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
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, morphism ...
between the objects. Thus, inverse limits can be defined 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) ...
although their existence depends on the category that is considered. They are a special case of the concept of
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
in category theory.
By working in the
dual category
In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields ...
, that is by reverting the arrows, an inverse limit becomes a
direct limit
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 cat ...
or ''inductive limit'', and a ''limit'' becomes a
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 ...
.
Formal definition
Algebraic objects
We start with the definition of an inverse system (or projective system) of
groups and
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" ...
. Let
be a
directed
Director may refer to:
Literature
* ''Director'' (magazine), a British magazine
* ''The Director'' (novel), a 1971 novel by Henry Denker
* ''The Director'' (play), a 2000 play by Nancy Hasty
Music
* Director (band), an Irish rock band
* ''D ...
poset
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
(not all authors require ''I'' to be directed). Let (''A''
''i'')
''i''∈''I'' be a
family
Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Idea ...
of groups and suppose we have a family of homomorphisms
for all
(note the order) with the following properties:
#
is the identity on
,
#
Then the pair
is called an inverse system of groups and morphisms over
, and the morphisms
are called the transition morphisms of the system.
We define the inverse limit of the inverse system
as a particular
subgroup
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 subgrou ...
of the
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
of the ''
''
's:
:
The inverse limit comes equipped with ''natural projections'' which pick out the th component of the direct product for each in . The inverse limit and the natural projections satisfy 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 ...
described in the next section.
This same construction may be carried out if the 's are sets,[John Rhodes & Benjamin Steinberg. The q-theory of Finite Semigroups. p. 133. .] semigroups,[ topological spaces,][ ]rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
, 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 ring), etc., and the 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 morphisms in the corresponding 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 inverse limit will also belong to that category.
General definition
The inverse limit can be defined abstractly 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) ...
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 be an inverse system of objects and 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 a category ''C'' (same definition as above). The inverse limit of this system is an object ''X'' in ''C'' together with morphisms ''i'': ''X'' → ''X''''i'' (called ''projections'') satisfying ''i'' = ∘ ''j'' for all ''i'' ≤ ''j''. The pair (''X'', ''i'') must be universal in the sense that for any other such pair (''Y'', ψ''i'') there exists a unique morphism ''u'': ''Y'' → ''X'' such that the diagram
commutes for all ''i'' ≤ ''j''. The inverse limit is often denoted
:
with the inverse system being understood.
In some categories, the inverse limit of certain inverse systems does not exist. If it does, however, it is unique in a strong sense: given any two inverse limits ''X'' and ''X of an inverse system, 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'' commuting with the projection maps.
Inverse systems and inverse limits in a category ''C'' admit 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 partially ordered set ''I'' 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 as ...
where the morphisms consist of arrows ''i'' → ''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'' ≤ ''j''. An inverse system is then just a contravariant 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 ...
''I'' → ''C''. Let be the category of these functors (with natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a na ...
s as morphisms). An object ''X'' of ''C'' can be considered a trivial inverse system, where all objects are equal to ''X'' and all arrow are the identity of ''X''. This defines a "trivial functor" from ''C'' to The direct limit, if it exists, is defined as a right adjoint
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kn ...
of this trivial functor.
Examples
* The ring of ''p''-adic integers is the inverse limit of the rings (see modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his bo ...
) with the index set being the natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s with the usual order, and the morphisms being "take remainder". That is, one considers sequences of integers such that each element of the sequence "projects" down to the previous ones, namely, that whenever product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...
with cylinder set In mathematics, the cylinder sets form a basis of the product topology on a product of sets; they are also a generating family of the cylinder σ-algebra.
General definition
Given a collection S of sets, consider the Cartesian product X = \prod_ ...
s as the open sets.
* The ''p''-adic solenoid is the inverse limit of the topological groups with the index set being the natural numbers with the usual order, and the morphisms being "take remainder". That is, one considers sequences of real numbers such that each element of the sequence "projects" down to the previous ones, namely, that whenever is the "remainder".
* The ring \textstyle Rt">, 1) is the "remainder".
* The ring \textstyle Rt of formal power series over a commutative ring ''R'' can be thought of as the inverse limit of the rings \textstyle R[t]/t^nR[t], indexed by the natural numbers as usually ordered, with the morphisms from \textstyle R[t]/t^R[t] to \textstyle R[t]/t^nR[t] given by the natural projection. In particular, when R = \Z/p\Z, this gives the ring of p-adic integers.
* Pro-finite group In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups.
The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups ...
s are defined as inverse limits of (discrete) finite groups.
* Let the index set ''I'' of an inverse system (''X''''i'', f_) have 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''. Then the natural projection ''m'': ''X'' → ''X''''m'' is an isomorphism.
* In the category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition ...
, every inverse system has an inverse limit, which can be constructed in an elementary manner as a subset of the product of the sets forming the inverse system. The inverse limit of any inverse system of non-empty finite sets is non-empty. This is a generalization of Kőnig's lemma
Kőnig's lemma or Kőnig's infinity lemma is a theorem in graph theory due to the Hungarian mathematician Dénes Kőnig who published it in 1927. It gives a sufficient condition for an infinite graph to have an infinitely long path. The computab ...
in graph theory and may be proved with Tychonoff's theorem
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is t ...
, viewing the finite sets as compact discrete spaces, and then applying the finite intersection property In general topology, a branch of mathematics, a non-empty family ''A'' of subsets of a set X is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of A is non-empty. It has the strong finite inte ...
characterization of compactness.
* 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 ...
, every inverse system has an inverse limit. It is constructed by placing the initial topology
In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' ...
on the underlying set-theoretic inverse limit. This is known as the limit topology.
** The set of infinite strings is the inverse limit of the set of finite strings, and is thus endowed with the limit topology. As the original spaces are discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a ...
