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

${\ displaystyle f_ {ii} = \ operatorname {id} _ {X_ {i}}}$for all the identical figure on and${\ displaystyle i}$${\ displaystyle X_ {i}}$

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