Direct limit
In mathematics, a direct limit is a way to construct a object from many 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 between those smaller objects. The direct limit of the objects, where ranges over some directed set, is denoted by.
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 with a given algebraic structure, such as groups, rings, modules, algebras, etc. With this in mind, homomorphisms are understood in the corresponding setting.Let be a directed set. Let be a family of objects indexed by and be a homomorphism for all with the following properties:
- is the identity of, and
- for all.
The direct limit of the direct system is denoted by and is defined as follows. Its underlying set is the disjoint union of the 's modulo a certain :
Here, if and, then iff there is some with and and 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 direct system, i.e. whenever.
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 homomorphisms. Formally, the direct limit of the direct system consists of the object together with the canonical homomorphisms.
Direct limits in an arbitrary 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 . A target is a pair where is an object in and are morphisms for each such that whenever . A direct limit of the direct system is a universally repelling target in the sense that is a target and for each target, there is a unique morphism such that for each i. The following diagramwill then commute for all i, j.
The direct limit is often denoted
with the direct system and the canonical morphisms 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
- A collection of subsets of a set can be partially ordered by inclusion. If the collection is directed, its direct limit is the union. The same is true for a directed collection of subgroups of a given group, or a directed collection of subrings of a given ring, etc.
- Let be any directed set with a greatest element. The direct limit of any corresponding direct system is isomorphic to and the canonical morphism is an isomorphism.
- Let K be a field. For a positive integer n, consider the general linear group GL consisting of invertible n x n - matrices with entries from K. We have a group homomorphism GL → GL which 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. An element of GL can be thought off as an infinite invertible matrix which differs from the infinite identity matrix in only finitely many entries. The group GL is of vital importance in algebraic K-theory.
- Let p be a prime number. Consider the direct system composed of the factor groups and the homomorphisms induced by multiplication by. The direct limit of this system consists of all the roots of unity of order some power of, and is called the Prüfer group.
- There is a injective ring homomorphism from the ring of symmetric polynomials in variables to the ring of symmetric polynomials in 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. The corresponding direct system is 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 → 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
An important property is that taking direct limits in the category of modules is an exact functor. This means that if you start with a directed system of short exact sequences and form direct limits, you obtain a short exact sequence.
Related constructions and generalizations
We note that a direct system in a category admits an alternative description in terms of functors. Any directed set can be considered as a small category whose objects are the elements and there is a morphisms if and only if. A direct system over 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. Here we start with a covariant functor from a filtered category to some 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.
Given an arbitrary category, there may be direct systems in which don't have a direct limit in . In this case, we can always embed into a category in which all direct limits exist; the objects of are called ind-objects of .
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.