CATEGORIES AND FUNCTORS FOR THE STRUCTURAL-PHENOMENOLOGICAL MODELING
Mihai DRÃGÃNESCU
Center for Machine
Learning, Natural Language Processing and Conceptual
Modeling of the
Romanian Academy
E-mail: dragam@racai.ro
A possible extension of the theory of categories to the structural-phenomenological
domains of science is presented. Notions like phenomenological categories
with phenomenological objects and morphisms and structural-phenomenological
categories with objects formed both of structural and phenomenological
parts are introduced. Concerning the functors, the most important are those
between structural and phenomenological categories. The morphisms and the
functors for the structural-phenomenological domains are also physical
and informational processes, having a role in the reality of nature. In
general, the physical and informational feasibility condition is essential
for the adaptation and development of the theory of category into a mathematical
structural-phenomenological theory of categories.
1. INTRODUCTION
Saunders Mac Lane and S. Eilenberg used the notions of categories
and functors for the first time in 1945. [1].
The category is a sort of mathematical universe that has brought
about a remarkable unification and simplification of mathematics [2].
The theory of categories is seen today as a convenient and generalized
language of mathematics. But because the concept of a category is so general,
it is to be expected that theorems provable for all categories will not
usually be very deep. Consequently, many theorems of category theory are
stated and proved for particular classes of categories? [3].
If in the domain of structural science the theory of categories
brings some simplicity and elegance, yet the entire mathematics can be
expressed only by the theory of sets, as it was observed by W.V.Quine (apud
[4],
p.221, rom.ed.). But the situation may be quite different for the structural-
phenomenological science [5][6];
in this case the theory of categories might become indeed the appropriate
language of description. In this paper is shown that adapting the theory
of categories for the description of structural-phenomenological domains
of reality is possible.
Recently, G.Kato and D.C.Struppa [7] and D.C.
Struppa et al [8] considered the use of the theory of
categories for both physics and the science of consciousness:
"Category theory as a generalized language of mathematics has
been shown to be widely applicable to physical theories and as such it
is not just a powerful mathematical language, it is also perhaps the foundation
of physical theories. Since consciousness may be difficult, if not impossible,
to formulate in terms of analytical methods, a powerful, "pre-analytical"
mathematical language may be appropriate for consciousness as it is for
physics" [8].
In a first period the structural-phenomenological modeling used
the theory of sets and automata theory together with symbols for non-formal
functions and processes [9] [10].
A second period for the structural-phenomenological modeling
seems to be more efficient by using categories and functors. Structural-phenomenological
theories may be "detailed theories" or "envelope theories" [6]).
The considerations that follow are exposed in the frame of envelope theories.
2. PHENOMENOLOGICAL CATEGORIES
One cannot speak about the theory of
consciousness without taking into account the mental senses (qualia, "experience")
which are phenomenological information (phenomenological senses). Also,
one cannot speak about a deep physical theory without phenomenological
information or phenomenological senses, these two terms being equivalent.
Because in the definition of a category, it is not required that
its objects should be sets with elements [11], that
is usual mathematical objects, a category with its objects being phenomenological
senses is called phenomenological category.
A collection of phenomenological senses is a category if there
exists morphisms among these objects, and if the composition map of morphisms
and the identity morphism, like for any category [11],
are respected.
A collection (family) of phenomenological senses cannot be a
set if there are morphisms among its objects.
A collection of phenomenological senses is a set if among these
objects there are no morphisms. But such sets of phenomenological senses
could be objects of a category.
Therefore two types of phenomenological categories may be envisaged:
- a collection of phenomenological senses with morphisms among them, composition maps and identity morphism;The objects of a phenomenological category will be named phenomenological objects, and the morhisms, correspondingly, phenomenological morphisms.
- a collection of sets of phenomenological senses (each such set being an object of the category), also with morphisms, composition maps and identity morphism.
3. THE OBJECTS AND MORPHISMS OF A PHENOMENOLOGICAL CATEGORY
A phenomenological category Cphen is a collection of phenomenological
objects s1, s2, s3, ... where each s is an
elementary
physical and informational object (phenomenological sense [4]
[9])
or a set of phenomenological senses. As an elementary object,
s
is not a mathematical object, s being a symbol in the formal description
of the category. But Cphen, even if it contains only elementary
objects, is a mathematical object.
However, every s has a physical-informational content
which may be imagined, in a way, in the frame of a structural-phenomenological
orthophysical theory [9].
It may happen that a part of the objects of Cphen to be
elementary objects, in the manner described above, but another part to
be objects composed of a group of phenomenological senses which do not
have morphisms among them. The composed objects of Cphen are then
sets of phenomenological senses: sj = {sj1, sj2,...}.
Cphen
is a category not only because is formed of phenomenological objects, elementary
or composed, but also because it has morphisms among these objects.
For instance, the collection of phenomenological senses which
stays at the basis of a universe, that comprises the "active information"
(David Bohm, see [6]) of a universe is a phenomenological
category which might be formed of elementary and especially composed objects.
