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 formula_1, where formula_2 ranges over some directed set formula_3, is denoted by formula_4. (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 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 (over a fixed ring), algebras (over a fixed field), etc. With this in mind, are understood in the corresponding setting (group homomorphisms, etc.).
Let formula_5 be a preordered set that is directed and partially ordered (not all authors require formula_3 to be directed or partially ordered). Let formula_7 be a family of objects indexed by formula_3 and formula_9 be a homomorphism for all formula_10 with the following properties:
Then the pair formula_16 is called a over formula_17 The maps formula_18 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 formula_19 be a direct system." 
The system is said to be injective (resp. surjective, etc.) if this is true of all bonding maps. 
If formula_3 is directed (resp. countable) then the system is said to be directed (resp. countable). 
Canonical direct limit.
The of the direct system formula_16 in the category of sets is denoted by formula_22 and is defined as follows. Its underlying set is the disjoint union of the formula_23's modulo a certain :
Here, if formula_25 and formula_26 then formula_27 if and only if there is some formula_28 with formula_29 and formula_30 and such that formula_31 
The most common way of defining the disjoint union is: 
where for every index formula_33 each element formula_25 is canonically identified with the element formula_35 in this disjoint union. 
For every index formula_33 denote the equivalence class containing formula_25 by:
which results in the formula_2th :
It is this pair formula_43 that forms the canonical direct limit, although often only the set formula_22 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, formula_45 whenever formula_46 In particular, given any formula_10 and any formula_26 the fiber formula_49 will be a subset of the equivalence class formula_50 said differently, any element of formula_49 is always equivalent to formula_52 
The algebraic operations on formula_53 are defined such that these maps become homomorphisms. Formally, the direct limit of the direct system formula_54 consists of the object formula_22 together with the canonical homomorphisms formula_56
Direct limits in an arbitrary category.
The direct limit can be defined in an arbitrary category formula_57 by means of a universal property. Let formula_54 be a direct system of objects and morphisms in formula_57 (as defined above). A target or cocone is a pair formula_60 where formula_61 is an object in formula_57 and formula_63 are morphisms for each formula_64 such that formula_65 whenever formula_46 A direct limit of the direct system formula_54 is a universally repelling target formula_60 in the sense that formula_60 is a target and for each target formula_70 there is a unique morphism formula_71 such that formula_72 for each formula_73 The following diagram
will then commute for all indices formula_74 
The direct limit is often denoted
with the direct system formula_54 and the canonical morphisms formula_77 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 formula_78 there exists a isomorphism formula_79 that commutes with the canonical morphisms.
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 if you start with a directed system of short exact sequences formula_130 and form direct limits, you obtain a short exact sequence formula_131.
Related constructions and generalizations.
We note that a direct system in a category formula_57 admits an alternative description in terms of functors. Any directed set formula_133 can be considered as a small category formula_134 whose objects are the elements formula_3 and there is a morphisms formula_136 if and only if formula_137. A direct system over formula_3 is then the same as a covariant functor formula_139. 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 formula_140 from a filtered category formula_141 to some category formula_57 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 formula_57, there may be direct systems in formula_57 which don't have a direct limit in formula_57 (consider for example the category of finite sets, or the category of finitely generated abelian groups). In this case, we can always embed formula_57 into a category formula_147 in which all direct limits exist; the objects of formula_147 are called ind-objects of formula_57. 
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.