PlanetPhysics/ETAS Interpretation

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 $$\mathcal{\S}_{n+1}$$ (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 S1 and 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: S2, S3  and 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 S1 to 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).

Simple examples of ETAS interpretation in supercategories
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 ); this concept is subject to all eight ETAC axioms, plus additional axioms related to the definition of the double groupoid (generally non-Abelian) structures;
 * 1) functor categories subject only to the eight ETAC axioms;
 * 2) \htmladdnormallink{functor {http://planetphysics.us/encyclopedia/TrivialGroupoid.html} supercategories}, $$\mathsf{\F_S}: \mathcal{A} \to \mathcal{B}$$,
 * 1) A \htmladdnormallink{topological groupoid {http://planetphysics.us/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;
 * 2) 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.)
 * 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 higher dimensional homotopy theory, especially in non-Abelian algebraic topology;

A supercategory is a diagram  of the form: $$\diamond \diamond Id_C \diamond C  \diamond \diamond s $$ in Cat --the category of categories and (homo-) functors between categories-- such that: $$\diamond \diamond \textsl{Id} \diamond \diamond Id_C \diamond C  \diamond C \diamond \diamond s \diamond \diamond s = \diamond  \diamond Id_C \diamond Id_C  \diamond  \diamond \textsl{Id},$$ (where the `diamond' symbol should be replaced by the symbol `square', as in the original Dr. Urs Schreiber's postings.)
 * 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 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 ( 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:

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 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, $$G(2)$$-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';

a commutative (and distributive) supercategory with the additional axioms to ETAC being those that define the Heyting logic algebra; non-commutative, algebraic $$n$$-valued logic diagrams that are subject to the axioms of \emph{$$LM_n$$ algebras of $$n$$-valued logics}; 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 topological structure;
 * 1) An 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 discrete topology organismic supercategory of algebraic theories (or supercategory only with discrete topology, e.g., a class  of objects);
 * 2) Any `standard' topos with a (commutative) Heyting logic algebra as a subobject classifier is an example of
 * 1) The generalized $$LM_n$$ (\L{}ukasiewicz- Moisil) toposes are supercatgeories of
 * 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

Organismic Supercategories  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 ; organismic supercategories were thus defined as superstructure interpretations of ETAS  (including ETAC, as appropriate) in terms of triples $$K  = (C, \Pi, N )$$, where C  is an arbitrary category (interpretation of ETAC axioms, formulas, etc.), $$\Pi$$ is a category of complete self--reproducing entities, $$\pi$$,  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 $$N $$ is a category of non-atomic expressions, defined as follows.

An 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 non-atomic expression.