ETAS interpretation
0.1 Introduction
ETAS is the acronym for the “Elementary Theory of Abstract Supercategories” as defined by the axioms of metacategories and supercategories.
The following are simple examples of supercategories^{} that are essentially interpretations^{} of the eight ETAC axioms reported by W. F. Lawvere (1968), with one or several ETAS axioms added as indicated in the examples listed. A family, or class, of a specific level (or ’order’) $(n+1)$ of a supercategory ${\mathbb{S}}_{n+1}$ (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 ${\mathbb{S}}_{1}$ 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 ${\mathbb{S}}_{2}$ is defined as an interpretation of ETAS axioms S1 and S2; for $n=3$, the supercategory ${\mathbb{S}}_{4}$ is defined as an interpretation of the eight ETAC axioms plus the additional three ETAS axioms: S2, S3 and S4. Any (proper) recursive formula^{} or ’function’ can be utilized to generate supercategories at levels $n$ higher than ${\mathbb{S}}_{4}$ by adding composition or consistency laws to the ETAS axioms S1 to S4, thus allowing a digital computer algorithm to generate any finite level supercategory ${\mathbb{S}}_{n}$ syntax, to which one needs then to add semantic interpretations (which are complementary to the computer generated syntax).
0.2 Simple examples of ETAS interpretation in supercategories

1.
Functor categories^{} subject only to the eight ETAC axioms;

2.
Functor^{} supercategories, ${\mathcal{F}}_{\U0001d5b2}:\mathcal{A}\to \mathcal{B}$, 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 functor categories);

3.
A 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;

4.
Supergroupoids (also definable as crossed complexes of groupoids^{}), and 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.)

5.
A double groupoid^{} category is a ‘simple’ example of a higher dimensional supercategory which is useful in higher dimensional homotopy theory, especially in nonAbelian^{} algebraic topology; this concept is subject to all eight ETAC axioms, plus additional axioms related to the definition of the double groupoid (generally nonAbelian) structures;

6.
An example of ‘standard’ supercategories was recently introduced in mathematical (or more specifically ‘categorified’) physics, on the web’s http://golem.ph.utexas.edu/category/2007/07/supercategories.htmlnCategory café’s web site under “Supercategories”. This is a rather ‘simple’ example of supercategories, albeit in a much more restricted sense as it still involves only the standard categorical^{} homomorphisms, homofunctors, and so on; it begins with a somewhat standard definiton of supercategories, 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 ([6] and references cited therein). The following is an almost exact quote from the above nCategory cafe’ s website posted mainly by Dr. Urs Schreiber: A supercategory is a diagram of the form:
$$\diamond \diamond I{d}_{C}\diamond \text{\mathbf{C}}\diamond \diamond s$$ in Cat–the category of categories and (homo) functors between categories– such that:
$$\diamond \diamond \text{\mathit{I}\mathit{d}}\diamond \diamond I{d}_{C}\diamond \text{\mathbf{C}}\diamond \text{\mathbf{C}}\diamond \diamond s\diamond \diamond s=\diamond \diamond I{d}_{C}\diamond I{d}_{C}\diamond \diamond \text{\mathit{I}\mathit{d}},$$ (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 one object– the above quoted superdiagram of diamonds, an arbitrary abstract category C (subject to all ETAC axioms), and the standard category identity^{} (homo) functor; it can be further specialized to the previously introduced concepts of supergroupoids (also definable as crossed complexes of groupoids), and 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 braided monoidal supercategory ?”. Dr. Urs Schreiber, suggested the following answer: “like an ordinary braided monoidal catgeory is a 3category which in lowest degrees looks like the trivial 2group, a braided monoidal supercategory is a 3category which in lowest degree looks like the strict 2group that comes from the crossed module $G(2)=(\diamond 2\diamond \text{\mathit{I}\mathit{d}}\diamond 2)$”. Urs called this generalization^{} of stabilization of ncategories, $G(2)$stabilization. Therefore, the claim would be that ‘braided monoidal supercategories come from $G(2)$stabilized 3categories, with $G(2)$ the above strict 2group’;

7.
An organismic set^{} of order $n$ can be regarded either as a category of algebraic theories representing organismic sets of different orders $o\le n$ or as a discrete topology organismic supercategory^{} of algebraic theories (or supercategory only with discrete topology, e.g. , a class of objects);

8.
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;

9.
The generalized $L{M}_{n}$ (Łukasiewicz–Moisil) toposes are supercatgeories of noncommutative, algebraic $n$valued logic diagrams that are subject to the axioms of $L{M}_{n}$ algebras of $n$valued logics;

10.
$n$categories are supercategories restricted to interpretations of the ETAC axioms;

