Direct limit

In various areas of mathematics, the category theoretical notion colimit (also direct limit or inductive limit ) used to generalize the set-theoretic concept of unification.

Basic definition ( for part -level index sets )

Let I be a fixed part ordered set.

An inductive system consists of the specification of objects ( eg, amounts, groups or topological spaces ) Xi for the elements i of I and transfer pictures

Which are compatible with the particular structure (ie lot of pictures, group homomorphisms, continuous mappings of topological spaces ).

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

Which are compatible with, i.e.,

With the following universal property:

This means that whenever pictures are given, for

Applies, there is a unique mapping

Of the images " come ", that is,

Construction for quantities

The inductive limit of an inductive system ( Xi, fi, j) of quantities can be constructed as a set of equivalence classes: In the disjoint union

Elements are to be equivalent, which are mapped by the fi, j to like elements.

• Category theory