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\begin{document}

 \subsection{Introduction}

\emph{\htmladdnormallink{ETAS}{http://planetphysics.org/encyclopedia/ETACAxioms.html}} is the acronym for the \emph{``Elementary Theory of Abstract Supercategories''}
as defined by the
\htmladdnormallink{axioms of metacategories and supercategories}{http://planetphysics.org/encyclopedia/AxiomsOfMetacategoriesAndSupercategories.html}.


The following are simple examples of \htmladdnormallink{supercategories}{http://planetphysics.org/encyclopedia/ETACAxioms.html} that are essentially interpretations
of the eight \emph{\htmladdnormallink{ETAC}{http://planetphysics.org/encyclopedia/Formula.html}} axioms reported by W. F. Lawvere (1968), with one or several \emph{ETAS} axioms added as indicated in the examples listed. A family, or class, of a specific level (or 'order') $(n+1)$ of a
supercategory $\mathcal{\S}_{n+1}$ (with $n$ being an integer) is defined by the specific \htmladdnormallink{ETAS axioms}{http://planetphysics.org/encyclopedia/ETACAxioms.html} added to the eight \htmladdnormallink{ETAC axioms}{http://planetphysics.org/encyclopedia/ETACAxioms.html}; thus, for $n=0$, there are no additional ETAS axioms and the supercategory $\mathcal{\S}_1$ is the limiting, lower type, currently defined as a category with only one composition law and any standard interpretation of the eight ETAC axioms. Thus, the first level of 'proper' supercategory $\mathcal{\S}_2$ is defined as an interpretation of ETAS axioms \textbf{S1} and \textbf{S2}; for $n=3$, the supercategory $\mathcal{\S}_4$ is defined as an interpretation of the eight ETAC axioms plus the additional three ETAS axioms: \textbf{S2}, \textbf{S3} and \textbf{S4}. Any (proper) recursive \htmladdnormallink{formula}{http://planetphysics.org/encyclopedia/Formula.html} or 'function' can be utilized to generate supercategories at levels $n$ higher than $\mathcal{\S}_4$ by adding composition or consistency laws to the ETAS axioms \textbf{S1} to \textbf{S4}, thus allowing a digital \htmladdnormallink{computer}{http://planetphysics.org/encyclopedia/Program3.html} \htmladdnormallink{algorithm}{http://planetphysics.org/encyclopedia/RecursiveFunction.html} to generate any finite level supercategory $\mathcal{\S}_n$ syntax, to which one needs then to add semantic interpretations (which are complementary to the computer generated syntax).


\subsection{Simple examples of ETAS interpretation in supercategories}

\begin{enumerate}

\item \emph{Functor categories} subject only to the eight \emph{ETAC} axioms;
\item {\em Functor supercategories}, $\mathsf{\F_S}: \mathcal{A} \to \mathcal{B}$,
with both $\mathcal{A}$ and $\mathcal{B}$ being 'large' categories (i.e.,
$\mathcal{A}$ does not need to be small as in the case of \emph{functor categories});
\item A \emph{\htmladdnormallink{topological groupoid}{http://planetphysics.org/encyclopedia/GroupoidHomomorphism2.html} category} is an example of a particular supercategory
with all invertible morphisms endowed with both a topological and an agebraic
structure, still subject to all ETAC axioms;
\item \emph{Supergroupoids} (also definable as crossed complexes of \htmladdnormallink{Groupoids}{http://planetphysics.org/encyclopedia/GroupoidHomomorphism2.html}), and \emph{\htmladdnormallink{supergroups}{http://planetphysics.org/encyclopedia/Paragroups.html}} --also definable as crossed \htmladdnormallink{modules}{http://planetphysics.org/encyclopedia/RModule.html} of groups-- seem to be of great interest to mathematicians currently involved in `categorified' \htmladdnormallink{mathematical physics}{http://planetphysics.org/encyclopedia/PhysicalMathematics2.html} or physical mathematics.)
\item A \emph{\htmladdnormallink{double groupoid}{http://planetphysics.org/encyclopedia/HigherDimensionalHomotopy.html} category} is a `simple' example of a higher dimensional supercategory which is useful in \htmladdnormallink{higher dimensional homotopy}{http://planetphysics.org/encyclopedia/HigherDimensionalHomotopy.html} theory, especially in non-Abelian algebraic topology;
this \htmladdnormallink{concept}{http://planetphysics.org/encyclopedia/PreciseIdea.html} is subject to all eight ETAC axioms, plus additional axioms related to the definition of the double groupoid (generally non-Abelian) structures;

