Let us consider an example of a system whose state space consists of a torus, $T^{3}$, such that the states of the system are contained inside the torus, and all transitions lead to states inside the torus. The homology theory offers in this case two intuitive examples of generators as the entire torus is generated by only two cycles. These two cycles generate two homology groups $H_{0}(T)=Z$ and $H_{1}(T)=Z\otimes Z$, where $Z$ is the group of integers, and $\otimes$ denotes the product operation. These homology groups give a characterization of the topological space represented by the torus. In this way a connection is established between a topological structure-that of the toroidal space, and algebraic structures, those of homology groups, for example. Even more, we can assign two numbers to a given complex $K$ : the Betti number-which is the number of repetitions of $Z$ in the homology group, $H_{p}(K)=Z\otimes Z\otimes...\otimes Z\otimes G_{{PT}}$ of a complex $K$-and a number $p$ which is the number of elements of a finite abelian group $G_{{PT}}$. The Betti number gives the number of $p$-dimensional holes of the complex $K$, and the number $p$ gives the number of $p$-dimensional turns of $K$. A natural physical interpretation that one can give to the holes inside of the state space of a system will be that of instability fields of the system under consideration. Consequently, the Betti number will give a coarse idea about the instability of the system, being the number of instability fields of the state space. However, homological as well as homotopy techniques would allow a much finer characterization of the local and global properties of the dynamics of a system, being able, for example, to locate singularities in a state space (Hwa and Teplitz, 1966). It must be mentioned here that the theory of categories and functors provides a natural framework for homological techniques, mostly for abelian structures in commutative homological algebra. In the example discussed above, a number was assigned to a quality, that is, a Betti number was assigned to a topological space. Another example of such an assignment is found in the theory of elementary particles, where one associates a probability with a Feynman diagram that qualitatively represents particle interactions via quantum fields or exchanged quanta. However, the general procedure is not to assign a single number to a quality, but an entire set of numbers or elements. In our second example the operation of addition induced a corresponding operation on diagrams. This fact suggests that operations which are used in metric, or quantitative, biology may induce corresponding operations in relational biology. Conversely, one can think of significant relational operations with notions which would permit us to obtain solutions of complicated problems of quantitative biology.

Observables of a biological system may be introduced as intensities of some (metabolic / biochemical / enzymatic, or genetic, etc.) activities of the living system (as suggested for example by Rashevsky and Rosen); alternatively, they may be considered either as parameters characterizing specific or global processes inside the system, or as variables (e.g., molecular set variables) that specify the quantities of certain bioproducts which are formed as a result of the activities of the system. Among observables, structural parameters as defined in (Rosen 1968a, b) and time observables play distinguished roles. Some observables are “linked”, in the sense that, a change in one of the observables implies a corresponding change of the others. Linked observables were represented as morphisms in a categorical diagram. Such a diagram may represent, for example, the linkage group of observables and is a part of the generating class of the system. Since diagrams can be regarded as functors (Mitchell, 1965) one can represent the linked observables that specify the generating classes of a system by ‘dynamic configuration’ functors between categories of possible system configurations at different times; therefore, qualitative changes in system dynamics may be represented by natural transformations of such dynamic configuration functors, whereas transition-state morphisms assign numerical values to observables in different system states. There is then a classification problem associated with defining, or determining through measurements the biodynamics of super-complex systems as the latter are endowed with variable state-space (or configuration space) topologies (Baianu et al, 2007a). This leads unavoidably to the consideration of supercategories of variable topological structures and their associated algebroids related to global stability and qualitative dynamics.

Let us define a state of a system or organismic set, biomolecular set, $(M,R)$-system,…, any theoretical model of a biosystem, at a given moment as an $n$-tuple of the values of essential observables at that moment. In this representation, either a state, or system ‘configuration’, is then defined by a dynamic functor from the category of generating classes of the system to $mathsf{\mathcal{R}}$–the set of real numbers regarded for example as a discrete category (or as a category whose objects are real numbers, and whose morphisms are mappings; the operations with real numbers in this category are induced by the structure of the category of generating classes). Let us also consider here a specific example from microbial genetics. An operon as defined by Jacob and Monod (1961) may be considered- in a very simple model- as having two states: an active state and an inactive one. In its inactive state the operon will not induce the synthesis of the corresponding enzyme, while in its active state, it will induce the synthesis of a determined quantity $C$ of synthesized enzyme per unit of time. Now, if we consider a linkage group of operons $O_{1},O_{2},...,O_{n}$, which are all active in the same time and, if the synthesized quantities of enzymes per unit of time are respectively, $C_{1},C_{2},...,C_{n}$, a state of the linked operons may be defined by the $n$-tuple $(C^{t}_{1},C^{t}_{2},...,C^{t}_{n})$ of the values of $C_{1},C_{2},...,C_{n}$ at a given instant $t$. However, suppose that only $C_{1},C_{2}$ and $C_{3}$ are essential, and all other observables can be expressed in terms of $C_{1},C_{2}$ and $C_{3}$. Moreover, let us suppose that we may find some operators such that $C_{2}=YC_{1}$ and $C_{3}=ZC_{2}$. In this case there exists a third operator $X$ such that $C_{3}=XC_{1}$, which makes the three operon diagram commutative. This is the generating diagram of the linkage group of operators. In this simple model, a genetic system, or organsimic set (discrete organismic supercategory) of zero-th order, $S_{0}$, will be then represented by a generating class, whose objects are generating diagrams of the linkage groups of operons, and whose morphisms are the functional connections among the activities of the operons. Suppose that a mutation takes place in such a genetic system, so that an operon will begin to induce the synthesis of an enzyme which was not induced previously by the genetic system. The state corresponding to the very moment when the change takes place will be considered as a singularity of the state space, because at that moment when the genetic mutation occurred one cannot characterize the state of the genetic system either by $C^{t}_{k}$ –the quantity of synthesized enzyme $E_{k}$ per unit of time, or by $C_{k}$ the quantity of the ‘mutated’ enzyme $E$ per unit of time (that is, $E$ is the enzyme which begins to be synthesized only after the mutation took place). It may happen that a mutation produces effects such as the complete inactivation of an essential operon. In this case, it is conceivable that the whole linkage group will become inactive. If the inactivated operon is the replicon (Jacob et al., 1963), then the cell will cease to divide, and eventually die. The singularity of the genomic state space in the case of a mutation would last much less than other states of the dynamical system, and may be therefore considered as an unstable state inside the state space of the system.

Generally, if the unstable state leads only to other unstable states, it may result in the destruction of the system-generating an unstable field outside the state space stable attractors of the system. Consequently, states in growth processes should have to be considered as metastable, and cannot be simply considered as unstable. The replacement of an observable by another, in the case of a mutation discussed here, is in fact a change of the structure of the genetic system– a structure which has both algebraic and topological representations in a supercatgeory, that may also be viewed as a family or class of multiple, (variable) categories; the latter type of generalized structure necessary for the representation of super-complex system biodynamics will be defined more precisely in the next section as an interpretation of ETAS axioms, a natural extension of Lawvere’s ETAC axioms for the foundation of the theory of categories and functors.