# Direct limit

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Special case of colimit in category theory

Algebraic structure → Ring theory Ring theory Basic concepts Rings • Subrings • Ideal • Quotient ring • Fractional ideal • Total ring of fractions • Product of rings • Free product of associative algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative algebra • Graded ring • Involutive ring • Category of rings • Initial ring Z {\displaystyle \mathbb {Z} } • Terminal ring 0 = Z / 1 Z {\displaystyle 0=\mathbb {Z} /1\mathbb {Z} } Related structures • Field • Finite field • Non-associative ring • Lie ring • Jordan ring • Semiring • Semifield Commutative algebra Commutative rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite field • Polynomial ring • Formal power series ring Algebraic number theory • Algebraic number field • Integers modulo n • Ring of integers • p-adic integers Z p {\displaystyle \mathbb {Z} _{p}} • p-adic numbers Q p {\displaystyle \mathbb {Q} _{p}} • Prüfer p-ring Z ( p ∞ ) {\displaystyle \mathbb {Z} (p^{\infty })} Noncommutative algebra Noncommutative rings • Division ring • Semiprimitive ring • Simple ring • Commutator Noncommutative algebraic geometry Free algebra Clifford algebra • Geometric algebra Operator algebra v t e

In [mathematics](/source/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](/source/Group_(mathematics)), [rings](/source/Ring_(mathematics)), [vector spaces](/source/Vector_space) or in general objects from any [category](/source/Category_(mathematics)). The way they are put together is specified by a system of [homomorphisms](/source/Homomorphism) ([group homomorphism](/source/Group_homomorphism), [ring homomorphism](/source/Ring_homomorphism), or in general [morphisms](/source/Morphism) in the category) between those smaller objects. The direct limit of the objects A i {\displaystyle A_{i}} , where i {\displaystyle i} ranges over some [directed set](/source/Directed_set) I {\displaystyle I} , is denoted by lim → ⁡ A i {\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](/source/Limit_(category_theory)) in [category theory](/source/Category_theory). Direct limits are [dual](/source/Dual_(category_theory)) to [inverse limits](/source/Inverse_limit), which are a special case of [limits](/source/Limit_(category_theory)) in category theory.

## Formal definition

We will first give the definition for [algebraic structures](/source/Algebraic_structure) like [groups](/source/Group_(mathematics)) and [modules](/source/Module_(mathematics)), and then the general definition, which can be used in any [category](/source/Category_(mathematics)).

### Direct limits of algebraic objects

In this section objects are understood to consist of underlying [sets](/source/Set_(mathematics)) equipped with a given [algebraic structure](/source/Algebraic_structure), such as [groups](/source/Group_(mathematics)), [rings](/source/Ring_(mathematics)), [modules](/source/Module_(mathematics)) (over a fixed ring), [algebras](/source/Algebra_over_a_field) (over a fixed [field](/source/Field_(mathematics))), etc. With this in mind, *[homomorphisms](/source/Homomorphism)* are understood in the corresponding setting ([group homomorphisms](/source/Group_homomorphism), etc.).

Let ⟨ I , ≤ ⟩ {\displaystyle \langle I,\leq \rangle } be a [directed set](/source/Directed_set). Let { A i : i ∈ I } {\displaystyle \{A_{i}:i\in I\}} be a family of objects [indexed](/source/Index_set) by I {\displaystyle I\,} and f i j : A i → A j {\displaystyle f_{ij}\colon A_{i}\rightarrow A_{j}} be a homomorphism for all i ≤ j {\displaystyle i\leq j} with the following properties:

1. f i i {\displaystyle f_{ii}\,} is the identity on A i {\displaystyle A_{i}\,} , and

1. f i k = f j k ∘ f i j {\displaystyle f_{ik}=f_{jk}\circ f_{ij}} for all i ≤ j ≤ k {\displaystyle i\leq j\leq k} .

Then the pair ⟨ A i , f i j ⟩ {\displaystyle \langle A_{i},f_{ij}\rangle } is called a **direct system** over I {\displaystyle I} .

