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 A i {displaystyle A_{i}} , where i {displaystyle i} ranges over some directed set I {displaystyle I} , is denoted by lim → ⁡ A i {displaystyle varinjlim A_{i}} . (This is a slight abuse of notation as it suppresses the system of homomorphisms that is crucial for the structure of the 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 A i {displaystyle A_{i}} , where i {displaystyle i} ranges over some directed set I {displaystyle I} , is denoted by lim → ⁡ A i {displaystyle varinjlim A_{i}} . (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 a special case of limits in category theory. 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. 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 ⟨ I , ≤ ⟩ {displaystyle langle I,leq angle } be a directed set. Let { A i : i ∈ I } {displaystyle {A_{i}:iin I}} be a family of objects indexed by I {displaystyle I,} and f i j : A i → A j {displaystyle f_{ij}colon A_{i} ightarrow A_{j}} be a homomorphism for all i ≤ j {displaystyle ileq j} with the following properties: Then the pair ⟨ A i , f i j ⟩ {displaystyle langle A_{i},f_{ij} angle } is called a direct system over I {displaystyle I} . The direct limit of the direct system ⟨ A i , f i j ⟩ {displaystyle langle A_{i},f_{ij} angle } is denoted by lim → ⁡ A i {displaystyle varinjlim A_{i}} and is defined as follows. Its underlying set is the disjoint union of the A i {displaystyle A_{i},} 's modulo a certain equivalence relation ∼ {displaystyle sim ,} : Here, if x i ∈ A i {displaystyle x_{i}in A_{i}} and x j ∈ A j {displaystyle x_{j}in A_{j}} , then x i ∼ x j {displaystyle x_{i}sim ,x_{j}} iff there is some k ∈ I {displaystyle kin I} with i ≤ k {displaystyle ileq k} and j ≤ k {displaystyle jleq k} and such that f i k ( x i ) = f j k ( x j ) {displaystyle f_{ik}(x_{i})=f_{jk}(x_{j}),} .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. x i ∼ f i j ( x i ) {displaystyle x_{i}sim ,f_{ij}(x_{i})} whenever i ≤ j {displaystyle ileq j} . One naturally obtains from this definition canonical functions ϕ i : A i → lim → ⁡ A i {displaystyle phi _{i}colon A_{i} ightarrow varinjlim A_{i}} sending each element to its equivalence class. The algebraic operations on lim → ⁡ A i {displaystyle varinjlim A_{i},} are defined such that these maps become homomorphisms. Formally, the direct limit of the direct system ⟨ A i , f i j ⟩ {displaystyle langle A_{i},f_{ij} angle } consists of the object lim → ⁡ A i {displaystyle varinjlim A_{i}} together with the canonical homomorphisms ϕ i : A i → lim → ⁡ A i {displaystyle phi _{i}colon A_{i} ightarrow varinjlim A_{i}} .