axiomatic theories of metacategories and supercategoriesAxiomaticTheoriesOfMetacategoriesAndSupercategories2010-05-09 23:28:312010-05-09 23:28:31AxiomOSsupercategorymetacategorymetagraphaxiomstheory of metacategoriestheory of supercategoriesorganismic supercategories (OS)% there are many more packages, add them here as you need
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\newcommand{\hr}{{\hookrightarrow}}\subsection{Introduction}
This is a topic on the axioms of categories, metacategories and supercategories
that are relevant, respectively, to mathematics and meta-mathematics.
Lawvere's Elementary Theory of Abstract Categories (ETAC) provides an axiomatic construction of the theory of categories and functors. Intuitively, with this terminology and axioms, a category is meant to be any structure which is a direct interpretation of ETAC. A functor is then understood to be a triple consisting of two such categories and of a rule F (`the functor') which assigns to each arrow or morphism x of the first category, a unique morphism, written as `F(x)' of the second category, in such a way that the usual two conditions on both objects and arrows in the standard functor definition are fulfilled --the functor is well behaved, i.e., it carries object identities to image object identities, and commutative diagrams to image commmutative diagrams of the corresponding image objects and image morphisms. At the next level, one then defines natural transformations or functorial morphisms between functors as meta-level abbreviated formulas and equations pertaining to commutative diagrams of the distinct images of two functors acting on both objects and morphisms. As the name indicates natural transformations are also well--behaved in terms of the ETAC equations that are satisfied by natural transformations.
\subsection{ETAS and ETAC}
Categories were defined in refs. \cite{LW1,LW2} as mathematical interpretations of the `elementary theory of abstract categories' (ETAC). One can generalize the theory of categories to higher dimensions-- as in higher dimensional algebra (HDA)-- by defining multiple composition laws and allowing higher dimensional, functorial morphisms of several variables to be employed in such higher dimensional structures. Thus, one can introduce an elementary theory of supercategories (ETAS;(\cite{ICB3,ICB1}) as a natural extension of Lawvere's ETAC theory to higher dimensions (\cite{BGB2}). Then, \PMlinkname{supercategories}{Supercategories3} can be defined as mathematical interpretations of the ETAS axioms as in ref.\cite{ICB3}.
\begin{remark}
Related concepts to the general notion of a supercategory recalled above can also be rendered graphically on a computer as a
\PMlinkname{multigraph}{Multigraph} or a \emph{hypergrap}.
On the other hand, a supercomputer architecture
and operating system software are examples of realizations of relatively simple, or lower dimensional supercategories,
as explained in further detail in the next subsections.
\end{remark}
\begin{thebibliography}{9}
\bibitem{ICB3}
I.C. Baianu: 1970, Organismic Supercategories: II. On Multistable Systems. \emph{Bulletin of Mathematical Biophysics}, \textbf{32}: 539-561.
\bibitem{ICB1}
I.C. Baianu : 1971a, Organismic Supercategories and Qualitative Dynamics of Systems. \emph{Bulletin of Mathematical Biophysics}, \textbf{33} (3), 339--354.
\bibitem{ICB6}
I.C. Baianu: 1977, A Logical Model of Genetic Activities in \L ukasiewicz Algebras: The Non-linear Theory. \emph{Bulletin of Mathematical Biophysics}, \textbf{39}: 249-258.
\bibitem{ICB7}
I.C. Baianu: 1980, Natural Transformations of Organismic Structures. \emph{Bulletin of Mathematical Biophysics}
\textbf{42}: 431-446.
\bibitem{BBGG1}
I.C. Baianu, Brown R., J. F. Glazebrook, and Georgescu G.: 2006, Complex Nonlinear Biodynamics in
Categories, Higher Dimensional Algebra and \L ukasiewicz--Moisil Topos: Transformations of
Neuronal, Genetic and Neoplastic networks, \emph{Axiomathes} \textbf{16} Nos. 1--2, 65--122.
\bibitem{BHS2}
R. Brown R, P.J. Higgins, and R. Sivera.: \emph{``Non-Abelian Algebraic Topology''},{\em (vol. 2. in preparation)}.
(2008).
\bibitem{BGB2}
R. Brown, J. F. Glazebrook and I. C. Baianu: A categorical and higher dimensional algebra framework for complex systems and spacetime structures, \emph{Axiomathes} \textbf{17}:409--493.
(2007).
\bibitem{BS}
R. Brown and C.B. Spencer: Double groupoids and crossed modules,
\emph{Cahiers Top. G\'eom.Diff.} \textbf{17} (1976), 343--362.
\bibitem{LW1}
W.F. Lawvere: 1963. Functorial Semantics of Algebraic Theories. \emph{Proc. Natl. Acad. Sci. USA}, 50: 869--872
\bibitem{LW2}
W. F. Lawvere: 1966. The Category of Categories as a Foundation for Mathematics. , In {\em Proc. Conf. Categorical Algebra--La Jolla}, 1965, Eilenberg, S et al., eds. Springer --Verlag: Berlin, Heidelberg and New York, pp. 1--20.
\bibitem{LO68}
L. L\"ofgren: 1968. On Axiomatic Explanation of Complete Self--Reproduction. \emph{Bull. Math. Biophysics},
\textbf{30}: 317--348.
\bibitem{SML2k}
S. Mac Lane. 2000. Ch.1: Axioms for Categories, in \emph{Categories for the Working Mathematician}. Springer: Berlin, 2nd Edition.
\end{thebibliography}