ROBERT MARTY
UniversitÚ de Perpignan
 
FORMAL CATEGORY-THEORETIC DEVELOPMENT OF OBJECTS IN THE WORLD
 

Using a methodological epistomological process, we now will associate specific mathematicaluniversals with these empirical universals. We will use elementary Category Theory in order toformalize Peircean phenomenology, because Category Theory allows us to treat in a principalway the objects of the world, and, at the same time, the relations that they have with otherobjects. (For introductions to Category Theory see [10-12]). The Category-Theoretic notationused in this paper is taken from Adamek's Theory of Mathematical Structures in which the study of categories of relational structures is developed.

DEFINITION 1. A relational structure of type n is a pair (X,a ) where X is a set and a is an n-adic relation on X, that is to say a subset of the Cartesian product Xn.

DEFINITION 2. More generally, a relational structure of type  where each  is a positive integer, is a pair where X is a set and  a family of  ni-adic relations.The type is the arity-size of all participating relations in the "family." The subscript allows there to be several different arity-sizes (each n-adic relation is like an n-ended directed hyperarcin a hypergaph).
   Indeed, the qualities of feeling caused by external stimuli or by mnemonic reminder corresponds to the elements of X, and each bundle of qualities of feeling linked together by a perceptual judgement corresponds to an -tuple of . In addition, the following definition corresponds to the unity of perceptual fact relative to an identified object of the exterior world (which itself appears always as an entity), by means of the introduction of a connectivity condition on the family .

DEFINITION 3. A relational structure (X, a ) of type  is connected if for all pairs of elements and  there exists a subset of the family called and elements of X such that and belong to an n-tuple of  and belong to an n-tuple of and to an n-tuple of a k. (This generalizes the connectivity of a graph to a directed hypergraph.)
   Now, the relations between physical or non-physical objects of the world (for instance, similarities, identities, type/token relations, etc.) can be naturally taken into account by morphisms between connected relational structures (CRS), which are defined as follows:

DEFINITION 4. Let (X, a ) and (Y, ▀) be two relational structures of the same type n. A function G of X into Y is said to be compatible if (xl ,x2, . .. , xn) a implies that (i.e., membership of a node in a relation hyperarc is preserved).
   Now we begin to use Category Theory considering a functor F of the category "Set" of the sets within itself, and defining a relational structure of type F as a pair (X, a ) where X is a set and ; if (Y, b ) is another relational structure of type F, the function G of X into Y is F-compatible if . The category of these relational structures with these morphisms will be called "relational category S(F)." Then the relational category of type n is S(Qn) where Qn is the functor defined by F(X) = Xn and F(G ) = G n. If  is the coproduct4 of the functors we obtain the relational category of which the objects are defined in Definition 3 and the morphisms in Definition 4. For instance, in S(Q2 + Q3), a relational structure on a set X is a set consisting of a dyadic relation and a triadic relation ; given the two relational structures of type Q2 + Q3the compatible maps are precisely those which preserve the dyadic relations as well as the triadic ones, i.e., implies and implies The specific mathematical universal chosen will be the category the exclusive framework of our formalisation: it contains, in fact, with a suitable set of index 1, all forms of all possible perceptual judgements of all subjects and also all possible relations that these forms maintain between themselves.

Consensus Reality as 'Initial Structure'

Before proceeding with the main development, let us consider the surmised reality which lies behind multiple (possibly different) perceptions of an object, event or situation. All the relational categories S(F) possess an interesting property which again requires two definitions:

DEFINITION 5. A source in S(F) on a set X is a familywhere the are objects of S(F) and the are maps; we usually denote sources as follows

DEFINITION 6. An initial structure of a source is a structure on X such that:

(i) is a morphism of S(F)

(ii) for each object and each function such that is a morphism for each i, then is also a morphism.

PROPOSITION 1. Adamek [10] The relational category S(F) is initially complete, that is to say every source possesses a unique initial structure.

In fact, is the required initial structure.

This has the following consequences for us: We have a subject (the perceiver) with a set of perceptual judgements relative to an external object (i.e., a network arrangement of bundles of simple qualities of feeling) formalized by objects (Yi , bi ) of the relational category Considering the intersection (supposed not to be empty) X of the Yi and the inclusion maps   we obtain a source on X. Proposition 1 assures us of the existence of a relational structure (X, a) which is incorporated in each . We naturally identify (X, a) with the structure of the object for this subject; corresponds formally to the " object behind the appearances" of the phenomenologists. Of course depends on the , that is to say the totality of the experience that this subject has of the perceived object. It is possible to build a new source associated with a set of perceiving subjects (a community) taking intersections of underlying sets X of each structure presumably built by every subject. Proposition 1 permits us again to assert the existence of a unique relational structure which can be interpreted as the structure of the object collectively elaborated by the community in question (see Figure 2). This is the "common knowledge" of the community members about the object, or the common conception of it.

This gives us considerable power, because from the phenomenological point of view developed in the next section, it is the Peircean notion of "skeleton-set" [8, CP 7-426 ] or the Husserlian notion of "eidos" which we algebraicize in that way. That is the reason why we call this initial structure an "eidetic structure." For example, in Figure 3, we have the eidetic structure of the French flag.


4 The symbol C indicates the coproduct of a series, which here means disjoint union, also represented by `plus' signs.

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