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 supercategoriesPlanetmathPlanetmath that are essentially interpretationsMathworldPlanetmathPlanetmath 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 𝕊n+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 𝕊1 is the limiting, lower type, currently defined as a categoryMathworldPlanetmath with only one compositionMathworldPlanetmathPlanetmath law and any standard interpretation of the eight ETAC axioms. Thus, the first level of ’proper’ supercategory 𝕊2 is defined as an interpretation of ETAS axioms S1 and S2; for n=3, the supercategory 𝕊4 is defined as an interpretation of the eight ETAC axioms plus the additional three ETAS axioms: S2, S3 and S4. Any (proper) recursive formulaMathworldPlanetmathPlanetmath or ’function’ can be utilized to generate supercategories at levels n higher than 𝕊4 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 𝕊n 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. 1.

    Functor categoriesPlanetmathPlanetmath subject only to the eight ETAC axioms;

  2. 2.

    FunctorMathworldPlanetmath supercategories, 𝖲:𝒜, with both 𝒜 and being ’large’ categories (i.e., 𝒜 does not need to be small as in the case of functor categories);

  3. 3.

    A topological groupoidPlanetmathPlanetmathPlanetmathPlanetmath category is an example of a particular supercategory with all invertible morphismsMathworldPlanetmath endowed with both a topological and an agebraic structure, still subject to all ETAC axioms;

  4. 4.

    Supergroupoids (also definable as crossed complexes of groupoidsPlanetmathPlanetmathPlanetmath), 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. 5.

    A double groupoidPlanetmathPlanetmathPlanetmath category is a ‘simple’ example of a higher dimensional supercategory which is useful in higher dimensional homotopy theory, especially in non-AbelianMathworldPlanetmathPlanetmath 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;

  6. 6.

    An example of ‘standard’ supercategories was recently introduced in mathematical (or more specifically ‘categorified’) physics, on the web’s 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 categoricalPlanetmathPlanetmath homo-morphisms, homo-functors, and so on; it begins with a somewhat standard definiton of super-categories, or ‘super categories’ from category theoryMathworldPlanetmathPlanetmathPlanetmathPlanetmath, but then it becomes more interesting as it is being tailored to supersymmetry and extensionsPlanetmathPlanetmathPlanetmath of ‘Lie’ superalgebras, or superalgebroids, which are sometimes called graded ‘Lie’ algebrasMathworldPlanetmathPlanetmath that are thought to be relevant to quantum gravity ([6] 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 supercategory is a diagram of the form:


    in Cat–the category of categories and (homo-) functors between categories– such that:


    (where the ‘diamondMathworldPlanetmath’ 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 identityPlanetmathPlanetmathPlanetmathPlanetmath (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 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)=(2𝐼𝑑2)”. Urs called this generalizationPlanetmathPlanetmath of stabilization of n-categories, G(2)-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’;

  7. 7.

    An organismic setPlanetmathPlanetmath of order n can be regarded either as a category of algebraic theories representing organismic sets of different orders on or as a discrete topology organismic supercategoryPlanetmathPlanetmath of algebraic theories (or supercategory only with discrete topology, e.g. , a class of objects);

  8. 8.

    Any ‘standard’ topos with a (commutativePlanetmathPlanetmathPlanetmath) Heyting logic algebraPlanetmathPlanetmath 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. 9.

    The generalized LMn (Łukasiewicz–Moisil) toposes are supercatgeories of non-commutative, algebraic n-valued logic diagrams that are subject to the axioms of LMn algebras of n-valued logics;

  10. 10.

    n-categories are supercategories restricted to interpretations of the ETAC axioms;

  11. 11.

    An organismic supercategory is defined as a supercategory subject to the ETAC axioms and also subject to the ETAS axiom of completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath self–reproduction involving π–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 variableMathworldPlanetmath 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 𝐊=(𝐶,Π,𝑁), where C is an arbitrary category (interpretation of ETAC axioms, formulas, etc.), Π is a category of complete self–reproducing entities, π, ([4]) subject to the negationMathworldPlanetmath of the axiom of restrictionPlanetmathPlanetmathPlanetmathPlanetmath (for elements of sets): S:(S)andu:[uS)v:(vu)and(vS)], (which is known to be independent from the ordinary logico-mathematical and biological reasoning), and N is a category of non-atomic expressions, defined as follows.

Definition 0.2.

An atomically self–reproducing entity is a unit class relationMathworldPlanetmathPlanetmathPlanetmath u such that πππ, which means “π stands in the relation π to π”, πππ,π, etc.

An expression that does not contain any such atomically self–reproducing entity is called a non-atomic expression.


  • 1 See references [13] to [26] in the Bibliography for Category Theory and Algebraic Topology (
  • 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). 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 EXT-2004-059. 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: TransformationsMathworldPlanetmathPlanetmath 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; Preprint No. EXT-2004-072.
Title ETAS interpretation
Canonical name ETASInterpretation
Date of creation 2013-03-22 18:16:04
Last modified on 2013-03-22 18:16:04
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 80
Author bci1 (20947)
Entry type Topic
Classification msc 81-00
Classification msc 92B05
Classification msc 03G30
Classification msc 18-00
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