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 in the category) between those smaller objects. The direct limit of the objects , where ranges over some directed set , is denoted by . (This is a slight abuse of notation as it suppresses the system of homomorphisms that is crucial for the structure of the limit.)
Direct limits of algebraic objectsEdit
In this section objects are understood to consist of underlying 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 preordered set that is directed and partially ordered (not all authors require to be directed or partially ordered). Let be a family of objects indexed by and be a homomorphism for all with the following properties:
- is the identity morphism of and
- Compatibility conditions (of direct systems): for all ; that is,
Then the pair is called a direct system over The maps are called the bonding, connecting, transition, or linking maps/morphisms of the system. If the bonding maps are understood or if there is no need to assign them symbols (e.g. as in the statements of some theorems) then the bonding maps will often be omitted (i.e. not written); for this reason it is common to see statements such as "let be a direct system."[note 1]
If is an object and is a collection of morphisms, each of the form then is said to be compatible or consistent with a direct system if for all indices the following compatibility condition of cones is satisfied:
In this case the pair is called a cocone from the direct system and is called its vertex.
Canonical direct limitEdit
The canonical direct limit of the direct system in the category of sets is denoted by and is defined as follows. Its underlying set is the disjoint union of the 's modulo a certain equivalence relation :
Here, if and then if and only if there is some with and and such that The most common way of defining the disjoint union is:
where for every index each element is canonically identified with the element in this disjoint union. For every index denote the equivalence class containing [note 2] by:
which results in the th canonical function:
- defined by
It is this pair that forms the canonical direct limit, although often only the set is mentioned.
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; that is, whenever In particular, given any and any the fiber will be a subset of the equivalence class said differently, any element of is always equivalent to
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 categoryEdit
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 or cocone 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 The following diagram
will then commute for all indices
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 there exists a unique isomorphism that commutes with the canonical morphisms.
- 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 be a field. For a positive integer consider the general linear group consisting of invertible - matrices with entries from We have a group homomorphism which enlarges matrices by putting a 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 written as An element of can be thought off as an infinite invertible matrix which differs from the infinite identity matrix in only finitely many entries. The group is of vital importance in algebraic K-theory.
- Let 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 (non-obvious) 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 be a -valued sheaf on a topological space Fix a point The open neighborhoods of form a directed set ordered by inclusion ( if and only if contains ). The corresponding direct system is where is the restriction map. The direct limit of this system is called the stalk of at denoted For each neighborhood of the canonical morphism associates to a section of over an element of the stalk called the germ of at
- 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.
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 if you start with a directed system of short exact sequences and form direct limits, you obtain a short exact sequence .
Related constructions and generalizationsEdit
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 (consider for example the category of finite sets, or the category of finitely generated abelian groups). 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.
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.
- This is abuse of notation and terminology since calling a direct system is technically incorrect.
- The equivalence classes of are subsets of and not of so actually denotes the equivalence class containing the element However, although should technically be written as this will only be done when such technical clarity is important.
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