The phenomenological morphisms are transformations from si to sk. These are non-formal processes realizable in a both physical and informational way in a phenomenological realm of reality (named informatter in the orthophysical ontological model of existence [4][9]).
For each pair (si , sk) there is, in principle,
a set of morphisms Mo (si , sk).
Every of these
uik # Mo (si , sk) (Note. The symbol # stands for membership of a set)being a transformation
uik : si -->sk.The physical-informational content of the morphism is a natural transfor-mation from a phenomenological sense to another. It does not matter where these two phenomenological senses are located. In fact, in the phenomenological realm there is no physical space, and still if we imagine these two phenomenological senses like two separated points, the agitation of one point -because it is a process, may be a sort of vibration - produces an excitation of another point which will agitate itself, vibrate, in a more or less different way. We consider, in such a case of excitation, that these two points (phenomenological senses), as processes, are "relatively neighbors" and if the phenomenological category has only such morphisms, then the category is said to be "not too large".
Concerning the composition map of morphisms, the situation is identical with the case of the classical structural categories, that is the composition map of two morphisms is given as
m : Mo (si , sk) x Mo (sk , sl) --> Mo (si , sl)If s1, s2, s3, s4 are phenomenological objects, the composition map of the morphisms
u : s1 --> s2 ; v : s2 --> s3 ; w : s3--> s4is associative:
w(vu) = (wv)uWhat is supposed for the case of the phenomenological category is that these compositions are realizable in a physical and informational way in the phe-nomenological realm.
The same is the case with the identity morphism,
(1s) : s --> sThe condition to be imposed for the morphisms of a phenomenological category is to be phenomenological realizable. Therefore, excepting the phenomenological content, all the general structure of the phenomenological category is identical with that of the classical structural category. Further, perhaps, not all the various properties of structural categories are to be found also for the phenomenological categories. The particular case of the last type of categories has to be explored in the context of their use for the structural-phenomenological modeling. It is known that there are various types of morphisms of a structural category [11]. If they are phenomenological feasible, they will be used also for phenomenological categories. For instance, the isomorphism between two objects A and B of a category of any type, means that(1s)u = u
v(1s) = v
u : A --> Bis an isomorphism if there exists the morphism
v : B --> Asuch that
vu = (1A)and
uv = (1B)In such a case A and B are not different. This is phenomenological feasible and the notion of isomorphism is also good for phenomenological categories. If two elementary particles (like two electrons) have the same phenomenological senses (because these senses are forming, as active information, using deep energy, such particles) are not different.
4. FUNCTORS BETWEEN STRUCTURAL AND PHENOMENOLOGICAL CATEGORIES
A functor is a map between categories [11][3]. If C1 and C2 are two categories, a functor F between these categories is a map
A --> FAthat associates to each object A of C1 an object FA (written also F(A)) of C2; and for each morphism
u : A --> Din C1 associates a morphism FA -->FD in C2, subject to the conditions of transport of structure
F(uv) = F(u)F(v),and
F(1A) = (1FA).But, as in general between each pair of objects A and D in C1, there is in principle a set of morphisms Mo (A,D), the map for the association of morphisms is
F(A,D) = Mo (A,D) --> Mo (FA, FB)The functors between structural classical categories are treated in well-known books [2][11][12]. The most important cases of functors for the structural-phenomenological modeling are the functors between structural and phenomenological categories.
Of course, functors between phenomenological categories may also be
envisaged.
Let a structural category Cstr and a phenomenological
category Cphen and among them a functor F. To the structure A in
the category Cstr is associated the phenomenological sense
FA = sAin the category Cphen. This may be seen in an abstract way, but also as a physical and informational process. For instance, in the case of the human mind was defined an explanation gap between the neurobiological structures and phenomena of qualia or "experience". Both are recognized, the association of structure and phenomenological sense is recognized, but without explanation [13]. Being a fact of reality, it may be said that the functor between the corresponding structural and phenomenological categories is a reality, not only a mathematical concept. How this functor is realized in detail is important, but not such important at the level of an envelope theory. The functor for the structural-phenomenological modeling represents a physical and informational process. The functor itself is such a reality. The functor is a feasible reality realizing the coupling between a structure and informatter (referred to in the previous chapter).
A series of problems are opened at this step of our reasoning.
A neurobiological structure may be a category of neuronal automata,
and in general categories of automata are also to be considered. An
automaton may be considered as a category, of which objects are its states.
Each state is a structure, a set, and the morphisms between the objects
are therefore also functions (from a functional point of view, relations
and functions among sets were named formal functions [10]).
A category of automata is then a category of categories. Each object is
a category with an automaton with many states that are the objects of this
automaton. The morhisms of the category of automata are maps from an automaton
to another. This may be the case of the maps among various parts of the
brain. In such a case, it might possible that the association of phenomenological
senses to depend, acting subject to some conditions, on more functors between
a structural and a phenomenological. By analogy with the functional architecture
[10],
a functorial architecture could be defined.
