Talk:PlanetPhysics/Supercategory

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\begin{document}

 \begin{definition}
A \emph{supercategory} is defined axiomatically in terms of ETAS (\cite{ICB3}) as a \htmladdnormallink{non-Abelian}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} structure consisting of classes of \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $\mathcal{O}$ and/or classes of \htmladdnormallink{categorical diagrams}{http://planetphysics.us/encyclopedia/CategoricalDiagramsDefinedByFunctors.html} $\mathcal{D}$ with multiple structures $S_0, S_1, ..., S_n$ (with $(n+1)$ being an integer called its \emph{structural dimension} or \emph{order})-- such as, \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html}, \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html}, geometric, and analytical-- that also have defined distinct \emph{ETAS \htmladdnormallink{composition laws}{http://planetphysics.us/encyclopedia/Identity2.html}} $\Gamma_1 := \circ, \Gamma_2 := *, \Gamma_3:= **, ...,\Gamma_n$, defined for each \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html} of structure, subject to the ETAS composition law axioms for $\Gamma$'s. Such composition laws form a finite collection denoted as
$\mathcal{L}_{Sn}$.
\end{definition}

\subsection{Examples}

\begin{example}

Let us consider first a simple example of an $\mathcal{S}_1$ supercategory of 1st order defined with a single composition law of \htmladdnormallink{functors}{http://planetphysics.us/encyclopedia/Functor.html} as in the standard definition of a functor category. Thus, a less \htmladdnormallink{general definition}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of supercategory was recently introduced in mathematical (`categorified') physics, on the web's \htmladdnormallink{n-category}{http://planetphysics.us/encyclopedia/InfinityGroupoid.html} caf\'e s web site under \htmladdnormallink{``Supercategories''}{http://golem.ph.utexas.edu/category/2007/07/supercategories.html}. This is a rather `simple' example of supercategory, albeit in a much more restricted sense as it still involves only one composition law for
the functors, and also its components are all \htmladdnormallink{small categories}{http://planetphysics.us/encyclopedia/Cod.html}; this \htmladdnormallink{functor category}{http://planetphysics.us/encyclopedia/Functor.html} is sometimes called a
`\emph{super-category}' in the published literature. Thus, the following specific example of an $\mathcal{S}_1$ supercategory begins with a standard definiton of such a super-category, or `super \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html}' (functor category) from \htmladdnormallink{category theory}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}; it becomes interesting as it is being tailored to \htmladdnormallink{supersymmetry}{http://planetphysics.us/encyclopedia/AntiCommutationRelations.html} and extensions of `\htmladdnormallink{Lie' superalgebras}{http://planetphysics.us/encyclopedia/AntiCommutationRelations.html}, or \htmladdnormallink{superalgebroids}{http://planetphysics.us/encyclopedia/GeneralizedSuperalgebras.html}, which are sometimes called graded `\htmladdnormallink{Lie' algebras}{http://planetphysics.us/encyclopedia/BilinearMap.html} that are thought to be relevant to \htmladdnormallink{quantum gravity}{http://planetphysics.us/encyclopedia/LQG2.html} (\cite{BGB2} and references cited therein). The following is an almost exact quote from the above n-Category cafe's website posted by Dr. Urs Schreiber:
\end{example}

