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 manner of the gluing process being specified by morphisms between the objects. Inverse limits can be defined in any category, and they are a special case of the concept of a limit in category theory.

Formal definition

Algebraic objects

We start with the definition of an inverse system (or projective system) of groups and homomorphisms. Let ( $I$ , ≤) be a directed set or a poset (not all authors require $I$ to be directed). Let $A • = (A i) i \in I$ be a family of groups and suppose we have a family of homomorphisms $f_{ij}:A_{j}\to A_{i}$ for all $i\leq j$ (note the order), called bonding maps, with the following properties:

$f_{ii}$ is the identity on $A_{i}$ ,[note 1]
Compatibility condition: $f_{ik}=f_{ij}\circ f_{jk}$ for all $i\leq j\leq k$ ; that is, $A_{k}\xrightarrow {f_{jk}} A_{j}\xrightarrow {f_{ij}} A_{i}\;\;{\text{ is equal to }}\;\;A_{k}\xrightarrow {f_{ik}} A_{i}.$

Then the pair $\left(A_{\bullet },\left(f_{ij}\right)_{i\leq j\in I}\right)$ is called an inverse system of groups and morphisms over $I$ . The maps $f_{ij}$ are called the bonding, connecting, transition, or linking maps/morphisms of the system. If the bonding maps are understood or if there is no need to assign them symbols (e.g. as in the statements of some theorems) then the bonding maps will often be omitted (i.e. not written); for this reason it is common to see statements such as "let $A_{\bullet }$ be an inverse system."[note 2]

The system is said to be injective (resp. surjective, etc.) if this is true of all bonding maps. If $I$ is directed (resp. countable) then the system is said to be directed (resp. countable).

We define the canonical inverse limit of the inverse system $((A_{i})_{i\in I},(f_{ij})_{i\leq j\in I})$ as a particular subgroup of the direct product of the $A_{i}$ 's:

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\}.

The inverse limit $A$ comes equipped with natural projections π_i: $A$ → $A_{i}$ which pick out the ith component of the direct product for each $i$ in $I$ . 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 $A_{i}$ 's are sets,[1] semigroups,[1] topological spaces,[1] rings, modules (over a fixed ring), algebras (over a fixed ring), etc., and the homomorphisms 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. Let $Sys X • := (X •, f ij)$ be an inverse system of objects and morphisms in a category C (same definition as above). We will not write $f ij$ unless $i$ and $j$ are indices satisfying $i \leq j$ .

A collection of morphisms $ψ • = (ψ i) i \in I$ from an object $Y$ in C is said to be compatible or consistent[2] with the system $Sys X •$ if for all indices $i \leq j$ , the morphism $ψ i$ has prototype $ψ i : Y \to X i$ and the following compatibility condition is satisfied:

f ij \circ ψ j = ψ i

,

in which case the pair $(Y, ψ •)$ is called a cone from $Y$ into the system. The object $Y$ is called the vertex of the cone $(Y, ψ •)$ .

An inverse limit[2][3] of this inverse system $Sys X • := (X •, f ij)$ is a cone $(X, π •)$ into $Sys X •$ (so these morphisms must satisfy $π i = f ij \circ π j$ ) for which the following condition holds:

Universal property: If

(Y, ψ •)

is any cone into this system (so by assumption these morphisms satisfy

ψ i = f ij \circ ψ j

) then there exists a unique morphism

u : Y \to X

such that

ψ i = π i \circ u

for every index

i

(this may be abbreviated as

ψ • = π • \circ u

);

In this case, the following diagram will commute for all indices $i \leq j$ ,

This unique map $u : Y \to X$ is called the limit of the cone $(Y, ψ •)$ and it may also be denoted by $ψ \infty$ , $\varprojlim \left(Y,\psi _{\bullet }\right)$ , $\varprojlim \psi _{\bullet }$ , or $lim ψ •$ .

