# Direct limit

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.

## Formal definition

### Algebraic objects

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:

1. ${\displaystyle f_{ii}\,}$ is the identity of ${\displaystyle A_{i}\,}$, 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 underlying set of the direct limit, ${\displaystyle A\,}$, of the direct system ${\displaystyle \langle A_{i},f_{ij}\rangle }$ is defined as 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 ${\displaystyle x_{j}\in A_{j}}$, ${\displaystyle x_{i}\sim \,x_{j}}$ if there is some ${\displaystyle k\in I}$ such that ${\displaystyle f_{ik}(x_{i})=f_{jk}(x_{j})\,}$. Heuristically, 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 directed system, i.e. ${\displaystyle x_{i}\sim \,f_{ik}(x_{i})}$.

One naturally obtains from this definition canonical functions ${\displaystyle \phi _{i}\colon A_{i}\rightarrow A}$ sending each element to its equivalence class. The algebraic operations on ${\displaystyle A\,}$ are defined such that these maps become morphisms.

An important property is that taking direct limits in the category of modules is an exact functor.

### Direct limit over a direct system in a 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 such that ${\displaystyle \phi _{i}=\phi _{j}\circ f_{ij}}$. A direct limit is a universally repelling target in the sense that for each target ${\displaystyle \langle Y,\psi _{i}\rangle }$, there is a unique morphism ${\displaystyle f\colon X\rightarrow Y}$ where ${\displaystyle f\circ \phi _{i}=\psi _{i}}$ for each i. The direct limit of ${\displaystyle \langle X_{i},f_{ij}\rangle }$ is often denoted

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

Unlike for algebraic objects, the direct limit may not exist in an arbitrary category. If it does, however, it is unique in a strong sense: given another direct limit X′ there exists a unique isomorphism X′ → X commuting with the canonical morphisms.

We note that a direct system in a category ${\displaystyle {\mathcal {C}}}$ admits an alternative description in terms of functors. Any directed poset ${\displaystyle \langle I,\leq \rangle }$ can be considered as a small category ${\displaystyle {\mathcal {I}}}$ where the morphisms consist of arrows ${\displaystyle i\rightarrow j}$ if and only if ${\displaystyle i\leq j}$. A direct system is then just a covariant functor ${\displaystyle {\mathcal {I}}\rightarrow {\mathcal {C}}}$. In this case a direct limit is a colimit.

## Examples

• A collection of subsets ${\displaystyle M_{i}}$ of a set M can be partially ordered by inclusion. If the collection is directed, its direct limit is the union ${\displaystyle \bigcup M_{i}}$.
• Let I be any directed set with a greatest element m. The direct limit of any corresponding direct system is isomorphic to Xm and the canonical morphism φm: XmX is an isomorphism.
• Let p be a prime number. Consider the direct system composed of the groups Z/pnZ and the homomorphisms Z/pnZZ/pn+1Z induced by multiplication by p. The direct limit of this system consists of all the roots of unity of order some power of p, and is called the Prüfer group Z(p).
• Let F be a C-valued sheaf on a topological space X. Fix a point x in X. The open neighborhoods of x form a directed poset ordered by inclusion (UV if and only if U contains V). The corresponding direct system is (F(U), rU,V) where r is the restriction map. The direct limit of this system is called the stalk of F at x, denoted Fx. For each neighborhood U of x, the canonical morphism F(U) → Fx associates to a section s of F over U an element sx of the stalk Fx called the germ of s at x.
• Direct limits in the category of topological spaces are given by placing the final topology on the underlying set-theoretic direct limit.
• Direct limits are linked to inverse limits via
${\displaystyle \mathrm {Hom} (\varinjlim X_{i},Y)=\varprojlim \mathrm {Hom} (X_{i},Y).}$
• Consider a sequence {An, φn} where An is a C*-algebra and φn : AnAn + 1 is a *-homomorphism. The C*-analog of the direct limit construction gives a C*-algebra satisfying the universal property above.

## Related constructions and generalizations

The categorical dual of the direct limit is called the inverse limit (or projective limit). More general concepts are the limits and colimits of category theory. The terminology is somewhat confusing: direct limits are colimits while inverse limits are limits.