, the limit space is totally disconnected
In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set ...
. This is one way of realizing the ''p''-adic numbers and the Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.
T ...
(as infinite strings).
Derived functors of the inverse limit
For an abelian category
In 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 prototypical example of an abelian category is the category of ...
''C'', the inverse limit functor
:\varprojlim:C^I\rightarrow C
is left exact. If ''I'' is ordered (not simply partially ordered) and countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
, and ''C'' is the category Ab of abelian groups, the Mittag-Leffler condition is a condition on the transition morphisms ''f''''ij'' that ensures the exactness of \varprojlim. Specifically, Eilenberg Eilenberg is a surname, and may refer to:
* Samuel Eilenberg (1913–1998), Polish mathematician
* Richard Eilenberg (1848–1927), German composer
Named after Samuel
* Eilenberg–MacLane space
* Eilenberg–Moore algebra
* Eilenberg–Steenro ...
constructed a functor
:\varprojlim^1:\operatorname^I\rightarrow\operatorname
(pronounced "lim one") such that if (''A''''i'', ''f''''ij''), (''B''''i'', ''g''''ij''), and (''C''''i'', ''h''''ij'') are three inverse systems of abelian groups, and
:0\rightarrow A_i\rightarrow B_i\rightarrow C_i\rightarrow0
is a short exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the conte ...
of inverse systems, then
:0\rightarrow\varprojlim A_i\rightarrow\varprojlim B_i\rightarrow\varprojlim C_i\rightarrow\varprojlim^1A_i
is an exact sequence in Ab.
Mittag-Leffler condition
If the ranges of the morphisms of an inverse system of abelian groups (''A''''i'', ''f''''ij'') are ''stationary'', that is, for every ''k'' there exists ''j'' ≥ ''k'' such that for all ''i'' ≥ ''j'' : f_(A_j)=f_(A_i) one says that the system satisfies the Mittag-Leffler condition.
The name "Mittag-Leffler" for this condition was given by Bourbaki in their chapter on uniform structures for a similar result about inverse limits of complete Hausdorff uniform spaces. Mittag-Leffler used a similar argument in the proof of Mittag-Leffler's theorem In complex analysis, Mittag-Leffler's theorem concerns the existence of meromorphic functions with prescribed poles. Conversely, it can be used to express any meromorphic function as a sum of partial fractions. It is sister to the Weierstrass facto ...
.
The following situations are examples where the Mittag-Leffler condition is satisfied:
* a system in which the morphisms ''f''''ij'' are surjective
* a system of finite-dimensional vector spaces or finite abelian groups or modules of finite length or Artinian modules.
An example where \varprojlim^1 is non-zero is obtained by taking ''I'' to be the non-negative integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s, letting ''A''''i'' = ''p''''i''Z, ''B''''i'' = Z, and ''C''''i'' = ''B''''i'' / ''A''''i'' = Z/''p''''i''Z. Then
:\varprojlim^1A_i=\mathbf_p/\mathbf
where Z''p'' denotes the p-adic integers
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extens ...
.
Further results
More generally, if ''C'' is an arbitrary abelian category that has enough injectives, then so does ''C''''I'', and the right derived functor
In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.
Motivation
It was noted in var ...
s of the inverse limit functor can thus be defined. The ''n''th right derived functor is denoted
:R^n\varprojlim:C^I\rightarrow C.
In the case where ''C'' satisfies Grothendieck's axiom (AB4*), Jan-Erik Roos
Jan-Erik Ingvar Roos (October 16, 1935 – December 15, 2017) was a Swedish mathematician.
He was born in Halmstad, in the province of Halland on the Swedish west coast. Roos enrolled at Lund University in 1954, and started studying mathematics ...
generalized the functor lim1 on Ab''I'' to series of functors limn such that
:\varprojlim^n\cong R^n\varprojlim.
It was thought for almost 40 years that Roos had proved (in ''Sur les foncteurs dérivés de lim. Applications. '') that lim1 ''A''''i'' = 0 for (''A''''i'', ''f''''ij'') an inverse system with surjective transition morphisms and ''I'' the set of non-negative integers (such inverse systems are often called " Mittag-Leffler sequences"). However, in 2002, Amnon Neeman and Pierre Deligne
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...
constructed an example of such a system in a category satisfying (AB4) (in addition to (AB4*)) with lim1 ''A''''i'' ≠ 0. Roos has since shown (in "Derived functors of inverse limits revisited") that his result is correct if ''C'' has a set of generators (in addition to satisfying (AB3) and (AB4*)).
Barry Mitchell has shown (in "The cohomological dimension of a directed set") that if ''I'' has cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
\aleph_d (the ''d''th infinite cardinal
In mathematics, transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to qua ...
), then ''R''''n''lim is zero for all ''n'' ≥ ''d'' + 2. This applies to the ''I''-indexed diagrams in the category of ''R''-modules, with ''R'' a commutative ring; it is not necessarily true in an arbitrary abelian category (see Roos' "Derived functors of inverse limits revisited" for examples of abelian categories in which lim''n'', on diagrams indexed by a countable set, is nonzero for ''n'' > 1).
Related concepts and generalizations
The categorical dual
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category ''C'' and the dual properties of the opposite category ''C''op. Given a statement regarding the category ''C'', by interchanging the ...
of an inverse limit is a direct limit
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 cat ...
(or inductive limit). More general concepts are the limits and colimits
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 s ...
of category theory. The terminology is somewhat confusing: inverse limits are a class of limits, while direct limits are a class of colimits.
Notes
References
*
*
*
*
*
*
*
* Section 3.5 of
{{Category theory
Limits (category theory)
Abstract algebra
de:Limes (Kategorientheorie)