structure preserving “models” of the category Mod. Mazzola is concerned with the contravariant functor-category having contravariant functors as its objects and natural transformations as its morphisms. We explain how these functors are related to Forms. We revisit the pointing ontology by saying that it is concerned with the contravariant functor-category The Pointer Category consits of one object ^. and no arrows besides its identity arrow (which we identify with ^.). The evaluation of the corresponding representable functor @ Sets at this one and only object yields ^.@^. = {^.} = . Recall that Simple Forms are coordinated by ^. and have as their FrameSet. The key to Mazzola’s ontology is to consider @^. as a variable FrameFunctor instead of its only value and to replace AmbientSets by their corresponding functors with repect to the isomorpy of categories SetsSets. A new phenomenon in Mazzola’s ontology is the possibility of Adress variation. Modules play a double role: Each Module A Mod provides a different viewpoint into a variable “Form”-Functor Fun(F) Mod^@ and gives access to a local AmbientSet A@Fun(F) of a Form F. Mazzola calls these functors FrameSpaces and AmbientSpaces highlighting the geometrical nature of his approach. Simple Forms are coordinated by Modules M and have the corresponding representable functors @M as their FrameSpaces. Identifiers are supposed to be natural functor monomorphsims. Limits, Colimits and Power – constructions are defined with respect to the functor-category Mod^@. The Coordinates of an A-adressed Denotator of a Form F are defined as an element of the Set A@Fun(F). The category Mod^@ is a Topos, i.e., it has good properties that allow to built Logics on it. On a metalevel of Metalanguage-Modeling we may consider the only functor ! : Mod sending all modules to the pointer ^.. It induces a natural transformation !^@ : ^@ Mod^@ which is an faithfull embedding of the pointer ontology into Mazzola’s one. AmbientSets in the pointer ontology correspond to constant AmbientSpaces, i.e., to constant functors in Mazzola’s ontology. The FrameSet for Simple @-Forms corresponds to the constant functor sending each module M to , it is isomorphic to the representable functor of the Zero Module. Hence all regular, i.e., non-circular, -Forms correspond to Forms having only one simple coordinator in their recursive construction: the Zero-Module.

Now recall the Form TwelveTone. With regard to some problems in the context of “American Set Theory” one might want to work with this Form. But note that the Minor-Third-Transposition t₃ is qualitatively not distingiushed from any other permutation of the 12 Cofactors. Hence, specific arithmetic operations on denotators are not supported by the pointer ontology.