PlanetPhysics/ETAS Interpretation

Introduction

[[../ETACAxioms/|ETAS]] is the acronym for the "Elementary Theory of Abstract Supercategories" as defined by the [[../AxiomsOfMetacategoriesAndSupercategories/|axioms of metacategories and supercategories]].

The following are simple examples of [[../SuperCategory6/|supercategories]] that are essentially interpretations of the eight [[../ETACAxioms/|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 ${\mathrm{\S }}_{n+1}$ (with $n$ being an integer) is defined by the specific [[../ETACAxioms/|ETAS axioms]] added to the eight [[../ETACAxioms/|ETAC axioms]]; thus, for $n=0$, there are no additional ETAS axioms and the supercategory ${\mathrm{\S }}_1$ is the limiting, lower [[../Bijective/|type]], currently defined as a [[../Cod/|category]] with only one [[../Identity2/|composition law]] and any standard interpretation of the eight ETAC axioms. Thus, the first level of 'proper' supercategory ${\mathrm{\S }}_2$ is defined as an interpretation of ETAS axioms S1 and S2 ; for $n=3$, the supercategory ${\mathrm{\S }}_4$ is defined as an interpretation of the eight ETAC axioms plus the additional three ETAS axioms: S2 , S3 and S4 . Any (proper) recursive [[../Formula/|formula]] or [[../Bijective/|'function]]' can be utilized to generate supercategories at levels $n$ higher than ${\mathrm{\S }}_4$ by adding [[../Cod/|composition]] or consistency laws to the ETAS axioms S1 to S4 , thus allowing a digital [[../Program3/|computer]] [[../RecursiveFunction/|algorithm]] to generate any finite level supercategory ${\mathrm{\S }}_n$ syntax, to which one needs then to add semantic interpretations (which are complementary to the computer generated syntax).

Simple examples of ETAS interpretation in supercategories

1. [[../TrivialGroupoid/|functor categories]] subject only to the eight ETAC axioms;
2. \htmladdnormallink{functor {http://planetphysics.us/encyclopedia/TrivialGroupoid.html} supercategories}, Failed to parse (unknown function "\F"): {\displaystyle \mathsf{\F_S}: \mathcal{A} \to \mathcal{B}} ,

with both $𝒜$ and $ℬ$ being [[../Cod/|'large' categories]] (i.e., $𝒜$ does not need to be small as in the case of functor categories );

1. A \htmladdnormallink{topological groupoid {http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} category} is an example of a particular supercategory with all invertible [[../TrivialGroupoid/|morphisms]] endowed with both a [[../CoIntersections/|topological]] and an agebraic structure, still subject to all ETAC axioms;
2. [[../Supergroup/|supergroupoids]] (also definable as [[../SingularComplexOfASpace/|crossed complexes]] of [[../GroupoidHomomorphism2/|groupoids]]), and [[../Paragroups/|supergroups]] --also definable as [[../CubicalHigherHomotopyGroupoid/|crossed modules]] of groups-- seem to be of great interest to mathematicians currently involved in `categorified' [[../PhysicalMathematics2/|mathematical physics]] or [[../NonNewtonian2/|physical mathematics]].)
3. A \htmladdnormallink{double groupoid {http://planetphysics.us/encyclopedia/ThinEquivalence.html} category} is a `simple' example of a higher dimensional supercategory which is useful in [[../ThinEquivalence/|higher dimensional homotopy]] theory, especially in [[../ModuleAlgebraic/|non-Abelian algebraic topology]];

this [[../PreciseIdea/|concept]] is subject to all eight ETAC axioms, plus additional axioms related to the definition of the double groupoid (generally [[../AbelianCategory3/|non-Abelian]]) structures;

1. An example of `standard' supercategories was recently introduced in mathematical (or more specifically `categorified') physics, on the web's n-Category caf\'e'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 homo-morphisms, homo-functors, and so on; it begins with a somewhat standard definiton of [[../SuperCategory6/|super-categories]], or `super categories' from [[../TrivialGroupoid/|category theory]], but then it becomes more interesting as it is being tailored to [[../Supersymmetry/|supersymmetry]] and extensions of `[[../AntiCommutationRelations/|Lie' superalgebras]], or [[../GeneralizedSuperalgebras/|superalgebroids]], which are sometimes called graded `[[../BilinearMap/|Lie' algebras]] that are thought to be relevant to [[../LQG2/|quantum gravity]] ( and references cited therein). The following is an almost exact quote from the above [[../InfinityGroupoid/|n-category]] cafe' s website posted mainly by Dr. Urs Schreiber:

A supercategory is a diagram of the form: $\diamond \diamond I{d}_C{\diamond }^{‴}{C}^{‴}\diamond \diamond s$ in Cat --the category of categories and (homo-) functors between categories-- such that: Failed to parse (unknown function "\textsl"): {\displaystyle \diamond \diamond \textsl{Id} \diamond \diamond Id_C \diamond '''C''' \diamond '''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 `[[../PiecewiseLinear/|square',]] as in the original Dr. Urs Schreiber's postings.)

This specific instance is that of a supercategory which has only one [[../TrivialGroupoid/|object']]-- the above quoted [[../SuperdiagramsAsHeterofunctors/|superdiagram]] of diamonds, an arbitrary abstract category C (subject to all ETAC axioms), and the standard category [[../Cod/|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 [[../TrivialGroupoid/|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 Failed to parse (unknown function "\textsl"): {\displaystyle G(2)=(\diamond 2 \diamond \textsl{Id} \diamond 2)} . Urs called this generalization 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';

1. An [[../TheoryOfOrganismicSets/|organismic set]] of order $n$ can be regarded either as a category of [[../CoIntersections/|algebraic]] theories representing [[../RSystemsCategory/|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);
2. Any `[[../GrothendieckTopos/|standard' topos]] with a (commutative) [[../SUSY2/|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;

1. The generalized $L{M}_n$ (\L{}ukasiewicz- Moisil) toposes are supercatgeories of

non-commutative , algebraic $n$-valued logic [[../TrivialGroupoid/|diagrams]] that are subject to the axioms of \emph{$L{M}_n$ algebras of $n$-valued logics};

1. $n$-categories are supercategories restricted to interpretations of the ETAC axioms;
2. 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 . L\"ofgren, 1968; ); its objects are classes representing organisms in terms of morphism (super) diagrams or equivalently as heterofunctors of organismic classes with variable [[../TrivialGroupoid/|topological structure]];

Organismic Supercategories () An example of a class of supercategories interpreting such ETAS axioms as those stated above was previously defined for [[../VariableCategory2/|organismic structures]] with different levels of [[../Complexity/|complexity]] (); organismic supercategories were thus defined as superstructure interpretations of ETAS (including ETAC, as appropriate) in terms of triples $K^{‴}={(}^{″}{C}^{″},\mathrm{\Pi }{,}^{″}{N}^{″})$, where C is an arbitrary category (interpretation of ETAC axioms, formulas, etc.), $\mathrm{\Pi }$ is a category of complete self--reproducing entities, $\pi$, () subject to the negation of the axiom of restriction (for elements of sets): $\mathrm{\exists }S:(S\ne \oslash )and\mathrm{\forall }u:[u\in S)\Rightarrow \mathrm{\exists }v:(v\in u)and(v\in S)]$, (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.

An atomically self--reproducing entity is a unit class [[../Bijective/|relation]] $u$ such that $\pi \pi ⟨\pi ⟩$, which means "$\pi$ stands in the relation $\pi$ to $\pi$", $\pi \pi ⟨\pi ,\pi ⟩$, etc.

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