The **direct limit** of the direct system ⟨ A i , f i j ⟩ {\displaystyle \langle A_{i},f_{ij}\rangle } is denoted by lim → ⁡ A i {\displaystyle \varinjlim A_{i}} and is defined as follows. Its underlying set is the [disjoint union](/source/Disjoint_union) of the A i {\displaystyle A_{i}} 's [modulo](/source/Modulo_(jargon)) a certain [equivalence relation](/source/Equivalence_relation) ∼ {\displaystyle \sim \,} :

- lim → ⁡ A i = ⨆ i A i / ∼ . {\displaystyle \varinjlim A_{i}=\bigsqcup _{i}A_{i}{\bigg /}\sim .}

Here, if x i ∈ A i {\displaystyle x_{i}\in A_{i}} and x j ∈ A j {\displaystyle x_{j}\in A_{j}} , then x i ∼ x j {\displaystyle x_{i}\sim \,x_{j}} if and only if there is some k ∈ I {\displaystyle k\in I} with i ≤ k {\displaystyle i\leq k} and j ≤ k {\displaystyle j\leq k} such that f i k ( x i ) = f j k ( x j ) {\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](/source/Inverse_limit) is that an element is equivalent to all its images under the maps of the direct system, i.e. x i ∼ f i j ( x i ) {\displaystyle x_{i}\sim \,f_{ij}(x_{i})} whenever i ≤ j {\displaystyle i\leq j} .

One obtains from this definition *canonical functions* ϕ j : A j → lim → ⁡ A i {\displaystyle \phi _{j}\colon A_{j}\rightarrow \varinjlim A_{i}} sending each element to its equivalence class. The algebraic operations on lim → ⁡ A i {\displaystyle \varinjlim A_{i}\,} are defined such that these maps become homomorphisms. Formally, the direct limit of the direct system ⟨ A i , f i j ⟩ {\displaystyle \langle A_{i},f_{ij}\rangle } consists of the object lim → ⁡ A i {\displaystyle \varinjlim A_{i}} together with the canonical homomorphisms ϕ j : A j → lim → ⁡ A i {\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](/source/Category_(mathematics)) C {\displaystyle {\mathcal {C}}} by means of a [universal property](/source/Universal_property). Let ⟨ X i , f i j ⟩ {\displaystyle \langle X_{i},f_{ij}\rangle } be a direct system of objects and morphisms in C {\displaystyle {\mathcal {C}}} (as defined above). A *target* is a pair ⟨ X , ϕ i ⟩ {\displaystyle \langle X,\phi _{i}\rangle } where X {\displaystyle X\,} is an object in C {\displaystyle {\mathcal {C}}} and ϕ i : X i → X {\displaystyle \phi _{i}\colon X_{i}\rightarrow X} are morphisms for each i ∈ I {\displaystyle i\in I} such that ϕ i = ϕ j ∘ f i j {\displaystyle \phi _{i}=\phi _{j}\circ f_{ij}} whenever i ≤ j {\displaystyle i\leq j} . A direct limit of the direct system ⟨ X i , f i j ⟩ {\displaystyle \langle X_{i},f_{ij}\rangle } is a *universally repelling target* ⟨ X , ϕ i ⟩ {\displaystyle \langle X,\phi _{i}\rangle } in the sense that ⟨ X , ϕ i ⟩ {\displaystyle \langle X,\phi _{i}\rangle } is a target and for each target ⟨ Y , ψ i ⟩ {\displaystyle \langle Y,\psi _{i}\rangle } , there is a unique morphism u : X → Y {\displaystyle u\colon X\rightarrow Y} such that u ∘ ϕ i = ψ i {\displaystyle u\circ \phi _{i}=\psi _{i}} for each *i*. The following diagram

will then [commute](/source/Commutative_diagram) for all *i*, *j*.

The direct limit is often denoted

- X = lim → ⁡ X i {\displaystyle X=\varinjlim X_{i}}

with the direct system ⟨ X i , f i j ⟩ {\displaystyle \langle X_{i},f_{ij}\rangle } and the canonical morphisms ϕ i {\displaystyle \phi _{i}} (or, more precisely, canonical injections ι i {\displaystyle \iota _{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](/source/Isomorphism) *X*′ → *X* that commutes with the canonical morphisms.

## Examples

- A collection of subsets M i {\displaystyle M_{i}} of a set M {\displaystyle M} can be [partially ordered](/source/Partial_order) by inclusion. If the collection is directed, its direct limit is the union ⋃ M i {\displaystyle \bigcup M_{i}} .