11.
An 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 (viz. Löfgren, 1968; [1]); its objects are classes representing organisms in terms of morphism (super) diagrams or equivalently as heterofunctors of organismic classes with variable^{} topological structure;
Definition 0.1.
Organismic Supercategories ([1]) 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 ([1]); organismic supercategories were thus defined as superstructure interpretations of ETAS (including ETAC, as appropriate) in terms of triplets $\text{\mathbf{K}}=(\text{\mathit{C}},\mathrm{\Pi},\text{\mathit{N}})$, where C is an arbitrary category (interpretation of ETAC axioms, formulas, etc.), $\mathrm{\Pi}$ is a category of complete self–reproducing entities, $\pi $, ([4]) subject to the negation^{} of the axiom of restriction^{} (for elements of sets): $\exists S:(S\ne \oslash )and\forall u:[u\in S)\Rightarrow \exists v:(v\in u)and(v\in S)]$, (which is known to be independent from the ordinary logicomathematical and biological reasoning), and N is a category of nonatomic expressions, defined as follows.
Definition 0.2.
An atomically self–reproducing entity is a unit class relation^{} $u$ such that $\pi \pi \u27e8\pi \u27e9$, which means “$\pi $ stands in the relation $\pi $ to $\pi $”, $\pi \pi \u27e8\pi ,\pi \u27e9$, etc.
An expression that does not contain any such atomically self–reproducing entity is called a nonatomic expression.
References
 1 See references [13] to [26] in the Bibliography for Category Theory and Algebraic Topology (http://planetmath.org/CategoricalOntologyABibliographyOfCategoryTheory)
 2 W.F. Lawvere: 1963. Functorial Semantics of Algebraic Theories., Proc. Natl. Acad. Sci. USA, 50: 869–872.
 3 W. F. Lawvere: 1966. The Category of Categories as a Foundation for Mathematics. , In Proc. Conf. Categorical Algebra–La Jolla, 1965, Eilenberg, S et al., eds. Springer –Verlag: Berlin, Heidelberg and New York, pp. 1–20.
 4 L. Löfgren: 1968. On Axiomatic Explanation of Complete Self–Reproduction. Bull. Math. Biophysics, 30: 317–348.
 5 R. Brown R, P.J. Higgins, and R. Sivera.: “Non–Abelian Algebraic Topology” (2008). http://www.bangor.ac.uk/mas010/nonabt/partI010604.pdfPDF file
 6 R. Brown, J. F. Glazebrook and I. C. Baianu: A categorical and higher dimensional algebra framework for complex systems and spacetime structures, Axiomathes 17:409–493. (2007).
 7 R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales–Bangor, Maths Preprint, 1986.
 8 R. Brown and C.B. Spencer: Double groupoids and crossed modules, Cahiers Top. Géom.Diff. 17 (1976), 343–362.
 9 I.C. Baianu: Łukasiewicz–Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN Preprint EXT2004059. Health Physics and Radiation Effects (June 29, 2004).
 10 I.C. Baianu, Brown R., J. F. Glazebrook, and Georgescu G.: 2006, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations^{} of Neuronal, Genetic and Neoplastic networks, Axiomathes 16 Nos. 1–2, 65–122.
 11 I.C. Baianu and M. Marinescu: 1974, A Functorial Construction of (M,R)– Systems. Revue Roumaine de Mathematiques Pures et Appliquees 19: 388–391.
 12 I.C. Baianu: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non–linear Theory. Bulletin of Mathematical Biophysics, 39: 249–258.
 13 I.C. Baianu: 1987a, Computer Models and Automata Theory in Biology and Medicine., in M. Witten (ed.), Mathematical Models in Medicine, vol. 7., Pergamon Press, New York, 1513–1577; http://doc.cern.ch//archive/electronic/other/ext/ext2004072.pdfCERN Preprint No. EXT2004072.
Title  ETAS interpretation 
Canonical name  ETASInterpretation 
Date of creation  20130322 18:16:04 
Last modified on  20130322 18:16:04 
Owner  bci1 (20947) 
Last modified by  bci1 (20947) 
Numerical id  80 
Author  bci1 (20947) 
Entry type  Topic 
Classification  msc 8100 
Classification  msc 92B05 
Classification  msc 03G30 
Classification  msc 1800 
Synonym  elementary theory of abstract supercategories 
Synonym  ETAS 
Related topic  Category 
Related topic  CategoryTheory 
Related topic  ETAS 
Related topic  CategoricalOntologyABibliographyOfCategoryTheory 
Related topic  AlgebraicComputation 
Related topic  CategoricalOntology 
Related topic  QuantumLogic 
Related topic  CategoryOfQuantumAutomata 
Related topic  FunctorCategory2 
Related topic  QuantumAutomataAndQuantumComputation2 
Related topic  SupercategoryOfVariableMolecularSets 
Related topic  ETA 
Defines  axioms of metacategories and supercategories 
Defines  examples of supercategories and metacategories 
Defines  ETAS interpretation 
Defines  ETAS axiom 
Defines  ETAS 