In natural autonomous systems, the dynamics is regulated by the 4 archetypal changes: birth/death, division/combination; for instance, for a cell, endocytosis, exocytosis, breakage and synthesis of macromolecules. This is modeled by the process of complexification of a category with respect to a strategy, which accounts for the emergence of more and more complex objects during the development of a system and the evolution.

A strategy S on a category K is the data of: a set of external elements 'to absorb'; a set of objects and of links of the category 'to suppress'; a set of patterns without a colimit 'to bind together', so that a colimit be added; a set of cones 'to transform into colimit-cones', so that the vertex of the cone becomes a colimit of its basis; a set of patterns with a colimit 'to decompose', so that the pattern loses its colimit.

The complexification of K with respect to the strategy S is a category in which the objectives of S are realized in the most economical way (solution of the universal problem). If the category models a natural system, economical means with the least material, temporal, computational and energetical cost. 

Construction of the complexification.

The complexification can be explicitly described:

- Its objects are: all the objects of K which are not to be suppressed by S, the elements that S requires to absorb, and, for each pattern P to bind, a new object becoming its colimit (denoted colimP), which emerges as the integration of the pattern into a higher order unit.

- Its links are of two types: simple links binding clusters between patterns of K, and complex links which are composites of simple links binding non-adjacent clusters. If G is a cluster from P to P', it binds into a simple link g in the following cases:

If P and P' admit colimits N and N' in K, there already exists in K a simple link g from N to N' binding G (by the properties of colimits).

If P and/or P' are bases of cones required to become colimit-cones, the simple link g between their vertices either already exists in K or emerges as a new link in the complexification..

Finally if S requires that P and P' be binded, g emerges as a link from colimP to colimP'.

In particular, each collective link from a pattern P acquiring a colimit to an object A (new or already in K) of the complexification is binded into a simple link from colimP to A

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The complexification process can be iterated, and successive complexifications lead to the emergence of a hierarchy of more and more complex objects. The stepwise evolution of a natural system can be modeled by an Evolutive System in which the transitions between successive state-categories correspond to functors from a category to one of its complexifications. It could be applied to the Evolution of the universe, as well as to the development of biological or social systems. 

Assemblies of Neurons

As an application, let us describe how the complexification process allows to model the formation of synchronous assemblies of neurons.

The response of a neural system to a simple stimulus is the activation of an individual, specialized neuron; for instance, in the visual area, there exist 'simple cells' representing a segment of a given direction, and 'complex cells' representing a particular angle. But more complex stimuli, except for some exceptions (e.g., a neuron activated by a hand holding a banana for a monkey), do not have their own 'grand-mother neuron'. Are they represented by the grouping of more elementary units?

The development of neuronal imaging seems to confirm this 'associationnist' idea: complex perceptual stimuli, or motor programs, are represented by the short-lived synchronized firing of a specific assembly of neurons. And learning would consist in the formation of such synchronous assemblies, through the strengthening of synapses between their neurons, following the rule already proposed by Hebb in 1947: a synapse between two neurons is strengthened if the two neurons fire at the same time, and depressed if one fires while the other does not.

An assembly of neurons is represented by a pattern P in the category of neurons Neur. The synchronization of this assembly will be modeled by the emergence of a colimit of this pattern in a complexification of Neur; this colimit, which operates as a unique 'higher order neuron' integrating the synchronous assembly, will be called a category-neuron (or simply a cat-neuron). The above construction determines what are the 'good' links between cat-neurons, hence between synchronous assemblies of neurons, thus solving a problem raised by neuroscientists. And it allows to define, by iterations of the complexification process, more and more complex cat-neurons, representing assemblies of assemblies of neurons, ... corresponding to the development of higher order mental objects and cognitive processes.