\begin{definition}
A \htmladdnormallink{super-category}{http://planetphysics.us/encyclopedia/2Category2.html} is a \emph{\htmladdnormallink{diagram}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}} 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 `\htmladdnormallink{square',}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} as in the original Dr.Urs Schreiber's postings.)
\end{definition}
This specific instance is that of a supercategory which has only \textbf{one object}-- the above quoted \htmladdnormallink{superdiagram}{http://planetphysics.us/encyclopedia/SuperdiagramsAsHeterofunctors.html} of diamonds, an arbitrary abstract category \textbf{C} (subject to all ETAC axioms), and the standard category \htmladdnormallink{identity}{http://planetphysics.us/encyclopedia/Cod.html} (homo-) functor; it can be further specialized to the previously introduced concepts of \textit{supergroupoids} (also definable as \htmladdnormallink{crossed complexes}{http://planetphysics.us/encyclopedia/SingularComplexOfASpace.html} of \htmladdnormallink{groupoids}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}), and \textit{supergroups} (also definable as \htmladdnormallink{crossed modules}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} of \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}), which seem to be of great interest to mathematicians involved in `Categorified' \htmladdnormallink{mathematical physics}{http://planetphysics.us/encyclopedia/PhysicalMathematics2.html} or Physical Mathematics. The geometric picture that one is tempted to associate with this `linear super-category' is that of
a `string of pearls'. This definition of a linear `supercategory' was then followed up with an even more interesting example. ``What, in this sense, is a \textit{braided monoidal supercategory ?}''. Urs, suggested the following answer: like an ordinary \htmladdnormallink{braided monoidal category}{http://planetphysics.us/encyclopedia/QuantumCategories.html} is a 3-category which in lowest degrees looks like the trivial 2-group, a
\htmladdnormallink{braided monoidal supercategory}{http://www.math.uni-hamburg.de/home/schreiber/scat.pdf} 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}. So the claim would be that braided monoidal supercategories come from
$G(2)$-stabilized 3-categories, with $G(2)$ the above strict 2-group. In view of the interpretation of this Urs type
of $\mathcal{S}_1$ supercategory as a category equipped with an ``odd vector field'', it may be more appropriately called a `\emph{\htmladdnormallink{vector field}{http://planetphysics.us/encyclopedia/NeutrinoRestMass.html} $\mathcal{S}_1$-supercategory}'.



\begin{remark}
One can readily extend the above definition of a \emph{one-object super-category} to a two-object, and...
\textbf{n-object}, proper supercategory using more that one category, more than one $Id_X$, and several types of diamond, square, and so on, to define an $n$--supercategory; thus, one ends up with several, connected `strings of pearls into a tangled strand'--the intuitive picture of a proper \htmladdnormallink{n-supercategory}{http://planetphysics.us/encyclopedia/SuperCategory6.html}, $\S_n$.
\end{remark}


\begin{example}
The \htmladdnormallink{2-category of double groupoids}{http://planetphysics.us/encyclopedia/2CategoryOfDoubleGroupoids.html} is another example of an $\mathcal{S}_1$ supercategory.
\end{example}

The following subsection lists examples of supercategories, including types of supercategories that
have applications in categorical logics and general \htmladdnormallink{system dynamics}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html} (i.e. beyond ODE's).

\subsection{A partial list of examples of supercategories}
\begin{enumerate}
\item functor categories;
\item \htmladdnormallink{2-categories}{http://planetphysics.us/encyclopedia/2Category.html}, \htmladdnormallink{super-categories}{http://planetphysics.us/encyclopedia/SuperCategory6.html} and n-categories; categories of categories (that are often considered for the foundation of mathematics);
\item \htmladdnormallink{higher dimensional algebras}{http://planetphysics.us/encyclopedia/InfinityGroupoid.html} (\htmladdnormallink{HDA}{http://planetphysics.us/encyclopedia/2Groupoid2.html}): \htmladdnormallink{double groupoids}{http://planetphysics.us/encyclopedia/WeakHomotopy.html}\htmladdnormallink{,double algebroids}{http://planetphysics.us/encyclopedia/GeneralizedSuperalgebras.html}, crossed complexes of groupoids, crossed complexes of \htmladdnormallink{algebroids}{http://planetphysics.us/encyclopedia/Algebroids.html}, superalgebroids, \htmladdnormallink{double categories}{http://planetphysics.us/encyclopedia/HorizontalIdentities.html}, multiple categories, $\omega$ and cubic structures,
\item organismic supercategories;
\item algebraic \htmladdnormallink{theories of organismic sets}{http://planetphysics.us/encyclopedia/TheoryOfOrganismicSets.html};
\item standard `super-categories' redefined as supercategories \cite{Urs2k7P};
\item super-categories of metabolic-repair, or $(M,R)$-systems;
\item super-categories of \L{}ukasiewicz algebraic (3-valued) logics;
\item supercategories of quantum automata;
\item $LM_n$-generalized toposes
\item toposes (or topoi) of genetic networks and human interactomes;
\item supercategories of Post algebraic logics;
\item supercategories of MV-algebraic logics.
\end{enumerate}

