Kolimes

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 . ${\displaystyle (I,\leq )}$

An inductive system consists of objects (e.g. sets, groups or topological spaces ) for the indices and transition maps ${\displaystyle (X_{i},f_{ij})}$ ${\displaystyle X_{i}}$ ${\displaystyle i\in I}$

${\displaystyle f_{ij}\colon X_{i}\to X_{j}}$for , ${\displaystyle i\leq j}$

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

1. ${\displaystyle f_{ii}=\operatorname {id} _{X_{i}}}$for all the identical figure on and ${\displaystyle i}$ ${\displaystyle X_{i}}$
2. ${\displaystyle f_{ik}=f_{jk}\circ f_{ij}}$for everyone . ${\displaystyle i\leq j\leq k}$

The inductive limit of an inductive system is an object together with images ${\displaystyle (X_{i},f_{ij})}$ ${\displaystyle \mathrm {colim} _{n}X_{n}}$

${\displaystyle u_{i}\colon X_{i}\to \mathrm {colim} _{n}\,X_{n}}$,

which are compatible with, d. H. ${\displaystyle f_{ij}}$

${\displaystyle u_{i}=u_{j}\circ f_{ij}}$ For ${\displaystyle i\leq j}$

with the following universal property:

Compatible systems of images in any test object correspond to images from to . ${\displaystyle X_{i}}$ ${\displaystyle T}$ ${\displaystyle \mathrm {colim} _{n}X_{n}}$ ${\displaystyle T}$

That means: Whenever images are given, for the ${\displaystyle t_{i}\colon X_{i}\to T}$

${\displaystyle t_{i}=t_{j}\circ f_{ij}}$ For ${\displaystyle i\leq j}$

holds, there is a clear mapping

${\displaystyle c\colon \mathrm {colim} _{n}\,X_{n}\to T}$,

from which the images "come", d. H. ${\displaystyle t_{i}}$

${\displaystyle t_{i}=c\circ u_{i}}$.

The inductive limit of an inductive system ( X _i ,  f _{i , j} ) of sets can be explicitly constructed as a set of equivalence classes

${\displaystyle \coprod _{i}X_{i}/\sim }$

in the disjoint union . Here elements and should be equivalent if one exists for which applies. ${\displaystyle \coprod _{i}X_{i}}$ ${\displaystyle x\in X_{i}}$ ${\displaystyle y\in X_{j}}$ ${\displaystyle k\in I}$ ${\displaystyle f_{ik}(x)=f_{jk}(y)\in X_{k}}$