Direct Limit
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In 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, rings, vector spaces or in general objects from any
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
. The way they are put together is specified by a system of
homomorphisms In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
( group homomorphism,
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition prese ...
, or in general
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphis ...
s in the category) between those smaller objects. The direct limit of the objects A_i, where i ranges over some
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty Set (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive binary relation \,\leq\, (that is, a preorder), with ...
I, is denoted by \varinjlim A_i . (This is a slight
abuse of notation In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors ...
as it suppresses the system of homomorphisms that is crucial for the structure of the limit.) Direct limits are a special case of the concept of
colimit In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions su ...
in category theory. Direct limits are
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
to inverse limits, which are also a special case of limits in category theory.


Formal definition

We will first give the definition for
algebraic structures In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
like groups and modules, and then the general definition, which can be used in any
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
.


Direct limits of algebraic objects

In this section objects are understood to consist of underlying sets equipped with a given
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
, such as groups, rings, modules (over a fixed ring),
algebras In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and additio ...
(over a fixed field), etc. With this in mind, ''
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...
s'' are understood in the corresponding setting ( group homomorphisms, etc.). Let \langle I,\le\rangle be a
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty Set (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive binary relation \,\leq\, (that is, a preorder), with ...
. Let \ be a family of objects indexed by I\, and f_\colon A_i \rightarrow A_j be a homomorphism for all i \le j with the following properties: # f_\, is the identity of A_i\,, and # f_= f_\circ f_ for all i\le j\le k. Then the pair \langle A_i,f_\rangle is called a direct system over I. The direct limit of the direct system \langle A_i,f_\rangle is denoted by \varinjlim A_i and is defined as follows. Its underlying set is the
disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ...
of the A_i's
modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is t ...
a certain : :\varinjlim A_i = \bigsqcup_i A_i\bigg/\sim. Here, if x_i\in A_i and x_j\in A_j, then x_i\sim\, x_j if and only if there is some k\in I with i \le k and j \le k and such that f_(x_i) = f_(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 In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ...
is that an element is equivalent to all its images under the maps of the direct system, i.e. x_i\sim\, f_(x_i) whenever i \le j. One obtains from this definition ''canonical functions'' \phi_j \colon A_j\rightarrow \varinjlim A_i sending each element to its equivalence class. The algebraic operations on \varinjlim A_i\, are defined such that these maps become homomorphisms. Formally, the direct limit of the direct system \langle A_i,f_\rangle consists of the object \varinjlim A_i together with the canonical homomorphisms \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 Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
\mathcal by means of a
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently ...
. Let \langle X_i, f_\rangle be a direct system of objects and morphisms in \mathcal (as defined above). A ''target'' is a pair \langle X, \phi_i\rangle where X\, is an object in \mathcal and \phi_i\colon X_i\rightarrow X are morphisms for each i\in I such that \phi_i =\phi_j \circ f_ whenever i \le j. A direct limit of the direct system \langle X_i, f_\rangle is a ''universally repelling target'' \langle X, \phi_i\rangle in the sense that \langle X, \phi_i\rangle is a target and for each target \langle Y, \psi_i\rangle, there is a unique morphism u\colon X\rightarrow Y such that 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 :X = \varinjlim X_i with the direct system \langle X_i, f_\rangle and the canonical morphisms \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 In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
''X''′ → ''X'' that commutes with the canonical morphisms.


Examples

*A collection of subsets M_i of a set M can be partially ordered by inclusion. If the collection is directed, its direct limit is the union \bigcup M_i. The same is true for a directed collection of
subgroups In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgro ...
of a given group, or a directed collection of subrings of a given ring, etc. *The
weak topology In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
of a
CW complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...
is defined as a direct limit. *Let X 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 (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an el ...
m. The direct limit of any corresponding direct system is isomorphic to X_m and the canonical morphism \phi_m: X_m \rightarrow X is an isomorphism. *Let ''K'' be a field. For a positive integer ''n'', consider the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible ...
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 off 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 Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
. *Let ''p'' be a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
. Consider the direct system composed of the factor groups \mathbb/p^n\mathbb and the homomorphisms \mathbb/p^n\mathbb \rightarrow \mathbb/p^\mathbb induced by multiplication by p. The direct limit of this system consists of all the
roots of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important i ...
of order some power of p, and is called the
Prüfer group In mathematics, specifically in group theory, the Prüfer ''p''-group or the ''p''-quasicyclic group or ''p''∞-group, Z(''p''∞), for a prime number ''p'' is the unique ''p''-group in which every element has ''p'' different ''p''-th roots. ...
\mathbb(p^\infty). *There is a (non-obvious) injective ring homomorphism from the ring of
symmetric polynomials In mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, is a ''symmetric polynomial'' if for any permutation of the subscripts one has ...
in n variables to the ring of symmetric polynomials in n + 1 variables. Forming the direct limit of this direct system yields the ring of symmetric functions. *Let ''F'' be a ''C''-valued sheaf on a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''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 Stalk or stalking may refer to: Behaviour * Stalk, the stealthy approach (phase) of a predator towards its prey * Stalking, an act of intrusive behaviour or unwanted attention towards a person * Deer stalking, the pursuit of deer for sport Biol ...
'' 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'' 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 con ...
are given by placing the
final topology In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a set X, with respect to a family of functions from topological spaces into X, is the finest topology on X that ...
on the underlying set-theoretic direct limit. *An ind-scheme is an inductive limit of schemes.


Properties

Direct limits are linked to
inverse limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ...
s via :\mathrm (\varinjlim X_i, Y) = \varprojlim \mathrm (X_i, Y). An important property is that taking direct limits in the category of modules is an
exact functor In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much ...
. This means that if you start with a directed system of short exact sequences 0 \to A_i \to B_i \to C_i \to 0 and form direct limits, you obtain a short exact sequence 0 \to \varinjlim A_i \to \varinjlim B_i \to \varinjlim C_i \to 0.


Related constructions and generalizations

We note that a direct system in a category \mathcal admits an alternative description in terms of
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...
s. Any directed set \langle I,\le \rangle can be considered as a small category \mathcal whose objects are the elements I and there is a morphisms i\rightarrow j
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...
i\le j. A direct system over I is then the same as a
covariant functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
\mathcal\rightarrow \mathcal. The
colimit In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions su ...
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. Here we start with a covariant functor \mathcal J \to \mathcal C from a filtered category \mathcal J to some category \mathcal 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. Given an arbitrary category \mathcal, there may be direct systems in \mathcal that don't have a direct limit in \mathcal (consider for example the category of finite sets, or the category of finitely generated abelian groups). In this case, we can always embed \mathcal into a category \text(\mathcal) in which all direct limits exist; the objects of \text(\mathcal) are called ind-objects of \mathcal. The categorical dual of the direct limit is called the
inverse limit In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ...
. 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


Notes


References

* * {{DEFAULTSORT:Direct Limit Limits (category theory) Abstract algebra