Said more succinctly and without indices, an inverse limit of an inverse system $Sys X •$ is a cone $(X, π •)$ into $Sys X •$ such that for any cone $(Y, ψ •)$ into this system, there exists a unique morphism $u : Y \to X$ such that $ψ • = π • \circ u$ .

The morphisms $π i : X \to X i$ are called projections from $X$ . An inverse limit is often denoted

X=\varprojlim X_{i}

with the inverse system $(X i, f ij)$ 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 $X 2$ of an inverse system, there exists a unique isomorphism $X 2 \to X$ commuting with the projection maps.

We note that an inverse system in a category C admits an alternative description in terms of functors. 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, and the inverse limit functor $\varprojlim :C^{I^{op}}\rightarrow C$ is a covariant functor.

Examples

The ring of p-adic integers is the inverse limit of the rings $\mathbb {Z} /p^{n}\mathbb {Z}$ (see modular arithmetic) with the index set being the natural numbers 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 $n_{i}\equiv n_{j}{\mbox{ mod }}p^{i}$ whenever $i<j.$ The natural topology on the p-adic integers is the one implied here, namely the product topology with cylinder sets as the open sets.
The p-adic solenoid is the inverse limit of the compact Hausdorff groups $\mathbb {R} /p^{n}\mathbb {Z}$ 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 $(n_{1},n_{2},\dots )$ such that each element of the sequence "projects" down to the previous ones, namely, that $n_{i}\equiv n_{j}{\mbox{ mod }}p^{i}$ whenever $i<j.$
The ring $\textstyle R[[t]]$ of formal power series over a commutative ring R can be thought of as the inverse limit of the rings $\textstyle R[t]/t^{n}R[t]$ , indexed by the natural numbers as usually ordered, with the morphisms from $\textstyle R[t]/t^{n+j}R[t]$ to $\textstyle R[t]/t^{n}R[t]$ given by the natural projection.
Pro-finite groups are defined as inverse limits of (discrete) finite groups.
Let the index set I of an inverse system (X_i, $f_{ij}$ ) have a greatest element m. Then the natural projection π_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 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, the limit space is totally disconnected. This is one way of realizing the p-adic numbers and the Cantor set (as infinite strings).

Smooth functions defined via limits without differentiation:

Let $Ω$ be a convex open subset of $ℝ$ and fix $a \in Ω$ . Let $M_{0}=C(\Omega )$ be the algebra of continuous $ℝ$ -valued functions on $Ω$ , and let $M_{n}=\mathbb {R} ^{n}\times C(\Omega )$ for all $n > 0$ . For each integer $n \geq 0$ define the bonding map $\mu _{n,n+1}:M_{n+1}\to M_{n}$ by

\left(p_{0},\dots ,p_{n},A\right)\mapsto \left(p_{0},\dots ,p_{n-1},\;p_{n}+(x-a){\frac {1}{n+1}}A(x)\right)

where by $p_{n}+(x-a){\frac {1}{n+1}}A(x)$ we mean the continuous function $x\mapsto p_{n}+(x-a){\frac {1}{n+1}}A(x)$ defined on $Ω$ (if $n = 0$ then $p_{0},\dots ,p_{n-1}$ is an empty list). For $0 \leq m < n$ , we define $\mu _{m,n}:M_{n}\to M_{m}$ in the usual way (e.g. $\mu _{2,5}:=\mu _{2,3}\circ \mu _{3,4}\circ \mu _{4,5}$ , etc.). Using Taylor's theorem, it is easily seen that each $\mu _{n,n+1}$ is injective so consequently, $\cap _{n=1}^{\infty }\operatorname {im} \mu _{0,n}$ is a limit of this system.

