In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a
direct limit (also called
inductive limit) is a
colimitIn category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits....
of a "directed family of objects". We will first give the definition for
algebraic structureIn abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
s like
groupsIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
and
modulesIn abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
, and then the general definition which can be used in any
categoryIn mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
.
Algebraic objects
In this section objects are understood to be sets with a given
algebraic structureIn abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
such as
groupsIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
,
ringsIn mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
,
modulesIn abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
(over a fixed ring),
algebrasIn mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
(over a fixed field), etc. With this in mind,
homomorphismIn abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
s are understood in the corresponding setting (
group homomorphismIn mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
s, etc.).
Start with the definition of a
direct system of objects and homomorphisms. Let
be a
directed setIn mathematics, a directed set is a nonempty set A together with a reflexive and transitive binary relation ≤ , with the additional property that every pair of elements has an upper bound: In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤...
. Let
be a family of objects
indexedIn mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index set...
by
and
be a homomorphism for all
with the following properties:
-
is the identity of 
, and
-
for all 
.
Then the pair
is called a direct system over
.
The underlying set of the
direct limit,
, of the direct system
is defined as the
disjoint unionIn mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a disjoint union...
of the
's
moduloThe word modulo is the Latin ablative of modulus which itself means "a small measure."It was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801...
a certain
equivalence relationIn mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
:
Here, if
and
,
if there is some
such that
.
Heuristically, two elements in the disjoint union are equivalent
if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
they "eventually become equal" in the direct system. An equivalent formulation that highlights the duality to the
inverse limitIn mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...
is that an element is equivalent to all its images under the maps of the directed system, i.e.
.
One naturally obtains from this definition
canonical morphisms 
sending each element to its equivalence class. The algebraic operations on
are defined via these maps in the obvious manner.
An important property is that taking direct limits in the category of
modulesIn abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
is an
exact functorIn homological algebra, an exact functor is a functor, from some category to another, which preserves exact sequences. Exact functors are very convenient in algebraic calculations, roughly speaking because they can be applied to presentations of objects easily...
.
Direct limit over a direct system in a category
The direct limit can be defined in an arbitrary
categoryIn mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
by means of a
universal propertyIn various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
. Let
be a direct system of objects and morphisms in
(same definition as above). The direct limit of this system is an object
in
together with morphisms
satisfying
. The pair
must be universal in the sense that for any other such pair
there exists a unique morphism
making the diagram
commuteIn mathematics, and especially in category theory, a commutative diagram is a diagram of objects and morphisms such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition...
for all
i,
j. The direct limit is often denoted
with the direct system
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 isomorphismIn abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
X′ →
X commuting with the canonical morphisms.
We note that a direct system in a category
admits an alternative description in terms of
functorIn category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
s. Any directed poset
can be considered as a small category
where the morphisms consist of arrows
if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
. A direct system is then just a covariant functor
.
General Definition
Let
and
categories. Let
be a constant functor with fixed object
. Define for every functor
the functor
which assigns to each
the set
of natural transformations from
F to
. If
is
representableIn mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures In mathematics, particularly category theory, a...
, the representing object in
is called the direct limit of
F and is also denoted by
.
If
is an
abelian categoryIn mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
where arbitrary (also infinite) direct sums of objects exists (this is Grothedieck's Axiom AB3). Then
is representable for every functor
and
is a
right-exactIn homological algebra, an exact functor is a functor, from some category to another, which preserves exact sequences. Exact functors are very convenient in algebraic calculations, roughly speaking because they can be applied to presentations of objects easily...
additive functor of abelian categories.
Examples
- A collection of subsets
of a set M can be partially ordered by inclusion. If the collection is directed, its direct limit is the union 
.
- Let I be any directed set with a greatest element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually...
m. The direct limit of any corresponding direct system is isomorphic to Xm and the canonical morphism φm: Xm → X is an isomorphism.
- Let p be a prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
. Consider the direct system composed of the groups Z/pnZ and the homomorphisms Z/pnZ → Z/pn+1Z which are 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 groupIn mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z, for a prime number p is the unique p-group in which every element has p pth roots. The group is named after Heinz Prüfer...
Z(p∞).
- Let F be a C-valued sheaf
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
on a topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X. Fix a point x in X. The open neighborhoods of x form a directed poset ordered by inclusion (U ≤ V 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 germIn mathematics, the notion of a germ of an object in/on a topological space captures the local properties of the object. In particular, the objects in question are mostly functions and subsets...
of s at x.
- Direct limits in the category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous...
are given by placing the final topologyIn general topology and related areas of mathematics, the final topology on a set X, with respect to a family of functions into X, is the finest topology on X which makes those functions continuous.- Definition :Given a set X and a family of topological spaces Y_i with functionsf_i: Y_i \to Xthe...
on the underlying set-theoretic direct limit.
- Inductive limits are linked to projective ones via
- Consider a sequence {An, φn} where An is a C*-algebra and φn : An → An + 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 dualIn category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category Cop...
of the direct limit is called the
inverse limitIn mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...
(or projective limit). More general concepts are the
limits and colimitsIn category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits....
of category theory. The terminology is somewhat confusing: direct limits are colimits while inverse limits are limits.