Multifold Objects

In complex systems, the same functions can be effected by several structurally different patterns. Multifold objects will model this notion.

While a pattern of linked objects in a category has at most one colimit (up to isomorphisms), conversely a complex object N may be the colimit of several decompositions into patterns, not always easily deduced one from the other at their own level.

We say that two decompositions P and Q of an object N are *equivalent* if there exists a cluster between P and Q which binds into the identity of N, so that this identity is a (P,Q)-simple link. Otherwise, we say that P and Q are *non-equivalent* decompositions of N.

A* multifold object *is an object which admits at least two non-equivalent decompositions. The 'switch' between these decompositions can be seen as a random fluctuation in its internal organization which does not modify its functionality at the higher level: several microstates lead to the same macroequilibrium. An example of such a switch is the passage between non-equivalent genotypes of a species (i.e., genotypes with differing alleles) leading to the same phenotype.

Complex links

The existence of multifold objects N implies that there may exist composites of simple links which are not simple.

Indeed, let *f *be a (P,P')-simple link from N to N', and *f' *a (Q',P")-simple link from N' to N", where P' and Q' are two non-equivalent decompositions of N'. Their composite *ff' *must exist in the category, but there is no reason for it to bind a cluster from P to P"; it is called a* *(P,P")*-complex link.*

More generally, if N and M are complex objects, a *complex link* from N to M is defined as a composite of simple links binding non-adjacent clusters and which is not simple (it does not bind a cluster debtween decompositions of N and M). A composite of complex links which is not simple is a complex link.

A complex link connects the objects not directly at the level of their more elementary components, but through the mediation of multifold objects each of which intervenes with two non-equivalent decompositions; the switch between their decompositions causes the emergence of new properties with respect to the clusters which are binded.

For instance, the communication between authors and subscribers of a Journal is represented by a complex link, mediated by the switch between the editors and the publishers of the Journal. Mathematically, a topological space admits several geometric realizations in simplicial complexes; the simple links correspond to simplicial maps and the complex links to continuous maps.

The existence of multifold objects is one of the characteristics of complex natural systems which ensures their flexibility.

We say that a category satisfies the *Multiplicity principle *if some of its objects are multifold. This principle is strengthened as follows in a hierarchical system:

1.There exist objects of level *n*+1 which are *n-multifold* in the sense that they admit at least two non-equivalent decompositions into patterns of level *n*.

2. An object of level *n* can belong to several patterns admitting different colimits of level *n*+1.

For instance, these conditions are satisfied in the HS modelling a neural system; indeed, in this case they have been pointed out by Edelman under the name of Degeneracy Principle (by analogy with the 'degeneracy' of the genetic code in which the same aminoacid can be coded by two different codons).

If N is a multifold object of level *n*+1, for each *k* £ *n*, we define the *k-entropy *of N as the number of its non-equivalent ramifications arriving to the level *k*.

It gives a measure of the flexibility of N, in the sense of the number of its functional internal organizations down to level *k.* But (opposing Rosen) we don't consider that the entropy determines its "real" complexity, which we measure by the complexity order of N.