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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
. The way they are put together is specified by a system of homomorphisms (
group homomorphism In mathematics, given two groups, (''G'',∗) and (''H'', ·), a group homomorphism from (''G'',∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) whe ...
,
ring homomorphism In mathematics, 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 that preserves addition, multiplication and multiplicative identity ...
, or in general
morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
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 preordered set in which every finite subset has an upper bound. In other words, it is a non-empty preordered set A such that for any a and b in A there exists c in A wit ...
I, is denoted by \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 in
category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
. Direct limits are dual to inverse limits, which are a special case of limits in category theory.


Formal definition

We will first give the definition for
algebraic structures In mathematics, an algebraic structure or algebraic system 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 multiplicatio ...
like groups and modules, and then the general definition, which can be used in any
category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
.


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 or algebraic system 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 multiplicatio ...
, such as groups, rings, modules (over a fixed ring), algebras (over a fixed field), etc. With this in mind, ''
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
s'' are understood in the corresponding setting (
group homomorphism In mathematics, given two groups, (''G'',∗) and (''H'', ·), a group homomorphism from (''G'',∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) whe ...
s, 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 preordered set in which every finite subset has an upper bound. In other words, it is a non-empty preordered set A such that for any a and b in A there exists c in A wit ...
. 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 on 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, the disjoint union (or discriminated union) A \sqcup B of the sets and is the set formed from the elements of and labelled (indexed) with the name of the set from which they come. So, an element belonging to both and appe ...
of the A_i's
modulo In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation. Given two positive numbers and , mo ...
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 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 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: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
\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 fro ...
. 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 (or, more precisely, canonical injections \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 In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 the ...
''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 of a given group, or a directed collection of subrings of a given ring, etc. *The weak topology of a
CW complex In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...
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 duality (order theory), dually ...
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 n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
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. *Let ''p'' be a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...
. 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 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 in number theory, the theory of group char ...
of order some power of p, and is called the Prüfer group \mathbb(p^\infty). *There is a (non-obvious) injective ring homomorphism from the ring of symmetric polynomials 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 Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
''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'' 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 on the underlying set-theoretic direct limit. *An ind-scheme is an inductive limit of schemes.


Properties

Direct limits are linked to inverse limits 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. This means that for any directed system of
short exact sequence In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ...
s 0 \to A_i \to B_i \to C_i \to 0, the sequence 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 \mathcal admits an alternative description in terms of
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
s. Any directed set \langle I,\le \rangle can be considered as a
small category In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows asso ...
\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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
i\le j. A direct system over I is then the same as a covariant functor \mathcal\rightarrow \mathcal. The colimit 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. 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