supercategory

Several examples of definitions of supercategories:

1. super-category or `category of categories' , subject to Lawvere's ETAC axioms designed for ``the foundation of Mathematics'';
2. meta-category and meta-theorems as defined by Mitchell (1965) and the corresponding meta-theories;
3. super-category of functor categories-as defined by Barry Mitchell (1965);
4. various n-categories published by Baez and his group;
5. supercategory defined at the n-category web `café' by Dr. Urs Schreiber from the Center for Mathematical Physics;
6. An elementary example of a supercategory is that of a quasigroup;
7. supercategory of organismic sets, and the theory of organismic sets;
8. supercategory of $(M,R)$ -systems and supercategory theory of $(M,R)$ -systems;
9. supercategory of quantum automata;

Examples of supercategories

1. functor categories;
2. 2-categories, super-categories and n-categories; categories of categories (that are often considered for the foundation of Mathematics);
3. higher dimensional algebras (HDA): double groupoids, double algebroids, crossed complexes of groupoids, crossed complexes of algebroids, superalgebroids, double categories, multiple categories, $\omega$ and cubic structures,
4. organismic supercategories: algebraic theories of organismic sets, super-categories of Metabolic-Repair, or $(M,R)$ -systems
5. super-categories of ukasiewicz algebraic (3-valued) logics
6. supercategories of quantum automata,
7. $LM_n$ -generalized toposes/topoi of genetic networks and interactomes, supercategories of Post algebraic logics, supercategories of MV-algebraic logics.

Supercategory theories

One of the major advantages of the ETAS axiomatic approach, which was inspired by the work of Lawvere [10,11], is that ETAS avoids all the antimonies/paradoxes previously reported for sets, sets of sets, involving for example the elementhood relation or infinite sets, the axiom of choice, and so on (Russell and Whitehead, 1925, and Russell, 1937; []). ETAS also provides an axiomatic approach to recent Higher Dimensional Algebra applications to Complex Systems Biology ([1] and references cited therein).

Ten axioms for a general ETAS example (See also [11] and the entry on axiomatic theory of supercategories.)

0. For any letters $x, y, u, A, B$ , and unary function symbols $\Delta_0$ and $\Delta_1$ , and composition law $\Gamma$ , the following are defined as formulas: $\Delta_0 (x) = A$ , $\Delta_1 (x) = B$ , $\Gamma (x,y;u)$ , 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 if and only if it follows from the above axioms 0 to 2.

A sentence 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 theorems 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. Associativity axiom: $\Gamma(x,y;u)$ 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 commutative diagrams can be now regarded as certain abbreviated formulas corresponding to systems 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 supercategory is meant to be any structure which is a direct interpretation of these ten ETAS axioms. A heterofunctor is then understood to be a 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 [])- 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 same type of structure, such as groupoid homomorphisms, or topological space homeomorphisms, automata homomorphisms, etc). At the next level, one then defines natural transformations or 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.

The general example presented above is an ETAS formulation closest to ETAC-axioms for categories and super-categories, or categories of categories in volved in the foundations of most Mathematics, (viz. Lawvere and others).

Notes on the general definition of a supercategory and a standard example

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. The new supercategory introduction was then continued with the following interesting example that aims to answer the following question. ``What, in this sense, is a braided monoidal supercategory ?''. Urs, suggested the following answer: like an ordinary braided monoidal category 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 $$G(2)=(\diamond 2 \diamond \emph{Id} \diamond 2)$$ . Urs called this generalization of stabilization of n-categories, $G(2)$ - 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.

Bibliography

1

References [14] to [34] in the Bibliography on category theory and algebraic topology

2

I.C. Baianu: 1973, Some Algebraic Properties of (M,R) - Systems. Bulletin of Mathematical Biophysics 35, 213-217.

3

I.C. Baianu and M. Marinescu: 1974, A Functorial Construction of (M,R)- Systems. Revue Roumaine de Mathematiques Pures et Appliquees 19: 388-391.

4

I.C. Baianu: 1977, A Logical Model of Genetic Activities in ukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biophysics, 39: 249-258.

5

I.C. Baianu: 1980, Natural Transformations of Organismic Structures. Bulletin of Mathematical Biophysics 42: 431-446.

6

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;

7

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).

8

R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales-Bangor, Maths Preprint, 1986.

9

R. Brown and C.B. Spencer: Double groupoids and crossed modules, Cahiers Top. Géom.Diff. 17 (1976), 343-362.

10

W.F. Lawvere: 1963. Functorial Semantics of Algebraic Theories., Proc. Natl. Acad. Sci. USA, 50: 869-872

11

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.