Between Cstr and Cphen the functor F associates
also morphisms of the first with morphisms of the second. If u is
a morphism in Cstr, then Fu is a morphism in Cphen.
This is feasible from a physical and informational point of view, because
to a transformation of structure, a transformation of phenomenological
sense is necessary.
The above-described functor may be called a structural-phenomenological
functor. Because also the reverse process is feasible, a phenomenological-structural
functor R may be defined between Cphen and Cstr. For
this case, to a phenomenological sense corresponds a structure, and to
a phenomenological morphism a structural morphism. Therefore, F and R have
to work together, the first one enhancing qualia and experience, the second
one bringing intuition and creation in the functioning of a mind. These
functors are present in any organism under forms that will be explored
in future papers. Two categories and two functors mainly de-scribe an organism
Org = <Cstr, Cphen, F, R> ,and of course other items will be necessary to be mentioned in this enumeration for various types of organisms.
5. STRUCTURAL-PHENOMENOLOGICAL CATEGORIES
The objects of a structural-phenomenological category are pairs of structural and phenomenological objects (A, s), if A and s correspond to each other by a functorial link. At the origin of a structural-phenomenological category there are, in such a case, two categories Cstr and Cphen among which there are functorial links.
If Cstr has the objects A, B, C, ..., the corresponding
Cphen
has the objects FA, FB, FC,... which are phenomenological senses, and F
is the functor from Cstr to Cphen.
The resulting structural-phenomenological category is not the
product Cstr x Cphen of the above two categories, but
only a subcategory C'str-phen of this product. Indeed, the product
[11]
of the mentioned categories (retaining, for clarity, only three objects)
contain the objects
(A, FA), (B, FB), (C, FC); (A, FB), (A, FC), (B, FA), (B, FC), (C,FA), (C,FB),but only the first three pairs are feasible in the reality of the organisms (Every structure has its phenomenological sense). The subcategory C'str-phen contains a part of the objects of the product category Cstr x Cphen , contains also all the morhisms of Cstr x Cphen among the three pairs of objects mentioned above, maintains the composition of the respective morphisms, and the identity morphisms. More, because C'str-phen contains all the morphisms of Cstr x Cphen among the three pairs of objects, it is a full subcategory [11].
It may be observed that although the product Cstr x Cphen
formally seems to be a structural-phenomenological category, this is not
true because the condition of feasibility in the real world is not fulfilled.
The full subcategory C'str-phen is then a structural-phenomenological
category.
The feasibility condition is essential for the adaptation
of the theory of categories to a mathematical structural-phenomenological
theory of categories.
REFERENCES
[1] EILENBERG S., MAC LANE S., General theory of natural equivalencies, Trans. Am. Math. Soc., 58, p.231-294, 1945.
[2] SAUNDERS MAC LANE et al, Categories for the Working Mathematician (2nd Ed), Springer Verlag, 1998.
[3] HILTON P.J., Categories, in The New Encyclopedia Britannica, vol.13, p.288-291, 1994.
[4] DRÃGÃNESCU M., Profunzimile lumii materiale (The depths of the material world), Bucharest, Editura Politica, 1979; in English, The Depths of Existence, 1997, on the Web: http://www.racai.ro/books/doe
[5] DRÃGÃNESCU M., L'universalité ontologique de l?informati-on (Ontological Universality of Information), préface et notes par Yves Kodratoff, Bucharest, Editura Academiei Române, 1996. Also on the Web: http://www.racai.ro/books/draganescu.html.
[6] DRÃGÃNESCU M., The Frontiers of Science and Self-organization, communication at the IV-th Conference " Structural-phenomenological Modeling", Academia Românã, June 20-21, 2000, to be published.
[7] KATO G. and STRUPPA D.C., A sheaf theoretic approach to consciousness, The Noetic Journal, 2, 1, p.1-3, 1999.
[8] STRUPPA D.C., KAFATOS M., ROY S., KATO G., AMOROSO R.L., Category theory as the language of consciousness, George Mason University, preprint, 2000.
[9] DRÃGÃNESCU M, Ortofizica (Orthophysics), Editura Stiintificã si Enciclopedicã, Bucharest, 1985.
[10] DRÃGÃNESCU M, STEFAN GH., BURILEANU C., Electronica Functionalã (Functional Electronics), Editura Tehnicã, Bucharest, 1991.
[11] BUCUR I., DELEANU A., Introduction to the Theory of Categories and Functors, London, John Wiley, 1968.
[12] LAWVERE S. H., SCHANUEL, W., LAWVERE W., Conceptual Mathematics: A First Introduction to Categories, Cambridge University Press, 1997.
[13] DRÃGÃNESCU M., Taylor's Bridge across the Explanatory Gap and its Extension, Consciousness and Cognition, 7, 165-168, 1998.
Communication at the Section for the Science and
Technology of Information of Romanian Academy, September 18, 2000.