These examples are representative of the systems which we want to study, namely of the self-regulated, evolutionary systems, able to manage their interactions with the environment and to memorize their experiences for a better adaptation. Such systems can be called 'living systems', and they comprise cells, biological organisms, as well as more generally ecological, social or cultural systems. Which are the main concepts to introduce for modeling such systems?

- The state of the system at a given time can be seen like a 'snapshot' of the system, showing its components and the interactions between them at this moment, be they of structural type (determining the form of the system), or of functional type such as transfers of information or constraints. With a large variability in the level of complexity of these components, in the links ensuring their horizontal and vertical interconnections and their possible relations with other objects external to the system, in their specific type of function within the system. With also a quantitative variability on the observables which measure the strength and the duration of the interactions.

- The structure and the organization of the system, like its components, are not invariant, but can change in the course of time. The changes come from exchanges between objects and with the environment; they lead to loss or acquisition of information, energy, matter, dissociation of certain components, formation of more complex components.

The system can evolve according to a natural mode, with maintenance of its homeostasis: the total structure remains stable although objects are renewed, repairs are carried out; for example an accountant will always exist in a company, although it is not always the same person. But if the environment varies and if the internal or external constraints become too important, they can result in a rupture of stability, at any level whatsoever, that simple repair mechanisms cannot control; we call it a fracture. To overcome it, a more or less important modification of the structure is needed, possibly requiring the formation of new complex components and links which acquire emerging new properties. This will be modeled by the process of vertical complexification.

The first case would be exemplified by the regular renewal of the macromolecules of a neuron, the second case by the formation of new synaptic links through learning. Or in the social field, we have the regular election of the representatives, possibly leading to political modifications of a more structural and conceptual nature if it involves a change of majority.

The vertical complexification may involve an enrichment of the hierarchy of the system, in particular with the formation of higher levels allowing to memorize more complex experiences, the successive modifications and their immediate or foreseeable consequences.

- The evolution is internally regulated. Because of the large variability of the objects and groups of different levels, a central regulating mechanism cannot exist, unless we introduce a divine process present everywhere, at all moments, all levels, overlooking all specific rhythms. It is thus necessary to consider a whole parallel distributed network of internal regulatory organs (we call them coregulators or CR), able to 'observe', analyze, evaluate and make decisions; these CRs acting in parallel horizontally as well as vertically in the hierarchical structure of the system. Each CR is a subsystem formed by a small group of components of a certain level of complexity, operating together in a stepwise mode at a specific timescale, using loops, feedback or feed-forward processes. Its specificity of function implies a specificity of the strategies which it can select, by taking account of the partial information received from other parts of the system or the outside (which form its 'landscape'), of the structural and temporal constraints which it must respect, of the results of its former experiences which are memorized for a better adaptation. The analysis of the situation will be very different depending on the CR. In particular their timescales are by nature heterogeneous: if we think of a biological system, from its atoms or even its particles to the molecules, macromolecules, organites, cells, organs and finally the entire organism, each one of these hierarchical levels will function at its own rhythm; it will take a much longer time to renew all the cells of a body than all the atoms of a proteinic chain. It is thus natural that each CR has its own strategy. It would not be possible to integrate in the same equation the analysis of the movement of the particles in the atoms of the stars, and the movement of these same stars within the universe. And yet we know the paradox of the wings of a butterfly in New Guinea which in a certain way contribute to the formation of a hurricane on the coasts of Florida.

- To explain such global effects, it should be understood that the strategies chosen by the various CRs must be realized not on the respective landscapes but on the system itself, where they compete for the common resources. Thus an balancing process, called the 'interplay among the strategies', must be carried out; this process is not directed centrally, but depends on the respective strengths and temporalities of the different CRs. If all the chosen strategies are compatible, they are all realized so that the global system maintains at least a certain homeostasis, and possibly develops new capacities. If not, their interplay will eliminate some of them, causing a fracture for the corresponding CRs. This fracture will have to be repaired later on; if there is no possibility to overcome it, it will ensue a dyschrony, and thus of loss of homeostasis. Aging and death can be interpreted as a failure of the regulating mechanisms which can no more adapt to new conditions.

In particular, a 'dialectics' is generated between CRs which are heterogeneous by their complexity levels and timescales. Indeed, small modifications of a lower level cannot be reflected in real time and one by one to a higher level CR with longer steps; it is only an accumulation of them which will be reflected with a delay, and it might cause a fracture; to repair it, this CR may impose a new strategy on the lower levels, possibly even a change of rhythm. Thus the risk of a cascade of 'de/resynchronizations' at increasingly higher levels, which we proposed as a characteristics of the aging of an organism.

- The system is able to learn and adapt, thanks to the development of an Internal memory. This memory records the successive experiences of the system, the choices of strategies by the CRs and their results. Each CR takes part in its development, and resorts to it in its choices. Moreover, in the case of more complex systems (e.g., the nervous system of a higher animal), there will be a classification of the records retained in the memory, leading to the formation of a semantic memory. And this can allow the formation of CRs equipped with specific properties such as intentionality or consciousness (cf. the article on consciousness in Articles).

To study these problems, we have recourse to the Theory of Categories, a subdomain of Mathematics introduced by Eilenberg and Mac Lane in 1945. This theory can be seen as a language unifying most of Mathematics, for it makes possible a general concept of structure, so that for example the passage from a group to its sub-groups, or from a topological space to its subspaces, resort to the 'same' general operation.

Our basic idea is that the evolution of living systems, and in particular of the human brain, rests on a small number of prototypical operations, which are exactly those Category Theory can model: synthesis through the aggregation of elementary objects to form more complex objects ('colimit' operation); conversely analysis through decomposition of complex objects; detection of the continuous identity of a composite in spite of the suppression or addition of some elements (progressive transformation of a colimit); recording of new objects and their later recognition (formation of and comparison with a colimit); classification of objects in classes of invariance leading to the definition of concepts (represented by a 'limit').

The Memory Evolutive Systems give a mathematical model, based on the Category Theory, for the 'living' systems considered above. Thanks to its construction of a mainly relational and qualitative nature, this model offers a method of analysis which covers at the same time the local, general, evolutionary and temporal aspects. It is developed in a series of articles (cf. List of publications) to which the article " Memory Evolutive Systems " (cf. Articles) can be used as an introduction.