In a natural system with components of several levels of complexity, a complex object has at least one decomposition into a pattern P of more elementary components. The problem arises then to characterize those links between two complex objects which are completely determined by the data of links between the objects of their decompositions (so that they are 'reducible' to the lower level); and to explain how more complex, non-reducible, links may emerge.

 

Clusters between patterns

The notion of a cluster models the interactions between objects of two patterns P and P' which are correlated by the distinguished links of these patterns.

If P and P' are two patterns in a category, a cluster G from P to P' is a maximal set of links from objects of P to objects of P' satisfying the following conditions:

 

1. For each object Pi of P, there exists at least one link in G from Pi to an object of P'; and if there are several such links, they are correlated by a zig-zag of distinguished links of P'.

 2. G also contains the links obtained by composing a link of G on the left with a link in P or on the right with a link in P'.

 

Thus a cluster is a family of links from objects of P to those of P', correlated by the distinguished links of each pattern and such that each object of P transmits uniquely compatible information or constraints to P'.  

Simple links

If the patterns P and P' admit colimits N and N' respectively in the category, a cluster from P to P' binds into a unique link from N to N' called a (P,P')-simple link.

Such a simple link from N to N' transmits information already mediated through components of N and N'. Roughly, it 'institutionalizes' the cluster, without adding any information which is not accessible at the level of the patterns. For instance, in Embryology, the induction of a population of cells by another corresponds to the formation of a simple link.

The composite of two simple links binding adjacent clusters is still a simple link: if f is (P,P')-simple and if f' is (P',P")-simple, then their composite ff' is a (P,P")-simple link. But a (P,P')-simple link might not be (Q,Q')-simple for other decompositions Q of N and Q' of N'.

 

Representative subpattern

If there exists a cluster from a pattern P to a subpattern Q of P, we say that Q is a representative subpattern of P. In this case, P and Q have both the same colimit if it exists, or none of them has a colimit. It means that the collective actions of P are completely determined by those of Q, no supplementary constraint being imposed by the objects of P which do not belong to Q. The typical example is that of the representatives of a nation: each elector votes for a list of candidates which endorse similar ideas.