Localization of Universal Problems. Local Colimits.

Category Theory has been applied in a series of papers by Ehresmann and Vanbremeersch (e.g., Bull. Math. Biol. 49(1): 13-50, 1987; SAMS 26: 81-117, 1997) to model complex systems such as biosystems, neural systems or social systems. For instance, in the category modelling a neural system, the internal representation of a physical object is described as the colimit of a diagram of receptors in the visual areas it activates. But there are several situations in which it would be useful to have a 'localized' notion of a colimit, for instance, to model the way an ambiguous object (which has two alternative readings) is memorized. The locally free diagrams and locally colimit diagrams, introduced by Guitart and Lair (Diagrammes 4: GL1-GL106,1980), could be used, but they are often too large and, anyway, are not uniquely determined. Particular locally free diagrams are introduced here, using the notion of the root of a category, which characterizes a minimal (in a precise sense) weakly coreflective subcategory.