Direct limit

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

Formal definitionEdit

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 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:

  1.   is the identity morphism of   and
  2. 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]

The system is said to be injective (resp. surjective, etc.) if this is true of all bonding maps. If   is directed (resp. countable) then the system is said to be directed (resp. countable).[1]

CoconesEdit

If   is an object and   is a collection of morphisms, each of the form   then   is said to be compatible or consistent[2] 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:

     or     

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.

ExamplesEdit

  • 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.

PropertiesEdit

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.[3]

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.

TerminologyEdit

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 alsoEdit

NotesEdit

  1. ^ Bierstedt 1988, pp. 41-56.
  2. ^ Mac Lane 1998, pp. 68-69.
  3. ^ Adamek, J.; Rosicky, J. (1994). Locally Presentable and Accessible Categories. Cambridge University Press. p. 15. ISBN 9780521422611.
  1. ^ This is abuse of notation and terminology since calling   a direct system is technically incorrect.
  2. ^ 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.

ReferencesEdit