{\em Several specific examples of \htmladdnormallink{organismic supercategories}{http://planetphysics.us/encyclopedia/SuperCategory6.html}} were developed in a series of mathematical biology publications listed in the bibliography.

\begin{remark}
Supercategories, and especially, {\em organismic supercategories} provide an unified conceptual framework for \htmladdnormallink{abstract relational biology}{http://planetphysics.us/encyclopedia/SystemsBiology.html} that utilizes flexible, algebraic and \htmladdnormallink{topological structures}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} which transform naturally under heteromorphisms or heterofunctors, and \htmladdnormallink{natural transformations}{http://planetphysics.us/encyclopedia/VariableCategory2.html}.


One of the major advantages of the ETAS axiomatic approach, which was inspired by the \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} of
Lawvere (\cite{LW1}, \cite{LW2}), is that ETAS avoids all the antimonies/paradoxes previously reported for sets,
sets of sets, involving for example the elementhood \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} or infinite sets, the axiom of choice, and so on (Russell and Whitehead, 1925, and Russell, 1937; \cite{BBGG1}). ETAS also provides an axiomatic approach to recent higher dimensional algebra (\cite{BHS2}, \cite{BGB2}) applications to \htmladdnormallink{complex systems biology}{http://planetphysics.us/encyclopedia/SystemsBiology.html} (\cite{Bgg2}, \cite{BBGG1}, and references cited therein).
\end{remark}

\subsection{A simple example of ETAS axioms (\cite{ICB3}):}

\begin{itemize}
\item (a). The eight ETAC axioms introduced by W. F. Lawvere for same type \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, functors, natural transformations
and other arrows in higher dimensions.
\item (b). A family of composition laws $\left\{\Gamma_i\right\}$, with $i= 1, 2, ..., n$, and
each $\Gamma_i$ being subject to ETAC axioms 5 to 7;
\item (c). The composition law conditions in the ETAC axioms 8a to 8c will apply only to (small) subclasses of homo-morphisms and homo-functors (but will not apply to hetero-morphisms and hetero-functors);
\item (d). Hetero-morphism and hetero-functor axioms specifying how two composition laws interact, involving also objects with different structure or type.
\end{itemize}

\subsection{Generic example of an ETAS with ten axioms:}


0. For any letters $x, y, u, A, B$, and \emph{unary \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html}} symbols $\Delta_0$ and $\Delta_1$,
and \emph{composition law} $\Gamma$, the following are defined as \emph{\htmladdnormallink{formulas}{http://planetphysics.us/encyclopedia/Formula.html}}: $\Delta_0 (x) = A$,
$\Delta_1 (x) = B$, $\Gamma (x,y;u)$, and $ x = y$; These formulas are to be, respectively, interpreted as
``$A$ is the domain of $x$", ``$B$ is the codomain, or range, of $x$", ``$u$ is the composition $x$ followed by $y$",
and ``$x$ equals $y$".

1. If $\Phi$ and $\Psi$ are formulas, then ``$[\Phi]$ and $[\Psi]$'' , ``$[\Phi]$ or$[\Psi]$'', ``$[\Phi] \Rightarrow [\Psi]$'', and ``$[not \Phi]$'' are also formulas.

