Editing Direct limit (section)

==Formal definition==
We will first give the definition for [[Algebraic structure|algebraic structures]] like [[Group (mathematics)|groups]] and [[Module (mathematics)|modules]], and then the general definition, which can be used in any [[Category (mathematics)|category]].

===Direct limits of algebraic objects===
In this section objects are understood to consist of underlying [[Set (mathematics)|sets]] equipped with a given [[algebraic structure]], such as [[group (mathematics)|groups]], [[ring (mathematics)|rings]], [[module (mathematics)|modules]] (over a fixed ring), [[algebra over a field|algebras]] (over a fixed [[field (mathematics)|field]]), etc. With this in mind, ''[[homomorphism]]s'' are understood in the corresponding setting ([[group homomorphism]]s, etc.).

Let <math>\langle I,\le\rangle</math> be a [[directed set]]. Let <math>\{A_i : i\in I\}</math> be a family of objects [[index set|indexed]] by <math>I\,</math> and  <math> f_{ij}\colon A_i \rightarrow A_j </math> be a homomorphism for all <math>i \le j</math> with the following properties:
# <math>f_{ii}\,</math> is the identity on <math>A_i\,</math>, and
# <math>f_{ik}= f_{jk}\circ f_{ij}</math> for all <math>i\le j\le k</math>.
Then the pair <math>\langle A_i,f_{ij}\rangle</math> is called a '''direct system''' over <math>I</math>.

The '''direct limit''' of the direct system <math>\langle A_i,f_{ij}\rangle</math> is denoted by <math>\varinjlim A_i</math> and is defined as follows. Its underlying set is the [[disjoint union]] of the <math>A_i</math>'s [[Modulo (jargon)|modulo]] a certain {{nowrap|[[equivalence relation]] <math>\sim\,</math>}}:

:<math>\varinjlim A_i = \bigsqcup_i A_i\bigg/\sim.</math>

Here, if <math>x_i\in A_i</math> and <math>x_j\in A_j</math>, then <math>x_i\sim\, x_j</math> if and only if there is some <math>k\in I</math> with <math>i \le k</math> and <math>j \le k</math> such that <math>f_{ik}(x_i) = f_{jk}(x_j)\,</math>.
Intuitively, 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. <math>x_i\sim\, f_{ij}(x_i)</math> whenever <math>i \le j</math>.

One obtains from this definition ''canonical functions'' <math>\phi_j \colon A_j\rightarrow \varinjlim A_i</math> sending each element to its equivalence class. The algebraic operations on <math>\varinjlim A_i\,</math> are defined such that these maps become homomorphisms. Formally, the direct limit of the direct system <math>\langle A_i,f_{ij}\rangle</math> consists of the object <math>\varinjlim A_i</math> together with the canonical homomorphisms <math>\phi_j \colon A_j\rightarrow \varinjlim A_i</math>.

=== Direct limits in an arbitrary category ===
The direct limit can be defined in an arbitrary [[category (mathematics)|category]] <math>\mathcal{C}</math> by means of a [[universal property]]. Let <math>\langle X_i, f_{ij}\rangle</math> be a direct system of objects and morphisms in <math>\mathcal{C}</math> (as defined above). A ''target'' is a pair <math>\langle X, \phi_i\rangle</math> where <math>X\,</math> is an object in <math>\mathcal{C}</math> and <math>\phi_i\colon X_i\rightarrow X</math> are morphisms for each <math>i\in I</math> such that <math>\phi_i =\phi_j \circ f_{ij}</math> whenever  <math>i \le j</math>. A direct limit of the direct system <math>\langle X_i, f_{ij}\rangle</math> is a ''universally repelling target'' <math>\langle X, \phi_i\rangle</math> in the sense that <math>\langle X, \phi_i\rangle</math> is a target and for each target <math>\langle Y, \psi_i\rangle</math>, there is a unique morphism <math> u\colon X\rightarrow Y</math> such that <math>u\circ \phi_i=\psi_i</math> for each ''i''. The following diagram
<div style="text-align: center;">[[Image:Direct limit category.svg|class=skin-invert]]</div>
will then [[commutative diagram|commute]] for all ''i'', ''j''. 

The direct limit is often denoted
:<math>X = \varinjlim X_i</math>
with the direct system <math>\langle X_i, f_{ij}\rangle</math> and the canonical morphisms <math>\phi_i</math> (or, more precisely, canonical injections <math>\iota_i</math>) 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.