ETASInterpretation 2009-03-19 00:05:52 2009-03-19 00:05:52 Topic % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym} \usepackage{xypic} \usepackage[mathscr]{eucal} \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote%%@ }}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\GL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} \newcommand{\G}{\mathcal G} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathbb G}} \newcommand{\dgrp}{{\mathbb D}} \newcommand{\desp}{{\mathbb D^{\rm{es}}}} \newcommand{\Geod}{{\rm Geod}} \newcommand{\geod}{{\rm geod}} \newcommand{\hgr}{{\mathbb H}} \newcommand{\mgr}{{\mathbb M}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathbb G)}} \newcommand{\obgp}{{\rm Ob(\mathbb G')}} \newcommand{\obh}{{\rm Ob(\mathbb H)}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\ghomotop}{{\rho_2^{\square}}} \newcommand{\gcalp}{{\mathbb G(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\glob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\oset}[1]{\overset {#1}{\ra}} \newcommand{\osetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \subsection{Introduction} \emph{ETAS} is the acronym for the \emph{``Elementary Theory of Abstract Supercategories''} as defined by the \PMlinkname{axioms of metacategories and supercategories}{AxiomsOfMetacategoriesAndSupercategories}. The following are simple examples of supercategories that are essentially interpretations of the eight \emph{ETAC} 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 ETAS axioms added to the eight ETAC axioms; 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 formula 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 computer algorithm 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{topological groupoid 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 groupoids), and \emph{supergroups} --also definable as crossed modules of groups-- seem to be of great interest to mathematicians currently involved in `categorified' mathematical physics or physical mathematics.) \item A \emph{double groupoid category} is a `simple' example of a higher dimensional supercategory which is useful in higher dimensional homotopy theory, especially in non-Abelian algebraic topology; this concept 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 \PMlinkexternal{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 super-categories, or `super categories' from category theory, but then it becomes more interesting as it is being tailored to supersymmetry and extensions of `Lie' superalgebras, or superalgebroids, 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 n-Category 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{organismic set} of order $n$ can be regarded either as a category of algebraic theories representing organismic sets 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 `standard' topos with a (commutative) Heyting logic algebra as a subobject classifier 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 complexity (\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 \PMlinkname{Bibliography for Category Theory and Algebraic Topology}{CategoricalOntologyABibliographyOfCategoryTheory} \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). \PMlinkexternal{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; \PMlinkexternal{CERN Preprint No. EXT-2004-072}{http://doc.cern.ch//archive/electronic/other/ext/ext-2004-072.pdf}. \end{thebibliography}