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

axiom, for we consider $Sets$ as a universe that contains $Mod$ ). $A@F$ is $F (A)$ , that is, the evaluation of the functor »at« the address $A$ , where $A (- Mod$ .

Consider the category $Mod$ that has as objects all left modules $/\M$ , over all associative rings $/\$ with unit. Its morphisms are the diaffine transformations $t e .f$ where $f$ es dilinear (it consists of a »compatible« pair of ring and module homomorphisms ( $r,u)$ and $t e$ is a translation in $N (- Mod$ , see Mazzola (2002). This category can also contain the empty set, a difference from the usual categories of modules with linear homomorphisms. The $e$ notation is purely formal and doesn’t have any relation to the exponential function. For example $t (e .f)(m) = f (m) + t$ with $m (- M,$ $f(m), t (- N$ (we will not specify the ring).

$@ Mod$ denotes the category of contravariant functors from the category $Mod$ to the category $Sets$ of sets (cf. Mac Lane, 1971):

Fu : Mod --> Sets.

Also, for every $F u (- Mod@$ , a module $M$ from the domain $Fu$ will be called the address of $F u$ , and a morphism $f : N --> M$ will be called an address change. The symbols $@M$ denote the representable functors $Mod(x, M )$ , while $M @F u$ denotes the evaluation of $F u$ at the address $M$ , that is, $F u(M )$ .

We know that $@ Mod$ is a topos, as it is a presheaf category. Accordingly, $@ Mod$ has a subobject classifier that we will denote, as usual, by $_O_$ . Its worth mentioning that the constructions we will make (with the exception of the fiber product in $Loc$ ) don’t make use of the specificity of $@ Mod$ , that is, of the category $Mod$ and could, at a given moment, be generalized to any presheaf category. Finally, we should emphasize that our article is centered around what’s known in MMT as Local Theory, which is the basis of the Global Theory.

2 >>Encylospace<< and Universal Concept Formats

Knowledge consists of information, but that information must be organized in the mind. Mere information, without a »coordinate system« that permits us to classify and find it when it is necessary, is not knowledge! For this reason, the paradigms of digital information({0, 1}, »off« and »on«, etc.) are not knowledge within themselves. It is necessary to establish a system of concepts that can endow us with a method of »navigation« and which functions in a complete concept space. The problem boils down to »...how to realize an architecture of such concepts without losing rigor and confiability« [3]. The denotator concept proposes a solution of this problem.

The Encyclospace is thought of as the modernization of the traditional concept of encyclopedia, developed by Diderot and D’Alembert in the enlightenment. With the arrival of the personal computer this modernization demands a dynamic way to structure knowledge in time and space, as opposed to the static environment that existed before; it also demands an interactive relation with this way of structuring information. Similarly, the alphabetical order