\item
An example of `standard' supercategories was recently introduced in mathematical (or more specifically `categorified') physics, on the web's \htmladdnormallink{n-Category caf\'e's web site}{http://golem.ph.utexas.edu/category/2007/07/supercategories.html} under \textit{``Supercategories''}. This is a rather `simple' example of supercategories, albeit in a much more restricted sense as it still involves only the standard categorical homo-morphisms, homo-functors, and so on; it begins with a somewhat standard definiton of \htmladdnormallink{super-categories}{http://planetphysics.org/encyclopedia/ETACAxioms.html}, or `super categories' from category theory, but then it becomes more interesting as it is being tailored to \htmladdnormallink{supersymmetry}{http://planetphysics.org/encyclopedia/GradedAlgebra.html} and extensions of `\htmladdnormallink{Lie' superalgebras}{http://planetphysics.org/encyclopedia/GradedAlgebra.html}, or \htmladdnormallink{superalgebroids}{http://planetphysics.org/encyclopedia/GeneralizedSuperalgebras.html}, which are sometimes called graded `Lie' algebras that are thought to be relevant to quantum gravity (\cite{BGB2} and references cited therein). The following is an almost exact quote from the above \htmladdnormallink{n-category}{http://planetphysics.org/encyclopedia/InfinityGroupoid.html} cafe' s website posted mainly by Dr. Urs Schreiber:
A \textit{supercategory} is a \textit{diagram} of the form:
$$\diamond  \diamond Id_C \diamond \textbf{C} \diamond \diamond s $$
in \textbf{Cat}--the category of categories and (homo-) functors between categories-- such that:
$$\diamond  \diamond \textsl{Id} \diamond \diamond Id_C \diamond \textbf{C} \diamond \textbf{C}\diamond \diamond s \diamond \diamond s = \diamond  \diamond Id_C \diamond Id_C  \diamond  \diamond \textsl{Id},$$
(where the `diamond' symbol should be replaced by the symbol `square', as in the original Dr. Urs Schreiber's postings.)

This specific instance is that of a supercategory which has only \textbf{one object}-- the above quoted superdiagram of diamonds, an arbitrary abstract category \textbf{C} (subject to all ETAC axioms), and the standard category identity (homo-) functor; it can be further specialized to the previously introduced concepts of \textit{supergroupoids} (also definable as crossed complexes of groupoids), and \textit{supergroups} (also definable as crossed modules of groups), which seem to be of great interest to mathematicians involved in `Categorified' mathematical physics or physical mathematics.) This was then continued with the following interesting example. ``What, in this sense, is a \textit{braided monoidal supercategory ?}''. Dr. Urs Schreiber, suggested the following answer: ``like an ordinary braided monoidal catgeory is a 3-category which in lowest degrees looks like the trivial 2-group, a braided monoidal supercategory is a 3-category which in lowest degree looks like the strict 2-group that comes from
the crossed module $G(2)=(\diamond 2 \diamond \textsl{Id} \diamond 2)$''. Urs called this generalization of stabilization of n-categories, $G(2)$-\textit{stabilization}. Therefore, the claim would be that `braided monoidal supercategories come from $G(2)$-stabilized 3-categories, with $G(2)$ the above strict 2-group';

