The colimit (or inductive limit) of a pattern actualizes the potentiality of the objects of the pattern to realize collective actions. Its limit (or projective limit) actualizes their capacity to collectively decode and classify information of which each receives only some part.

 

Let P be a pattern of linked objects Ni in a category. A common message from an object M of the category to P is a family of individual links gi from M to the Ni, which are correlated by the distinguished links of P. The limit of P, if it exists, is an object L which 'classifies' the common messages in the sense that the family (gi) uniquely factors into a link g from M to L. (Formally, the limit is defined as the colimit of the pattern looked at as a pattern in the 'opposite' category which has the same objects but in which the direction of all the links is inverted.)

 

The limit L detects the objects able to send a common message to the pattern, namely all the objects M for which there exists a link from M to L; roughly, they are the objects which have the property classified by L. These objects M and the links between them form a subcategory, called the classification domain of L.

All the properties of the colimit are transposed to the limit. In particular, iterated limits are inductively defined. And the construction of the complexification, with its simple and complex links, extends to the case where the given strategy fixes also patterns 'to classify', to which a limit will be added.