2. If $\Phi$ is a formula and $x$ is a letter, then ``$ \forall x[\Phi]$'',
``$ \exists x[\Phi]$'' are also formulas.

3. A string of symbols is a formula in ETAS iff it follows from the above axioms 0 to 2.

A \emph{\htmladdnormallink{sentence}{http://planetphysics.us/encyclopedia/Formula.html}} is then defined as any formula in which every occurrence of each letter $x$ is within the scope of a quantifier, such as $\forall x$ or $\exists x $. The \emph{\htmladdnormallink{theorems}{http://planetphysics.us/encyclopedia/Formula.html}} of ETAS are defined as all those sentences which can be derived through logical inference from the following ETAS axioms:

4. $\Delta_i(\Delta_j(x))=\Delta_j(x)$ for $i,j = 0, 1$.

5. $\Gamma(x,y;u)$ and $\Gamma(x,y;u')\Rightarrow u = u'$.

5. $ \exists u [\Gamma(x,y;u)] \Rightarrow \Delta_1(x) = \Delta_0(y)$;

7. $\Gamma(x,y;u) \Rightarrow \Delta_0(u) = \Delta_0(x)$ and $\Delta_1(u) = \Delta_1(y)$.

8. Identity axiom:
$ \Gamma(\Delta_0 (x), x;x)$ and $ \Gamma(x, \Delta_1 (x);x)$ yield always the same result.

7. \htmladdnormallink{Associativity axiom}{http://planetphysics.us/encyclopedia/Cod.html}: $\Gamma(x,y;u)$ and $\Gamma(y,z;w)$ and $\Gamma(x,w;f)$ and $\Gamma(u,z;g)\Rightarrow f = g $.
With these axioms in mind, one can see that \htmladdnormallink{commutative diagrams}{http://planetphysics.us/encyclopedia/Commutativity.html} can be now regarded as certain
\textit{abbreviated} formulas corresponding to \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} of equations such as:
$\Delta_0(f) = \Delta_0(h) = A$, $\Delta_1(f) = \Delta_0(g) = B$, $\Delta_1(g) = \Delta_1(h) = C$
and $\Gamma(f,g;h)$, instead of $g\circ f = h$ for the arrows f, g, and h, drawn respectively between the
`objects' A, B and C, thus forming a `triangular commutative super-diagram` in the usual sense of category theory. Compared with the ETAS formulas such diagrams have the advantage of a geometric--intuitive image of their equivalent underlying equations. The common property of A of being an object is written in shorthand as the abbreviated formula Obj(A) standing for the following three equations:

8. $A = \Delta_0(A) = \Delta_1(A)$,

9. $ \exists x[A = \Delta_0 (x)] \exists y[A = \Delta_1 (y)]$,

and

10. $\forall x \forall u [\Gamma (x,A; u)\Rightarrow x = u]$ and
$ \forall y \forall v [\Gamma (A,y; v)] \Rightarrow y = v$ .

Intuitively, with this terminology and axioms a \textit{supercategory} is meant to be any structure which is a direct interpretation of these ten ETAS axioms. A \textit{heterofunctor} is then understood to be a \textit{triple} consisting of two such supercategories (or classes of objects, diagrams, etc.) and of a set, or proper class, of rules $\S_H$ (`the hetero-functor') which assigns to each arrow or morphism $x$ of the first supercategory,
a unique morphism, written as `$\S_H (x)$' of the second category, in such a way that the usual conditions on objects (but not on arrows) are fulfilled (see for example \cite {ICBM})-- the functor is well behaved, it carries object identities to image object identities, and commutative super-diagrams (or classes) to image commmutative super-diagrams of the corresponding image objects and image (hetero)morphisms betwen objects with different structures, as well as homo-morphisms (between objects with the {\em same type of structure}, such as \htmladdnormallink{groupoid homomorphisms}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism.html}, or topological space \htmladdnormallink{homeomorphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, \htmladdnormallink{automata homomorphisms}{http://planetphysics.us/encyclopedia/AAT.html}, etc). At the next level, one then defines \emph{natural transformations} or \emph{functorial morphisms} between functors as metalevel abbreviated formulas and equations pertaining to commutative diagrams of the distinct images of two functors acting on both objects and morphisms. As the name indicates natural transformations are also well--behaved in terms of the ETAC equations satisfied. \\
{\em Remark:} The example shown above is the ETAS formulation closest to ETAC--
axioms for categories and super-categories, or categories of categories in volved in the
foundations of most Mathematics, (cf. Lawvere and others); see also \cite{LW2}
and the entry on axiomatic theory of supercategories. \\

