Inverse limit

2025-07-22T22:10:27Z

146.155.158.152: fixed typo

{{Short description|Construction in category theory}}
In [[mathematics]], the '''inverse limit''' (also called the '''projective limit''') is a construction that allows one to "glue together" several related [[mathematical object|objects]], the precise gluing process being specified by [[morphisms]] between the objects. Thus, inverse limits can be defined in any [[category (mathematics)|category]] although their existence depends on the category that is considered. They are a special case of the concept of [[Limit (category theory)|limit]] in category theory.

By working in the [[dual category]], that is by reversing the arrows, an inverse limit becomes a [[direct limit]] 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 [[group (mathematics)|groups]] and [[group homomorphism|homomorphisms]]. Let <math>(I, \leq)</math> be a [[directed set|directed]] [[poset]] (not all authors require ''I'' to be directed). Let (''A''''i'')''i''∈''I'' be a [[indexed family|family]] of groups and suppose we have a family of homomorphisms <math>f_{ij}: A_j \to A_i</math> for all <math>i \leq j</math> (note the order) with the following properties:
# <math>f_{ii}</math> is the identity on <math>A_i</math>,
# <math>f_{ik} = f_{ij} \circ f_{jk} \quad \text{for all } i \leq j \leq k.</math>
Then the pair <math>((A_i)_{i\in I}, (f_{ij})_{i\leq j\in I})</math> is called an ''inverse system'' of groups and morphisms over <math>I</math>, and the morphisms <math>f_{ij}</math> are called the transition morphisms of the system.

The '''inverse limit''' of the inverse system <math>((A_i)_{i\in I}, (f_{ij})_{i\leq j\in I})</math> is the [[subgroup]] of the [[direct product]] of the {{tmath|A_i}}'s defined as
:<math>A = \varprojlim_{i\in I}{A_i} = \left\{\left.\vec a \in \prod_{i\in I}A_i \;\right|\; a_i = f_{ij}(a_j) \text{ for all } i \leq j \text{ in } I\right\}.</math>

The definition above of an inverse system implies, that <math>A</math> is closed under pointwise multiplication, and therefore a group, since
:<math>f_{i,j}(a_j\cdot b_j)=f_{i,j}(a_j)\cdot f_{i,j}(b_j)=a_i\cdot b_i</math>
for all {{tmath|i<j}} and every {{tmath|\vec a, \vec b\in A}}

The inverse limit <math>A</math> comes equipped with ''natural projections'' {{math|{{pi}}''i'': ''A'' → ''A''''i''}} which pick out the {{math|''i''}}th component of the direct product for each <math>i</math> in <math>I</math>. The inverse limit and the natural projections satisfy a [[universal property]] described in the next section.

This same construction may be carried out if the <math>A_i</math>'s are [[Set (mathematics)|sets]], [[semigroup]]s, [[topological space]]s, [[ring (mathematics)|rings]], [[module (mathematics)|modules]] (over a fixed ring), [[algebra over a field|algebras]] (over a fixed ring), etc., and the [[homomorphism]]s are morphisms in the corresponding [[category theory|category]]. The inverse limit will also belong to that category.<ref name="same-construction">John Rhodes & Benjamin Steinberg. The q-theory of Finite Semigroups. p. 133. {{ISBN|978-0-387-09780-0}}.</ref> More generally, this construction applies when the {{tmath|A_i}} belong to a [[variety (universal algebra)|variety]] in the sense of [[universal algebra]], that is, a type of algebraic structures, whose axioms are unconditional ([[field (mathematics)|field]]s do not form an algebra, since zero does not have a [[multiplicative inverse]]).

=== General definition ===

The inverse limit can be defined abstractly in an arbitrary [[category (mathematics)|category]] by means of a [[universal property]]. Let <math display=inline> (X_i, f_{ij})</math> be an inverse system of objects and [[morphism]]s in a category ''C'' (same definition as above). The '''inverse limit''' of this system is an object ''X'' in ''C'' together with morphisms {{pi}}''i'': ''X'' → ''X''''i'' (called ''projections'') satisfying {{pi}}''i'' = <math>f_{ij}</math> ∘ {{pi}}''j'' for all ''i'' ≤ ''j''. The pair (''X'', {{pi}}''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

<div style="text-align: center;">[[File:InverseLimit-01.svg|175px|class=skin-invert]]</div>

[[commutative diagram|commutes]] for all ''i'' ≤ ''j''. The inverse limit is often denoted
:<math>X = \varprojlim X_i</math>
with the inverse system <math display=inline>(X_i, f_{ij})</math> and the canonical projections <math>\pi_i</math> 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]] ''X''′ → ''X'' commuting with the projection maps.

