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 N* _{i}* in a category, and let us assume that each N

* Iterated colimits and ramifications*

* *In a category, if N is the colimit of a pattern P of linked objects N* _{i}* and if each N

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 A* _{i}* is itself a (

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,(P* _{i}*)). We say that the iterated colimit can be

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 P* _{i}* ; 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.** **