# 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

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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.).

Start with the definition of a **direct system** of objects and homomorphisms. Let be a directed set. Let be a family of objects indexed by and be a homomorphism for all with the following properties:

Then the pair is called a direct system over .

The underlying set of the **direct limit**, , of the direct system is defined as the disjoint union of the 's modulo a certain equivalence relation :

Here, if and , if there is some such that . 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. .

One naturally obtains from this definition *canonical morphisms* sending each element to its equivalence class. The algebraic operations on 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

The direct limit can be defined in an arbitrary category by means of a universal property. Let be a direct system of objects and morphisms in (same definition as above). The direct limit of this system is an object in together with morphisms satisfying . The pair must be universal in the sense that for any other such pair there exists a unique morphism making the diagram

commute for all *i*, *j*. The direct limit is often denoted

with the direct system being understood.

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.

We note that a direct system in a category admits an alternative description in terms of functors. Any directed poset can be considered as a small category where the morphisms consist of arrows if and only if . A direct system is then just a covariant functor . In this case a direct limit is a colimit.

## Examples

- A collection of subsets of a set
*M*can be partially ordered by inclusion. If the collection is directed, its direct limit is the union . - Let
*I*be any directed set with a greatest element*m*. The direct limit of any corresponding direct system is isomorphic to*X*_{m}and the canonical morphism φ_{m}:*X*_{m}→*X*is an isomorphism. - Let
*p*be a prime number. Consider the direct system composed of the groups**Z**/*p*^{n}**Z**and the homomorphisms**Z**/*p*^{n}**Z**→**Z**/*p*^{n+1}**Z**induced by multiplication by*p*. The direct limit of this system consists of all the roots of unity of order some power of*p*, and is called the Prüfer group**Z**(*p*^{∞}). - Let
*F*be a*C*-valued sheaf on a topological space*X*. Fix a point*x*in*X*. The open neighborhoods of*x*form a directed poset ordered by inclusion (*U*≤*V*if and only if*U*contains*V*). The corresponding direct system is (*F*(*U*),*r*_{U,V}) where*r*is the restriction map. The direct limit of this system is called the*stalk*of*F*at*x*, denoted*F*_{x}. For each neighborhood*U*of*x*, the canonical morphism*F*(*U*) →*F*_{x}associates to a section*s*of*F*over*U*an element*s*_{x}of the stalk*F*_{x}called the*germ*of*s*at*x*. - Direct limits in the category of topological spaces are given by placing the final topology on the underlying set-theoretic direct limit.
- Direct limits are linked to inverse limits via

- Consider a sequence {
*A*,_{n}*φ*} where_{n}*A*is a C*-algebra and_{n}*φ*:_{n}*A*→_{n}*A*_{n + 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.

## See also

## References

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