\subsubsection{Additional explanation of the examples of supercategories listed above: }


In the usual sense, super-categories are defined as categories of categories,
and the process is repeated in higher dimensions in n-categories. There is however
a `more geometric', or `gluing' construction of double categories, double groupoids (\cite{BS}, and
double algebroids \cite{BM}) that involves additional conditions or axioms leading to non-Abelian higher dimensional structures and Higher Dimensional Algebra (HDA;\cite{BS},\cite{BM} and \cite{BGB2}). Similarly, the construction of \textit{supercategories} (\cite{ICB3}), unlike that of $n$-categories, allows for `gluing' together distinct structures
and/or their corresponding diagrams (such as algebraic and topological ones), through hetero-morphisms or hetero-functors, which are arrows (not necessarily subject to all of the ETAC axioms) linking distinct structures into the \htmladdnormallink{superstructure}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html} called a \textit{`supercategory}'; the latter is a generalized type of double groupoid, double category, \htmladdnormallink{2-category}{http://planetphysics.us/encyclopedia/2Category.html},..., n-category or super--category/meta--category, that always involves only diagrams of homo-morphisms at each level, but also includes hetero-morphisms or hetero-functors, and so on, between different types of structures.
Proper supercategories could also be called \textit{`n-ary'} super-categories, or \textit{multi-categories}, in the sense of extending double groupoid, double algebroid and double category structures to higher dimensions.
Note however that even in a general, abstract supercategory at the level of diagrams of homo-morphisms, homo-functors, natural transformations, or any n-categories with only one type of arrows at each level, all the ETAC axioms still hold. On the other hand, at the levels of supercategorical diagrams, or superdiagrams, involving several types of morphisms, or hetero-morphisms and hetero-functors, several of the rules for connecting diagrams are weakened, and the result is a superstructure which does not have all \htmladdnormallink{naturality conditions}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} satisfied by all arrows, and one has additional composition laws, such as $\Gamma', \Gamma'', ...$, and so on, satisfying new ETAS axioms that are not allowed in ETAC. Thus, additional ETAS axioms are needed to also specify how such distinct composition laws are combined within the same superstructure. Any interpretation of ETAS axioms (that may include also the ETAC axioms for the special cases of categories, n-categories, double categories, etc) then defines a \textit{supercategory}.

\begin{definition} \emph{: Organismic supercategory (\cite{ICBM},\cite{ICB3} and \cite{ICB1}.}
An example of a supercategory interpreting such ETAS axioms as those stated above
was previously defined for \htmladdnormallink{organismic structures}{http://planetphysics.us/encyclopedia/VariableCategory2.html} with different levels of \htmladdnormallink{complexity}{http://planetphysics.us/encyclopedia/Complexity.html} (\cite{ICB3}); an \textit{organismic supercategory} was thus defined as a \textit{superstructure interpretation 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. 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}

\textbf{Note:}
Supercategories are mapped ('morphed', transformed or linked) by hetero-functors (or 'heterofunctors'); {\em hetero-functors} are defined as a natural extension of the concept of functor between categories, in the sense that they link objects from different categorical diagrams or categories with different structures following rules and axioms specified by the elementary theory of abstract supercategories, ETAS.

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\end{document}