Direct system (mathematics)

In mathematics, a direct limit (also called inductive limit) is a colimit of a "directed family of objects". 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 be 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 ${\displaystyle \langle I,\leq \rangle }$ be a directed set. Let ${\displaystyle \{A_{i}:i\in I\}}$ be a family of objects indexed by ${\displaystyle I\,}$ and ${\displaystyle f_{ij}\colon A_{i}\rightarrow A_{j}}$ be a homomorphism for all ${\displaystyle i\leq j}$ with the following properties:

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