at what is outside the coconut, let us confront its epidermis with what is beyond the object’s boundary, and we shall understand what is within the coconut. In his Logic (Hegel, 1963, 2. Kapitel, A.b)), Hegel, citing Spinoza, states that all determination is by an affirmative negation, and that something is negation of negation. It is obvious that the solution of Grothendieck’s coconut problem is precisely this insight: Something is the dissolution of its conceptual epidermis, the solution of its conceptual negation (what is impossible within a given concept?) in the negation of what it excludes. To put in in the French wording, it is all about »dévissage de l’identité«.

It may seem that such philosophical far-out mysteries are not what formal and effective science is about, but this is erroneous: Once we have understood the conceptual epidermis, the boundary of a concept’s power, we can transcend it and offer solutions to the present conceptual limitations, solutions which help overcome the inherent limitations. Recall that Galois’ solution of old questions--such as the trisection of an angle by use of ruler and compasses--is in fact the result of a thorough analysis of the conditions for such a solution, showing that any solution space has properties not shared with a specific ruler- and compasses-aided construction method.

In other words, Galois shows that any conceptual extension by ruler and compasses must contradict the targeted extension by trisection. To our mind it is not by chance that Grothendieck’s more recent research (his unpublished manuscript La Longue Marche à travers la Théorie de Galois, written in 1981) is about >great Galois unification theories< (unifying Galois and fundamental groups, but see Schneps and Lochak (1997) for more details).

The following technical sections should be viewed in this light in order to understand why we so strongly insist on conceptual extensions and on the related Galois theory. However, a certain familiarity with models and examples from Mathematical Music Theory as exposed in (Mazzola, 2002) is recommended for a deeper understanding of the meaning of the formalism.

2 Form Semiotics

We refer to (Mazzola, 2002) regarding the development and musicological motivation for the structure of a form semiotic. Here, we want to give a slightly more elegant definition of a form semiotic. To this end, we suppose given a topos with subobject classifier , together with a subcategory such that the Yoneda map into the topos of presheaves over yields a fully faithful functor if the presheaves are restricted to . Recall that this is the case, for example, for any presheaf topos over a category together with its full subcategory of represented -objects (this is Yoneda’s Lemma), or for the topos , together with any singleton subcategory defined by any singleton . For reasons stemming from the context of mathematical music theory (Mazzola, 2002), we call such a subcategory an address subcategory of , and its objects are called addresses. Denote by the subcategory of monomorphisms of , where it is understood that objects in a category are identified with their identity morphisms. We further need