In a natural system with components of several levels of complexity (e.g., atoms, molecules, cells,... in an organism), lower level components are binded together to form more complex components; and the process is iterated, such complex components being binded together to form still more complex ones. The notion of an iterated colimit allows to model this situation.

Let P be a pattern of linked objects Ni in a category, and let us assume that each Ni is itself a complex object admitting a decomposition in a pattern Pi (of which it is the colimit). If P also admits a colimit N in the category, what are the relations between N and the objects of the various patterns Pi? In particular, is it possible to transcribe the constraints imposed on the objects Ni by the distinguished links of P into constraints on their own components, so that N can be 'reduced' to the colimit of an appropriate pattern connecting these components? 

 

  Iterated colimits and ramifications

In a category, if N is the colimit of a pattern P of linked objects Ni and if each Ni is the colimit of a pattern Pi, we say that N is the 2-iterated colimit of (P,(Pi)), and (P,(Pi)) is called a ramification of N of length 2.

The ramification represents an internal organization of N specifying two levels, which determines in 2 steps the links of N to the other objects of the category. Since a complex object may have several decompositions, an iterated colimit can also have several non-equivalent ramifications.

 More generally, we inductively define a k-iterated colimit as the colimit A of a pattern of which each object Ai is itself a (k-1)-iterated colimit, and a k-ramification of A is the data of a decomposition of A and of a (k-1)-ramification of each Ai .

Roughly, A has a kind of fractal structure, whose components at each intermediate step are themselves ramified, but moreover with correlations between these ramifications introduced by the 'horizontal' distinguished links between the components at each level. The multiplicity of ramifications makes this structure very flexible. Indeed, if we think that a decomposition of A attributes particular values ('variables') to some characteristic features of A, as the slots in a frame (in the sense of Minsky), a k-ramification amplifies the choice since successive choices can be done at each of the k steps.

For example, A could represent the menu in a restaurant; the first decomposition describes the general composition of the menu: entree, meat, cheese and dessert, or (for another decomposition of A) soup, fish, fruits. At the second step, we get finer choices, e.g. tomatoes or ham as an entree, beef or veal as a meat, etc...

  

 

  Reduction of an iterated colimit

Let N be an object admitting a 2-ramification (P,(Pi)). We say that the iterated colimit can be reduced if it is possible to 'skip' the intermediate step represented by P and to directly represent N as the simple colimit of a 'large' pattern R whose objects are those of the various patterns Pi with the distinguished links of these Pi and supplementary links transcribing the constraints imposed by the links of P into constraints on the objects of the patterns Pi.

 

It is proved (in Ehresmann & Vanbremeersch, 1996) that this reduction is possible if all the distinguished links of P are simple links, binding adjacent clusters between the Pi ; in this case the supplementary links added in R are those of these clusters. Conversely, if some links of P are not simple, the iterated colimit cannot be reduced to a simple colimit.

 

As a concrete example, in the system modelling the occidental society, with its members and their various social groups, Europe constructed as a 'Europe of nations' would be modeled by a non-reducible 2-colimit, in which institutional links must be mediated by the nations. But a 'Europe of people' could be reduced to the simple colimit of the citizens of its various nations.