Autofunctors and Their Meaning

Kato and Struppa (1999) proposed the notion of autofunctor. In the present paper, the meanings of this notion are analysed in the light of a proposed structural-phenomenological interpretation of the theory of categories (Draganescu 2000). The autofunctor may become important for phenomenological categories. It represents an informational and physical process, not only a mathematical notion. It might offer, for instance, a better understanding of the generation of a universe from the deep underlying reality consisting of phenomenological information and orthoenergy. I. Preliminary considerations. For an integrative science is necessary to develop an integrative mathematics [1]. This embraces both the structural and phenomenological aspects of reality. It seems that the categories and functors are very suitable [2] [3] for such a development. In [3] were introduced new types of categories and functors: phenomenological categories, with phenomenological objects and morphisms; structural-phenomenological categories with objects and morphisms of structuralphenomenological nature; structural-phenomenological functors between a structural category (classical categories of mathematics) and a phenomenological category, and respectively phenomenologicalstructural functors between a phenomenological and a structural category. In [4] the morphisms are seen not only as transformations, in a category, from one object to another, but also as communications between the objects of a category. This is a very interesting perspective for the structural-phenomenological modeling. Therefore the categories may be: Categories with , that is with effective morphisms; Categories with , where the arrow between two objects means a process of communication. Between two phenomenological categories there are phenomenological functors. For instance, between the category of the phenomenological senses of a human mind and the category of all the phenomenological senses of existence such functors are feasible. Kato and Struppa introduced [4] the notion of autofunctor observing that "of particular interest is the problem of analysing the meaning of autofunctors (i.e. functors from U to itself)". U, in their paper, is a category, a universe defined as a model for a conscious universe. In principle, an autofunctor might be structural, but as far as we know it was not defined and used for classical mathematical categories. An autofunctor might be phenomenological, or structuralphenomenological depending on the category it refers to. The principle of feasibility [3] seems to restrict the possibility of autofunctors somewhat only for phenomenological categories, because of the nature of phenomenological processes. Because a structuralphenomenological category is always a sub-category of a product between a structural category and a phenomenological category [3], and if the phenomenological category has autofunctors, then changes produced by the phenomenological autofunctor, are transferred in changes of the structural-phenomenological category. In such a case, perhaps a structural-phenomenological autofunctor could be defined as a secondary or enhanced autofunctor for such a type of category, because it is only a consequence of the phenomenological autofunctor. II. The informational character of phenomenological categories and of their autofunctors Let us consider the deep underlying reality (orthoexistence) as an out of space and out of time see of deep energy (orthoenergy) and of the fundamental orthosense which is a phenomenological information (called also the infraconsciousness of existence [5]). The phenomenological category Cphe!1! is the part of the deep underlying reality (Fig.1) which contains the fundamental orthosense , which is the fundamental "experience" of informatter, the last one being the phenomenological informational matter of the deep reality [5] [6]. The phenomenological sense (noted <1>) is supposed to have three components [5]: 1. to exist in itself (which expresses also the unity of the entire existence); 2. to exist from itself (which may be seen also as a carrier of an autofunctor which associates to <1> a generated family of orthosenses that might be the basic phenomenological information (active information after David Bohm) of an universe); 3. to exist into itself (with some possible effects, one of them to be mentioned at the end of this section. The autofunctor FA from Fig.1 is not a simple functor that associates to an object, from a category, an object of another category, because the second category is the same with the first category. If the category has many objects, an association of each of its objects with any other of its objects, in principle, is possible. But in the case of Fig.1 there is more. The object <1> in the initial category Cphe!1! is associated, in the same category, with the object <1> and an object S that is a family of orthosenses. The autofunctor FA is a physical and informational process that generates the active information of a possible universe (note 1) . It is to be observed that S is itself a phenomenological category with phenomenological objects and morphisms, every of its objects containing also the sense <1> (the part contains the whole). In order to become a real universe it is necessary a phenomenological-energetic coupling, between S and the orthoenergy, which gives birth to the structure of a universe with its spacetime and quanta [5] (perhaps strings). This coupling is also a process, i.e. is a functor FSU from the phenomenological category S to the category U of the structural universe. If the autofunctor FA is inherent in the nature of deep reality, as was suggested before, the functor FSU is also a property of the deep reality (Fig.2). It may intervene or not, these possibilities (coupling or not coupling) being necessary for an internal play of the phenomenological senses in the deep reality, for a phenomenological processing of its internal information. When FSU is acting, a universe is born. When it is not acting , the former category S disappears or is transformed. The functor FSU is not acting at every generated phenomenological category S. When it does acting, due to the coupled orthoenergy, S is maintained together with the structural universe U, in the frame of the structural-phenomenological universe U (see also the next section). Two possibilities may be envisaged for FSU : a) the component of <1> is acting the functor FSU expressing the potential tendency to receive senses from the developments of a real universe; b) the functor FSU is controlled by the Fundamental Consciousness of existence [6] who may decide if it the case to apply or not this functor in a specific case of a generated phenomenological category S. III. The role of autofunctors Which is the possible place of the Fundamental Consciousness? To the object <1> in Cphe!1! , an autofunctor based on the orthosense associates a family of orthosenses that are essential for the Fundamental Consciousness. For instance, beside the sense of existence, a sense of knowing, of self, of will and others (i.e. senses that are usual for a consciousness, although some other existential senses may be present). This family of orthosenses, which is a phenomenological category, will be noted with G, and the autofunctor from Cphe!1! to Cphe!1! with FG as in Fig.3. One observes that <1> is not necessarily a category. <1> is rather a set [3] with three elements (note 2). On the contrary, S in the previous case (Fig.1) and G in the second case (Fig.3) are categories with many objects and with morhisms among them. The categories S and G could not maintain themselves without coupling with orthoenergy, i.e. without generating a real universe or, after the case, the selfconstitution of the Fundamental Consciousness. The category S after coupling with orthoenergy becomes the structural universe U. The category G after coupling with orthoenergy becomes the Fundamental Consciousness G. Between S and U there is an association of the objects of the first category with objects of the second category, and the same for the respective morphisms. Therefore between these two categories there is a phenomenological-structural functor H1 and inversely [3] a structuralphenomenological functor H2 . Together, the structural universe U and the phenomenological universe S constitute the structuralphenomenological universe U, or the universe U. This may be named also the integrative universe. Therefore, the universe U, U = < S, U, H1 , H2 > (1) is formed by two categories of objects and two functors (may be two families of functors) between them. The dynamics of a universe depends on the characteristics of all these four items (Fig.4).