\item An \emph{\htmladdnormallink{organismic set}{http://planetphysics.org/encyclopedia/TheoryOfOrganismicSets.html}} of order $n$ can be regarded either as a category of algebraic
theories representing \htmladdnormallink{organismic sets}{http://planetphysics.org/encyclopedia/RSystemsCategory.html} of different orders $o \leq n$ or as a \emph{discrete topology} organismic supercategory of algebraic theories (or supercategory only with discrete topology, e.g. , a \emph{class} of objects);
\item Any `\htmladdnormallink{standard' topos}{http://planetphysics.org/encyclopedia/GrothendieckTopos.html} with a (commutative) \htmladdnormallink{Heyting logic algebra as a subobject classifier}{http://planetphysics.org/encyclopedia/HeytingLogicAlgebraAsASubobjectClassifier.html} is an example of
a commutative (and distributive) supercategory with the additional axioms to ETAC being those that
define the Heyting logic algebra;
\item The generalized $LM_n$ (\L{}ukasiewicz- Moisil) toposes are supercatgeories of
\emph{non-commutative}, algebraic $n$-valued logic diagrams that are subject to the axioms of \emph{$LM_n$ algebras of
$n$-valued logics};
\item $n$-categories are supercategories restricted to interpretations of the ETAC axioms;
\item An \emph{organismic supercategory} is defined as a supercategory subject to the ETAC axioms
and also subject to the ETAS axiom of complete self-reproduction involving
$\pi$-entities (\emph{viz}. L\"ofgren, 1968; \cite{Refs-13to26}); its objects are classes representing organisms
in terms of morphism (super) diagrams or equivalently as heterofunctors of organismic classes
with variable topological structure;
\end{enumerate}

\begin{definition}
\emph{Organismic Supercategories (\cite{Refs-13to26})}
An example of a class of supercategories interpreting such ETAS axioms as those stated above
was previously defined for organismic structures with different levels of \htmladdnormallink{complexity}{http://planetphysics.org/encyclopedia/Complexity.html} (\cite{Refs-13to26}); \textit{organismic supercategories} were thus defined as \textit{superstructure interpretations of ETAS} (including ETAC, as appropriate) in terms of triples $\textbf{K} = (\emph{C}, \Pi, \textit{N})$, where \emph{C} is an arbitrary category (interpretation of ETAC axioms, formulas, etc.), $\Pi$ is a category of complete self--reproducing entities, $\pi$, (\cite{LO68}) subject to the negation of the axiom of restriction (for elements of sets):
$ \exists S: (S \neq \oslash) ~ and ~ \forall u: [u \in S) \Rightarrow \exists v: (v \in u)~ and ~( v \in S)]$, (which is known to be independent from the ordinary logico-mathematical and biological reasoning),
and $\textit{N}$ is a category of non-atomic expressions, defined as follows.
\end{definition}

\begin{definition}

An \textit{atomically self--reproducing entity} is a unit class relation $u$ such that $\pi \pi \left\langle \pi \right\rangle$, which means
``$\pi$ stands in the relation $\pi$ to $\pi$'', $\pi \pi \left\langle \pi , \pi \right\rangle$, etc.

An expression that does not contain any such atomically self--reproducing entity is called a \textit{non-atomic expression}.
\end{definition}


\begin{thebibliography}{9}

\bibitem{Refs-13to26}
See references [13] to [26] in the Bibliography for Category Theory and Algebraic Topology

\bibitem{LW1}
W.F. Lawvere: 1963. Functorial Semantics of Algebraic Theories., \emph{Proc. Natl. Acad. Sci. USA}, \textbf{50}: 869--872.

\bibitem{LW2}
W. F. Lawvere: 1966. The Category of Categories as a Foundation for Mathematics. , In \emph{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{BHS2}
R. Brown R, P.J. Higgins, and R. Sivera.: \textit{``Non--Abelian Algebraic Topology''} (2008).
\htmladdnormallink{PDF file}{http://www.bangor.ac.uk/mas010/nonab--t/partI010604.pdf}

\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{BM}
R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales--Bangor, Maths Preprint, 1986.

\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{ICB04b}
I.C. Baianu: \L ukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN Preprint EXT-2004-059. \textit{Health Physics and Radiation Effects} (June 29, 2004).

\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{ICBm2}
I.C. Baianu and M. Marinescu: 1974, A Functorial Construction of \emph{\textbf{(M,R)}}-- Systems. \emph{Revue Roumaine de Mathematiques Pures et Appliquees} \textbf{19}: 388-391.

\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{ICB2}
I.C. Baianu: 1987a, Computer Models and Automata Theory in Biology and Medicine.,  in M. Witten (ed.),
\emph{Mathematical Models in Medicine}, vol. 7., Pergamon Press, New York, 1513-1577;
\htmladdnormallink{CERN Preprint No. EXT-2004-072}{http://doc.cern.ch//archive/electronic/other/ext/ext-2004-072.pdf}.
\end{thebibliography} 

\end{document}
