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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, morphisms ...
between the objects. Thus, inverse limits can be defined in any category although their existence depends on the category that is considered. They are a special case of the concept of limit 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 t ...
, 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 categ ...
or ''inductive limit'', and a ''limit'' becomes a colimit.


Formal definition


Algebraic objects

We start with the definition of an inverse system (or projective system) of groups and homomorphisms. Let (I, \leq) 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 * ''Di ...
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 ...
(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. Ideal ...
of groups and suppose we have a family of homomorphisms f_: A_j \to A_i for all i \leq j (note the order) with the following properties: # f_ is the identity on A_i, # f_ = f_ \circ f_ \quad \text i \leq j \leq k. Then the pair ((A_i)_, (f_)_) is called an inverse system of groups and morphisms over I, and the morphisms f_ are called the transition morphisms of the system. We define the inverse limit of the inverse system ((A_i)_, (f_)_) 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 ''A_i'''s: :A = \varprojlim_ = \left\. The inverse limit A comes equipped with ''natural projections'' which pick out the th component of the direct product for each i in I. 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 fr ...
described in the next section. This same construction may be carried out if the A_i'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 an ...
,
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
(over a fixed ring), algebras (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. The inverse limit will also belong to that category.


General definition

The inverse limit can be defined abstractly in an arbitrary category 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 fr ...
. Let (X_i, f_) 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, morphisms ...
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'' = f_''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 :X = \varprojlim X_i with the inverse system (X_i, f_) 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 is ...
''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, and ma ...
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 asso ...
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 bicond ...
''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 C^ be the category of these functors (with natural transformations 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 C^. 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 kno ...
of this trivial functor.


Examples

* The ring of ''p''-adic integers is the inverse limit of the rings \mathbb/p^n\mathbb (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 (n_1, n_2, \dots) such that each element of the sequence "projects" down to the previous ones, namely, that n_i\equiv n_j \mbox p^ whenever i The natural topology on the ''p''-adic integers is the one implied here, namely the
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-seemi ...
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 \mathbb/p^n\mathbb 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 (x_1, x_2, \dots) such that each element of the sequence "projects" down to the previous ones, namely, that x_i\equiv x_j \mbox p^ whenever i Its elements are exactly of form n + r, where n is a p-adic integer, and r\in \textstyle_R \textstyle_Rt">,_1)_is_the_"remainder". *_The_ring_\textstyle_Rt_of_formal_power_series.html" ;"title="t.html" ;"title=", 1) is the "remainder". * The ring \textstyle Rt">, 1) is the "remainder". * The ring \textstyle Rt of formal power series">t.html" ;"title=", 1) 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 groups 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 elem ...
''m''. Then the natural projection ''m'': ''X'' → ''X''''m'' is an isomorphism. * In the category of sets, 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 computa ...
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 trans ...
, 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 inter ...
characterization of compactness. * In the category of topological spaces, 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'' t ...
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 g ...
, 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 (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 a ...
''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 numbe ...
, 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 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 contex ...
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 languag ...
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 extension ...
.


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 vari ...
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 w ...
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), 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 s ...
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 categ ...
(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 suc ...
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)