Inverse systems and inverse limits in a category ''C'' admit an alternative description in terms of [[functor]]s. Any partially ordered set ''I'' can be considered as a [[small category]] where the morphisms consist of arrows ''i'' → ''j'' [[if and only if]] ''i'' ≤ ''j''. An inverse system is then just a [[contravariant functor]] ''I'' → ''C''. Let <math>C^{I^\mathrm{op}}</math> be the category of these functors (with [[natural transformation]]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 <math>C^{I^\mathrm{op}}.</math> The inverse limit, if it exists, is defined as a [[right adjoint]] of this trivial functor.

== Examples ==
* The ring of [[p-adic number|''p''-adic integers]] is the inverse limit of the rings <math>\mathbb{Z}/p^n\mathbb{Z}</math> (see [[modular arithmetic]]) with the index set being the [[natural number]]s with the usual order, and the morphisms being "take remainder". That is, one considers sequences of integers <math>(n_1, n_2, \dots)</math> such that each element of the sequence "projects" down to the previous ones, namely, that <math>n_i\equiv n_j \mbox{ mod } p^{i}</math> whenever <math>i<j.</math> The natural topology on the ''p''-adic integers is the one implied here, namely the [[product topology]] with [[cylinder set]]s as the open sets.
* The [[Solenoid (mathematics)|''p''-adic solenoid]] is the inverse limit of the topological groups <math>\mathbb{R}/p^n\mathbb{Z}</math> 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 <math>(x_1, x_2, \dots)</math> such that each element of the sequence "projects" down to the previous ones, namely, that <math>x_i\equiv x_j \mbox{ mod } p^{i}</math> whenever <math>i<j.</math> Its elements are exactly of form <math>n + r</math>, where <math>n</math> is a ''p''-adic integer, and <math>r\in [0, 1)</math> is the "remainder".
* The ring <math>\textstyle R[[t]]</math> of [[formal power series]] over a commutative ring ''R'' can be thought of as the inverse limit of the rings <math>\textstyle R[t]/t^nR[t]</math>, indexed by the natural numbers as usually ordered, with the morphisms from <math>\textstyle R[t]/t^{n+j}R[t]</math> to <math>\textstyle R[t]/t^nR[t]</math> given by the natural projection.
* [[Pro-finite group]]s are defined as inverse limits of (discrete) finite groups.
* Let the index set ''I'' of an inverse system (''X''''i'', <math>f_{ij}</math>) have a [[greatest element]] ''m''. Then the natural projection {{pi}}''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]] in graph theory and may be proved with [[Tychonoff's theorem]], viewing the finite sets as compact discrete spaces, and then applying the [[finite intersection property]] characterization of compactness.
* In the [[category of topological spaces]], every inverse system has an inverse limit. It is constructed by placing the [[initial topology]] on the underlying set-theoretic inverse limit. This is known as the '''limit topology'''.
** The set of infinite [[String (computer science)|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 topology|discrete]], the limit space is [[totally disconnected]]. This is one way of realizing the [[p-adic|''p''-adic numbers]] and the [[Cantor set]] (as infinite strings).

==Derived functors of the inverse limit==

For an [[abelian category]] ''C'', the inverse limit functor
:<math>\varprojlim:C^I\rightarrow C</math>
is [[Exact functor|left exact]]. If ''I'' is ordered (not simply partially ordered) and [[countable]], 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 <math>\varprojlim</math>. Specifically, [[Samuel Eilenberg|Eilenberg]] constructed a functor
:<math>\varprojlim{}^1:\operatorname{Ab}^I\rightarrow\operatorname{Ab}</math>
(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
:<math>0\rightarrow A_i\rightarrow B_i\rightarrow C_i\rightarrow0</math>
is a [[short exact sequence]] of inverse systems, then
:<math>0\rightarrow\varprojlim A_i\rightarrow\varprojlim B_i\rightarrow\varprojlim C_i\rightarrow\varprojlim{}^1A_i</math>
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'' :<math> f_{kj}(A_j)=f_{ki}(A_i)</math> 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]].

