Direct limits of algebraic objects

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:

1. ${\displaystyle f_{ii}\,}$ is the identity on ${\displaystyle A_{i}\,}$, and
2. ${\displaystyle f_{ik}=f_{jk}\circ f_{ij}}$ for all ${\displaystyle i\leq j\leq k}$.

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 ${\displaystyle \sim \,}$:

${\displaystyle \varinjlim A_{i}=\bigsqcup _{i}A_{i}{\bigg /}\sim .}$

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 ${\displaystyle i\leq j}$.

One obtains from this definition canonical functions ${\displaystyle \phi _{j}\colon A_{j}\rightarrow \varinjlim A_{i}}$ sending each element to its equivalence class. The algebraic operations on ${\displaystyle \varinjlim A_{i}\,}$ are defined such that these maps become homomorphisms. Formally, the direct limit of the direct system ${\displaystyle \langle A_{i},f_{ij}\rangle }$ consists of the object ${\displaystyle \varinjlim A_{i}}$ together with the canonical homomorphisms ${\displaystyle \phi _{j}\colon A_{j}\rightarrow \varinjlim A_{i}}$.

Direct limits in an arbitrary category

The direct limit can be defined in an arbitrary category ${\displaystyle {\mathcal {C}}}$ by means of a universal property. Let ${\displaystyle \langle X_{i},f_{ij}\rangle }$ be a direct system of objects and morphisms in ${\displaystyle {\mathcal {C}}}$ (as defined above). A target is a pair ${\displaystyle \langle X,\phi _{i}\rangle }$ where ${\displaystyle X\,}$ is an object in ${\displaystyle {\mathcal {C}}}$ and ${\displaystyle \phi _{i}\colon X_{i}\rightarrow X}$ are morphisms for each ${\displaystyle i\in I}$ such that ${\displaystyle \phi _{i}=\phi _{j}\circ f_{ij}}$ whenever ${\displaystyle i\leq j}$. A direct limit of the direct system ${\displaystyle \langle X_{i},f_{ij}\rangle }$ is a universally repelling target ${\displaystyle \langle X,\phi _{i}\rangle }$ in the sense that ${\displaystyle \langle X,\phi _{i}\rangle }$ is a target and for each target ${\displaystyle \langle Y,\psi _{i}\rangle }$, there is a unique morphism ${\displaystyle u\colon X\rightarrow Y}$ such that ${\displaystyle u\circ \phi _{i}=\psi _{i}}$ for each i. The following diagram

will then commute for all i, j.

The direct limit is often denoted

${\displaystyle X=\varinjlim X_{i}}$

with the direct system ${\displaystyle \langle X_{i},f_{ij}\rangle }$ and the canonical morphisms ${\displaystyle \phi _{i}}$ 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.