In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any category. The way they are put together is specified by a system of homomorphisms (group homomorphism, ring homomorphism, or in general morphisms ranges over some directed set. This notation suppresses the system of homomorphisms; however, the limit depends on the system of homomorphisms.

Direct limits are a special case of the concept of colimit in category theory. Direct limits are dual to inverse limits, which are a special case of limits in category theory.

Formal definition

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.

Direct limits of algebraic objects

In this section objects are understood to consist of underlying sets equipped 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.).

be a directed set be a family of objects indexed with the following properties:

1. , and

2. .

is called a direct system.

The direct limit and is defined as follows. Its underlying set is the disjoint union's modulo a certain :

. Intuitively, 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.

One obtains from this definition canonical functions.

Direct limits in an arbitrary category

The direct limit can be defined in an arbitrary category by means of a universal property (as defined above). A target is a universally repelling target for each i. The following diagram will then commute for all i, j.

The direct limit is often denoted

) being understood.

Unlike for algebraic objects, not every direct system in an arbitrary category has a direct limit. If it does, however, the direct limit is unique in a strong sense: given another direct limit X′ there exists a unique isomorphism X′ → X that commutes with the canonical morphisms.

Examples

• can be partially ordered.

• . An analogous result holds for rings, modules, algebras, etc. Note the requirement of finite generation may be weakened, as long as the index set remains directed.

• The weak topology of a CW complex is defined as a direct limit.

• be any directed set with a greatest element is an isomorphism.

• Let K be a field. For a positive integer n, consider the general linear group GL(n;K) consisting of invertible n x n - matrices with entries from K. We have a group homomorphism GL(n;K) → GL(n+1;K) that enlarges matrices by putting a 1 in the lower right corner and zeros elsewhere in the last row and column. The direct limit of this system is the general linear group of K, written as GL(K). An element of GL(K) can be thought of as an infinite invertible matrix that differs from the infinite identity matrix in only finitely many entries. The group GL(K) is of vital importance in algebraic K-theory.

• Let p be a prime number. Consider the direct system composed of the factor groups. The direct limit of this system consists of all the roots of unity, and is called the Prüfer group.

• There is a (non-obvious) injective ring homomorphism from the ring of symmetric polynomials variables. Forming the direct limit of this direct system yields the ring of symmetric functions.

• 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 set ordered by inclusion (U ≤ V if and only if U contains V). The corresponding direct system is (F(U), rU,V) where r is the restriction map. The direct limit of this system is called the stalk of F at x, denoted Fx. For each neighborhood U of x, the canonical morphism F(U) → Fx associates to a section s of F over U an element sx of the stalk Fx 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.

• An ind-scheme is an inductive limit of schemes.

Properties

Direct limits are linked to inverse limits via

An important property is that taking direct limits in the category of modules is an exact functor. This means that for any directed system of short exact sequences of direct limits is also exact.

Related constructions and generalizations

admits an alternative description in terms of functors can be considered as a small category if and only if is then the same as a covariant functor. The colimit of this functor is the same as the direct limit of the original direct system.

A notion closely related to direct limits are the filtered colimits from a filtered category and form the colimit of this functor. One can show that a category has all directed limits if and only if it has all filtered colimits, and a functor defined on such a category commutes with all direct limits if and only if it commutes with all filtered colimits.

(consider for example the category of finite sets, or the category of finitely generated abelian groups are called ind-objects.

The categorical dual of the direct limit is called the inverse limit. As above, inverse limits can be viewed as limits of certain functors and are closely related to limits over cofiltered categories.

Terminology

In the literature, one finds the terms "directed limit", "direct inductive limit", "directed colimit", "direct colimit" and "inductive limit" for the concept of direct limit defined above. The term "inductive limit" is ambiguous however, as some authors use it for the general concept of colimit.

See also

• Direct limits of groups

Notes

References