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, "homomorphisms" are understood in the corresponding setting (group homomorphisms, etc.). Let formula_5 be a directed set. Let formula_6 be a family of objects indexed by formula_7 and formula_8 be a homomorphism for all formula_9 with the following properties: Then the pair formula_14 is called a direct system over formula_3. The direct limit of the direct system formula_14 is denoted by formula_17 and is defined as follows. Its underlying set is the disjoint union of the formula_11's modulo a certain : Here, if formula_20 and formula_21, then formula_22  iff there is some formula_23 with formula_24 and formula_25 and such that formula_26. 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. formula_27 whenever formula_9. One obtains from this definition "canonical functions" formula_29 sending each element to its equivalence class. The algebraic operations on formula_30 are defined such that these maps become homomorphisms. Formally, the direct limit of the  direct system formula_14 consists of the object formula_17 together with the canonical homomorphisms formula_29. Direct limits in an arbitrary category. The direct limit can be defined in an arbitrary category formula_34 by means of a universal property. Let formula_35 be a direct system of objects and morphisms in formula_34 (as defined above). A "target" is a pair formula_37 where formula_38 is an object in formula_34 and formula_40 are morphisms for each formula_41 such that formula_42 whenever formula_9. A direct limit of the direct system formula_35 is a "universally repelling target" formula_37 in the sense that formula_37 is a target and  for each target formula_47, there is a unique morphism formula_48 such that formula_49 for each "i". The following diagram will then commute for all "i", "j". The direct limit is often denoted with the direct system formula_35 and the canonical morphisms formula_52 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. 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_68 and form direct limits, you obtain a short exact sequence formula_69. Related constructions and generalizations. We note that a direct system in a category formula_34 admits an alternative description in terms of functors. Any directed set formula_71 can be considered as a small category formula_72 whose objects are the elements formula_3 and there is a morphisms formula_74 if and only if formula_75. A direct system over formula_3  is then the same as a covariant functor formula_77. 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_78 from a filtered category formula_79 to some category formula_34 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_34, there may be direct systems in formula_34 which don't have a direct limit in formula_34 (consider for example the category of finite sets, or the category of finitely generated abelian groups). In this case, we can always embed formula_34 into a category formula_85 in which all direct limits exist; the objects of formula_85 are called ind-objects of formula_34. 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.