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Collective link, and colimit of a pattern P The Memory Evolutive Systems give a model based on a 'dynamic' Category Theory, incorporating time and durations, for complex multi-scale systems, with the following characteristics: (i) The system is evolutionary, its components and their links varying over time. The few models of complex systems using category theory (e.g., inspired by [24]) only consider one category representing the invariant structure of system. On the contrary in MES, the system is not represented by a unique category but by an Evolutive System consisting in: a family of categories K t , representing the successive configurations of the system at each time t, and partial transition functors from K t to K t' accounting for the change from t to t'. (ii) The system is hierarchical, with a tangled hierarchy of components varying over time. A component C of a certain level 'binds' at least one pattern P of interacting components of lower levels so that C, and P acting collectively, have the same functional role. Modeling this hierarchy raises the Binding Problem: how do simple objects bind together to form "a whole that is greater than the sum of its parts" [1] and how can such "wholes" interact? In the categorical setting, the 'whole' C is represented by the colimit of the pattern P of interacting simple objects; and the interactions between wholes are described. (iii) There is emergence of complex multiform components, with development of a flexible central memory. Whence the Emergence Problem: how to measure the 'real' complexity of an object and what is the condition making possible the emergence over time of increasingly complex structures and processes? We characterize this condition as the Multiplicity Principle [12], a kind of 'flexible redundancy' which ensures the existence of multiform components. And we prove that it is necessary for the emergence of increasingly complex objects and processes with multiform presentations, constructed by iterated complexification processes [11]. (iv) The system has a multi-agent self-organization. Its global dynamic is modulated by the cooperation/competition of a network of internal functional subsystems, the co-regulators, with the help of a long-term memory. Each co-regulator operates locally with its own rhythm, logic and complexity, but their different commands can be conflicting and must be harmonized. While the local dynamics are amenable to conventional computations, the problem is different for the global one. MENS is a MES the level 0 of which represents the 'physical' neural system (neurons and synapses), while its higher
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