- Similarly, the collection of finitely generated subgroups H i {\displaystyle H_{i}} of a given group G {\displaystyle G} can be partially ordered by inclusion. Finite sets of finitely generated subgroups { ⟨ X 1 ⟩ , ⟨ X 2 ⟩ , … ⟨ X n ⟩ } {\displaystyle \{\langle X_{1}\rangle ,\langle X_{2}\rangle ,\dots \langle X_{n}\rangle \}} are containined in the finitely generated subgroup ⟨ ∪ X i ⟩ {\displaystyle \langle \cup X_{i}\rangle } , so the index set is indeed directed. With the inclusion morphisms f i , j : H i → H j {\displaystyle f_{i,j}:H_{i}\to H_{j}} , the direct limit is simply (isomorphic to) G {\displaystyle G} . An analogous result holds for rings, modules, algebras, etc. Note the requirement of finite generation may be weakened, as long as the index set remains directed.

- The [weak topology](/source/Weak_topology) of a [CW complex](/source/CW_complex) is defined as a direct limit.

- Let I {\displaystyle I} be any directed set with a [greatest element](/source/Greatest_element) m {\displaystyle m} . The direct limit X {\displaystyle X} of any corresponding direct system is isomorphic to X m {\displaystyle X_{m}} and the canonical morphism ϕ m : X m → X {\displaystyle \phi _{m}:X_{m}\rightarrow X} is an isomorphism.

- Let *K* be a field. For a positive integer *n*, consider the [general linear group](/source/General_linear_group) GL(*n;K*) consisting of invertible *n* x *n* - matrices with entries from *K*. We have a group homomorphism GL(*n;K*) → GL(*n*+1;*K*) that enlarges matrices by putting a 1 in the lower right corner and zeros elsewhere in the last row and column. The direct limit of this system is the general linear group of *K*, written as GL(*K*). An element of GL(*K*) can be thought of as an infinite invertible matrix that differs from the infinite identity matrix in only finitely many entries. The group GL(*K*) is of vital importance in [algebraic K-theory](/source/Algebraic_K-theory).

- Let *p* be a [prime number](/source/Prime_number). Consider the direct system composed of the [factor groups](/source/Quotient_group) Z / p n Z {\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} } and the homomorphisms Z / p n Z → Z / p n + 1 Z {\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} \rightarrow \mathbb {Z} /p^{n+1}\mathbb {Z} } induced by multiplication by p {\displaystyle p} . The direct limit of this system consists of all the [roots of unity](/source/Roots_of_unity) of order some power of p {\displaystyle p} , and is called the [Prüfer group](/source/Pr%C3%BCfer_group) Z ( p ∞ ) {\displaystyle \mathbb {Z} (p^{\infty })} .

- There is a (non-obvious) injective ring homomorphism from the ring of [symmetric polynomials](/source/Symmetric_polynomial) in n {\displaystyle n} variables to the ring of symmetric polynomials in n + 1 {\displaystyle n+1} variables. Forming the direct limit of this direct system yields the [ring of symmetric functions](/source/Ring_of_symmetric_functions).

- Let *F* be a *C*-valued [sheaf](/source/Sheaf_(mathematics)) on a [topological space](/source/Topological_space) *X*. Fix a point *x* in *X*. The open neighborhoods of *x* form a directed set ordered by inclusion (*U* ≤ *V* if and only if *U* contains *V*). The corresponding direct system is (*F*(*U*), *r**U*,*V*) where *r* is the restriction map. The direct limit of this system is called the *[stalk](/source/Stalk_(mathematics))* of *F* at *x*, denoted *F**x*. For each neighborhood *U* of *x*, the canonical morphism *F*(*U*) → *F**x* associates to a section *s* of *F* over *U* an element *s**x* of the stalk *F**x* called the *[germ](/source/Germ_(mathematics))* of *s* at *x*.

- Direct limits in the [category of topological spaces](/source/Category_of_topological_spaces) are given by placing the [final topology](/source/Final_topology) on the underlying set-theoretic direct limit.

- An [ind-scheme](/source/Ind-scheme) is an inductive limit of schemes.

