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

{{#invoke:see also|seealso}} 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.).

Start with the definition of a direct system of objects and homomorphisms. 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 ${\displaystyle f_{ij}:A_{i}\rightarrow A_{j}}$ 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 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 morphisms ${\displaystyle \phi _{i}:A_{i}\rightarrow A}$ sending each element to its equivalence class. The algebraic operations on ${\displaystyle A\,}$ are defined via these maps in the obvious manner.

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 ${\displaystyle {\mathcal {C}}}$ (same definition as above). The direct limit of this system is an object ${\displaystyle X\,}$ in ${\displaystyle {\mathcal {C}}}$ together with morphisms ${\displaystyle \phi _{i}:X_{i}\rightarrow X}$ satisfying ${\displaystyle \phi _{i}=\phi _{j}\circ f_{ij}}$. The pair ${\displaystyle \langle X,\phi _{i}\rangle }$ must be universal in the sense that for any other such pair ${\displaystyle \langle Y,\psi _{i}\rangle }$ there exists a unique morphism ${\displaystyle u:X\rightarrow Y}$ making the diagram

commute for all i, j. The direct limit is often denoted

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

with the direct system ${\displaystyle \langle X_{i},f_{ij}\rangle }$ being understood.

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

${\displaystyle \mathrm {Hom} (\varinjlim X_{i},Y)=\varprojlim \mathrm {Hom} (X_{i},Y).}$
• 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.