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 ${\displaystyle A_{i}}$, where ${\displaystyle i}$ ranges over some directed set ${\displaystyle I}$, is denoted by ${\displaystyle \varinjlim A_{i}}$. This notation suppresses the system of homomorphisms; however, the limit depends on the system of homomorphisms.

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 equipped 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:

Then the pair ${\displaystyle \langle A_{i},f_{ij}\rangle }$ is called a direct system over ${\displaystyle I}$.

The direct limit of the direct system ${\displaystyle \langle A_{i},f_{ij}\rangle }$ is denoted by ${\displaystyle \varinjlim A_{i}}$ and is defined as follows. Its underlying set is the disjoint union of the ${\displaystyle A_{i}}$'s modulo a certain equivalence relation ${\displaystyle \sim \,}$:

Here, if ${\displaystyle x_{i}\in A_{i}}$ and ${\displaystyle x_{j}\in A_{j}}$, then ${\displaystyle x_{i}\sim \,x_{j}}$ if and only if there is some ${\displaystyle k\in I}$ with ${\displaystyle i\leq k}$ and ${\displaystyle j\leq k}$ such that ${\displaystyle f_{ik}(x_{i})=f_{jk}(x_{j})\,}$. 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. ${\displaystyle x_{i}\sim \,f_{ij}(x_{i})}$ whenever ${\displaystyle i\leq j}$.