## Properties

Direct limits are linked to [inverse limits](/source/Inverse_limit) via

- H o m ( lim → ⁡ X i , Y ) = lim ← ⁡ H o m ( X i , Y ) . {\displaystyle \mathrm {Hom} (\varinjlim X_{i},Y)=\varprojlim \mathrm {Hom} (X_{i},Y).}

An important property is that taking direct limits in the category of [modules](/source/Module_(mathematics)) is an [exact functor](/source/Exact_functor). This means that for any directed system of [short exact sequences](/source/Short_exact_sequence) 0 → A i → B i → C i → 0 {\displaystyle 0\to A_{i}\to B_{i}\to C_{i}\to 0} , the sequence 0 → lim → ⁡ A i → lim → ⁡ B i → lim → ⁡ C i → 0 {\displaystyle 0\to \varinjlim A_{i}\to \varinjlim B_{i}\to \varinjlim C_{i}\to 0} of direct limits is also exact.

## Related constructions and generalizations

We note that a direct system in a category C {\displaystyle {\mathcal {C}}} admits an alternative description in terms of [functors](/source/Functor). Any directed set ⟨ I , ≤ ⟩ {\displaystyle \langle I,\leq \rangle } can be considered as a [small category](/source/Small_category) I {\displaystyle {\mathcal {I}}} whose objects are the elements I {\displaystyle I} and there is a morphism i → j {\displaystyle i\rightarrow j} [if and only if](/source/If_and_only_if) i ≤ j {\displaystyle i\leq j} . A direct system over I {\displaystyle I} is then the same as a [covariant functor](/source/Covariant_functor) I → C {\displaystyle {\mathcal {I}}\rightarrow {\mathcal {C}}} . The [colimit](/source/Limit_(category_theory)) of this functor is the same as the direct limit of the original direct system.

A notion closely related to direct limits are the [filtered colimits](/source/Filtered_category). Here we start with a covariant functor J → C {\displaystyle {\mathcal {J}}\to {\mathcal {C}}} from a [filtered category](/source/Filtered_category) J {\displaystyle {\mathcal {J}}} to some category C {\displaystyle {\mathcal {C}}} and form the colimit of this functor. One can show that a category has all directed limits if and only if it has all filtered colimits, and a functor defined on such a category commutes with all direct limits if and only if it commutes with all filtered colimits.[1]

Given an arbitrary category C {\displaystyle {\mathcal {C}}} , there may be direct systems in C {\displaystyle {\mathcal {C}}} that don't have a direct limit in C {\displaystyle {\mathcal {C}}} (consider for example the category of finite sets, or the category of [finitely generated abelian groups](/source/Finitely_generated_abelian_group)). In this case, we can always embed C {\displaystyle {\mathcal {C}}} into a category Ind ( C ) {\displaystyle {\text{Ind}}({\mathcal {C}})} in which all direct limits exist; the objects of Ind ( C ) {\displaystyle {\text{Ind}}({\mathcal {C}})} are called [ind-objects](/source/Ind-object) of C {\displaystyle {\mathcal {C}}} .

The [categorical dual](/source/Dual_(category_theory)) of the direct limit is called the [inverse limit](/source/Inverse_limit). As above, inverse limits can be viewed as limits of certain functors and are closely related to limits over cofiltered categories.

## Terminology

In the literature, one finds the terms "directed limit", "direct inductive limit", "directed colimit", "direct colimit" and "inductive limit" for the concept of direct limit defined above. The term "inductive limit" is ambiguous however, as some authors use it for the general concept of colimit.

## See also

- [Direct limits of groups](/source/Direct_limit_of_groups)

## Notes

1. **[^](#cite_ref-1)** Adamek, J.; Rosicky, J. (1994). [*Locally Presentable and Accessible Categories*](https://books.google.com/books?id=iXh6rOd7of0C). Cambridge University Press. p. 15. [ISBN](/source/ISBN_(identifier)) [9780521422611](https://en.wikipedia.org/wiki/Special:BookSources/9780521422611).

## References

- [Bourbaki, Nicolas](/source/Nicolas_Bourbaki) (1968), *Elements of mathematics. Theory of sets*, Translated from French, Paris: Hermann, [MR](/source/MR_(identifier)) [0237342](https://mathscinet.ams.org/mathscinet-getitem?mr=0237342)

- [Mac Lane, Saunders](/source/Saunders_Mac_Lane) (1998), *[Categories for the Working Mathematician](/source/Categories_for_the_Working_Mathematician)*, [Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics), vol. 5 (2nd ed.), Springer-Verlag

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Adapted from the Wikipedia article [Direct limit](https://en.wikipedia.org/wiki/Direct_limit) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Direct_limit?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.