# Direct limit

In mathematics, a direct limit (also called inductive limit) is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition, which can be used in any category.

## Formal definition

### Algebraic objects

{{#invoke:see also|seealso}} In this section objects are understood to be sets with a given algebraic structure such as groups, rings, modules (over a fixed ring), algebras (over a fixed field), etc. With this in mind, homomorphisms are understood in the corresponding setting (group homomorphisms, etc.).

$\varinjlim A_{i}=\bigsqcup _{i}A_{i}{\bigg /}\sim .$ Here, if $x_{i}\in A_{i}$ and $x_{j}\in A_{j}$ , $x_{i}\sim \,x_{j}$ if there is some $k\in I$ such that $f_{ik}(x_{i})=f_{jk}(x_{j})\,$ . Heuristically, two elements in the disjoint union are equivalent if and only if they "eventually become equal" in the direct system. An equivalent formulation that highlights the duality to the inverse limit is that an element is equivalent to all its images under the maps of the directed system, i.e. $x_{i}\sim \,f_{ik}(x_{i})$ .

One naturally obtains from this definition canonical morphisms $\phi _{i}:A_{i}\rightarrow A$ sending each element to its equivalence class. The algebraic operations on $A\,$ are defined via these maps in the obvious manner.

An important property is that taking direct limits in the category of modules is an exact functor.

### Direct limit over a direct system in a category commute for all i, j. The direct limit is often denoted

$X=\varinjlim X_{i}$ Unlike for algebraic objects, the direct limit may not exist in an arbitrary category. If it does, however, it is unique in a strong sense: given another direct limit X′ there exists a unique isomorphism X′ → X commuting with the canonical morphisms.

## Examples

$\mathrm {Hom} (\varinjlim X_{i},Y)=\varprojlim \mathrm {Hom} (X_{i},Y).$ • Consider a sequence {An, φn} where An is a C*-algebra and φn : AnAn + 1 is a *-homomorphism. The C*-analog of the direct limit construction gives a C*-algebra satisfying the universal property above.

## Related constructions and generalizations

The categorical dual of the direct limit is called the inverse limit (or projective limit). More general concepts are the limits and colimits of category theory. The terminology is somewhat confusing: direct limits are colimits while inverse limits are limits.