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In various areas of mathematics , the category-theoretical term Kolimes (also direct Limes or inductive Limes ) is used to generalize the set- theoretical concept of union .

Elementary definition (for partially ordered index sets)

The index set is a fixed directed set .

An inductive system consists of objects (e.g. sets, groups or topological spaces ) for the indices and transition maps

for ,

which are compatible with the respective structure (i.e. set maps, group homomorphisms, continuous maps of topological spaces) and meet the following conditions

  1. for all the identical figure on and
  2. for everyone .

The inductive limit of an inductive system is an object together with images


which are compatible with, d. H.


with the following universal property:

Compatible systems of images in any test object correspond to images from to .
Diagram for Kolimes.png

That means: Whenever images are given, for the


holds, there is a clear mapping


from which the images "come", d. H.


The inductive limit of an inductive system ( X if i , j ) of sets can be explicitly constructed as a set of equivalence classes

in the disjoint union . Here elements and should be equivalent if one exists for which applies.