Editing Direct limit (section)

=== 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.