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 space]]s or finite abelian groups or modules of finite [[length of a module|length]] or [[Artinian module]]s.

An example where <math>\varprojlim{}^1</math> is non-zero is obtained by taking ''I'' to be the non-negative [[integer]]s, letting ''A''''i'' = ''p''''i'''''Z''', ''B''''i'' = '''Z''', and ''C''''i'' = ''B''''i'' / ''A''''i'' = '''Z'''/''p''''i'''''Z'''. Then
:<math>\varprojlim{}^1A_i=\mathbf{Z}_p/\mathbf{Z}</math>
where '''Z'''''p'' denotes the [[p-adic integers]].

===Further results===

More generally, if ''C'' is an arbitrary abelian category that has [[Injective object#Enough injectives|enough injectives]], then so does ''C''''I'', and the right [[derived functor]]s of the inverse limit functor can thus be defined. The ''n''th right derived functor is denoted
:<math>R^n\varprojlim:C^I\rightarrow C.</math>
In the case where ''C'' satisfies [[Grothendieck]]'s axiom [[Abelian category#Grothendieck's axioms|(AB4*)]], [[Jan-Erik Roos]] generalized the functor lim1 on '''Ab'''''I'' to series of functors limn such that
:<math>\varprojlim{}^n\cong R^n\varprojlim.</math>
It was thought for almost 40 years that Roos had proved (in {{lang|fr|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]] 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 (mathematician)|Barry Mitchell]] has shown (in "The cohomological dimension of a directed set") that if ''I'' has [[cardinality]] <math>\aleph_d</math> (the ''d''th [[Aleph number|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 [[dual (category theory)|categorical dual]] of an inverse limit is a [[direct limit]] (or inductive limit). More general concepts are the [[limit (category theory)|limits and colimits]] 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 />

==References==
*{{citation|first=Nicolas|last=Bourbaki|authorlink=Nicolas Bourbaki|title=Algebra I|publisher=Springer|year=1989|isbn=978-3-540-64243-5|oclc=40551484}}
*{{citation|first=Nicolas|last=Bourbaki|authorlink=Nicolas Bourbaki|title=General topology: Chapters 1-4|publisher=Springer|year=1989|isbn=978-3-540-64241-1|oclc=40551485}}
*{{citation|first=Saunders |last=Mac Lane |authorlink=Saunders Mac Lane|title=[[Categories for the Working Mathematician]] | edition=2nd |date=September 1998 |publisher=Springer|isbn=0-387-98403-8}}
*{{Citation | last=Mitchell | first=Barry | author-link=Barry Mitchell (mathematician) | title=Rings with several objects | journal=[[Advances in Mathematics]] | mr=0294454 | year=1972 | volume=8 | pages=1–161 | doi=10.1016/0001-8708(72)90002-3| doi-access=free }}
*{{Citation | last=Neeman | first=Amnon | author-link=Amnon Neeman | title=A counterexample to a 1961 "theorem" in homological algebra (with appendix by Pierre Deligne) | journal=[[Inventiones Mathematicae]] | mr=1906154 | year=2002 | volume=148 | issue=2 | pages=397–420 | doi=10.1007/s002220100197 | doi-access=}}
*{{Citation | last=Roos | first=Jan-Erik | author-link=Jan-Erik Roos | title=Sur les foncteurs dérivés de lim. Applications | journal=C. R. Acad. Sci. Paris | mr=0132091 | year=1961 | volume=252 | pages=3702–3704}}
*{{Citation | last=Roos | first=Jan-Erik | author-link=Jan-Erik Roos | title=Derived functors of inverse limits revisited | journal=[[London Mathematical Society|J. London Math. Soc.]] |series=Series 2 | mr=2197371 | year=2006 | volume=73 | issue=1 | pages=65–83 | doi=10.1112/S0024610705022416}}
* Section 3.5 of {{Weibel IHA}}

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Inverse limit