a discrete diagram without arrows, consisting of $n$ copies of $F$ . This enables us to introduce a list form for all list lengths up to $N$ , say. We take the colimit of all the forms $L (F ),n = 0,1,2,3,...N n$ .

A more elegant way without using an indetermined number of cofactors for our list form works as follows: Consider the form

List(F ) : Id.Colimit(Item(F ),T erminal),

where $Item(F ) : Id.Limit(F,List(F ))$ and where $T erminal : Id.Simple(Z)$ . Here we denote the vertex forms for limits and colimits if we deal with discrete diagram schemes. The form $Terminal$ is made for terminating the list entries and writing down the list’s length. This however is not a complete definition since we do not know whether such a form exists! This definition is circular, and the existence of the corresponding presheaves must be proven. This is in fact true, see (Mazzola, 2002, G.2.2.1), but the proof uses more than finite completeness.

Despite the missing list type, the given types can be used to define more general list forms in the sense that general index sets can be used. To this end, let $I$ be an index set, together with a linear ordering relation <. Call a subset $J < I$ an initial interval iff there is either $x (- I$ such that $J = {i| i < x}$ or $J = I$ . Suppose further that $F$ is a given form over $@ M od ,@Mod$ , and that we want to define a form $List(I,F)$ such that its elements at address module $A$ are precisely the »lists« $(li)J$ , i.e., the sequences of $A@space(F )$ -elements for initial intervals $J$ of $I$ . To this end, consider the family $[i] : Id.Simple(@0)$ of simple forms with the zero module $0$ over the zero ring as trivial diagram, $Id = Id@0$ , and name keys $[i]$ over $N ameF orm$ . Let $X(I, 0) : [i] '--> @0,i (- I$ the diagram of these simple forms²

²: This diagramm is generally infinite, but the construction works as our topos is complete, not only finitely complete.

. Consider the >pure list< form $[I] : Id.Colimit(X(I, 0))$ , $Id$ being the identity on the frame space $colim X(I,0)$ . Observe that for any address module $A$ , we have $~ A@colim X(I, 0) --> I$ since $0$ is final in $Mod$ . We then define the auxiliary form $G : Id.Limit([I],F )$ and finally set

List(I,F ) : Id.Power(G)