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

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.).

Let be a directed set. Let be a family of objects indexed by and be a homomorphism for all with the following properties:

  1. is the identity of , and
  2. for all .

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 functions sending each element to its equivalence class. The algebraic operations on are defined such that these maps become morphisms.

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 (as defined above). A target is a pair where is an object in and are morphisms such that . A direct limit is a universally repelling target in the sense that for each target , there is a unique morphism where for each i. The direct limit of is often denoted


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.


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


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