We now identify this limit as the set of smooth functions. Suppose $f \in C(Ω)$ . Then $f \in im 𝜇 0,1$ if and only if $f = p 0 + (x - a) A 1 (x)$ for some real $p 0$ and some $A 1 \in C(Ω)$ , which by Taylor's theorem happens if and only if $f$ is continuously differentiable. By induction, $f \in im 𝜇 0, n$ if and only if

f=p_{0}+p_{1}(x-a)+{\frac {p_{2}}{2!}}(x-a)^{2}+\cdots +{\frac {p_{n-1}}{(n-1)!}}(x-a)^{n-1}+{\frac {(x-a)^{n}}{n!}}A_{n}(x)

for some real $p 0, ..., p n -1$ and some $A n \in C(Ω)$ , which by Taylor's theorem is true if and only if $f \in C n (Ω)$ (in this case, $p i$ is the $i th$ derivative of $f$ at $a$ ). Thus the limit of the above system is $\cap _{n=1}^{\infty }\operatorname {im} \mu _{0,n}=\cap _{n=1}^{\infty }C^{n}(\Omega )=C^{\infty }(\Omega )$ . Note that this construction of the smooth functions on $Ω$ does not use (or even need) the definition of a derivative (Taylor's theorem was only used to identify the resulting limit as the set of smooth functions on $Ω$ and to prove that the bonding maps were injective; it was not used in the definition of the inverse system nor in the definition of the limit $\cap _{n=1}^{\infty }\operatorname {im} \mu _{0,n}$ ).

ℝ k

k < \infty

Derived functors of the inverse limit

For an abelian category C, the inverse limit functor

\varprojlim :C^{I}\rightarrow C

is 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 $\varprojlim$ . Specifically, Eilenberg constructed a functor

\varprojlim {}^{1}:\operatorname {Ab} ^{I}\rightarrow \operatorname {Ab}

(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}\rightarrow 0

is a short exact sequence of inverse systems, then

0\rightarrow \varprojlim A_{i}\rightarrow \varprojlim B_{i}\rightarrow \varprojlim C_{i}\rightarrow \varprojlim {}^{1}A_{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 : $f_{kj}(A_{j})=f_{ki}(A_{i})$ 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.

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[4]^{pg 83} where $\varprojlim {}^{1}$ is non-zero is obtained by taking I to be the non-negative integers, letting A_i = pⁱZ, B_i = Z, and C_i = B_i / A_i = Z/pⁱZ. Then

\varprojlim {}^{1}A_{i}=\mathbf {Z} _{p}/\mathbf {Z}

where Z_p denotes the p-adic integers.

Further results

More generally, if C is an arbitrary abelian category that has enough injectives, then so does C^I, and the right derived functors of the inverse limit functor can thus be defined. The nth 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 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 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 $\aleph _{d}$ (the dth infinite cardinal), then Rⁿlim 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 limⁿ, on diagrams indexed by a countable set, is nonzero for n > 1).

Related concepts and generalizations

The categorical dual of an inverse limit is a direct limit (or inductive limit). More general concepts are the 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

Some authors (e.g. Dugundji) do note require that $f_{ii}$ be the identity map. The definition of the inverse limit of such a generalized system is identical to the definition given below.
This is abuse of notation and terminology since calling $A_{\bullet }$ an inverse system is technically incorrect.

References

John Rhodes & Benjamin Steinberg. The q-theory of Finite Semigroups. p. 133. ISBN 978-0-387-09780-0.
Mac Lane 1998, pp. 68-69.
Dugundji 1966, pp. 427-435.
Dugger, Daniel. "A Primer on Homotopy Colimits" (PDF). Archived (PDF) from the original on 3 Dec 2020.

Bibliography

Bierstedt, Klaus-Dieter (1988). An Introduction to Locally Convex Inductive Limits. Functional Analysis and Applications. Singapore-New Jersey-Hong Kong: Universitätsbibliothek. pp. 35–133. MR 0046004. Retrieved 20 September 2020.
Bourbaki, Nicolas (1989), Algebra I, Springer, ISBN 978-3-540-64243-5, OCLC 40551484
Bourbaki, Nicolas (1989) [1966]. General Topology: Chapters 1–4 [Topologie Générale]. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129.
Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485.
Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098.
Mac Lane, Saunders (September 1998), Categories for the Working Mathematician (2nd ed.), Springer, ISBN 0-387-98403-8
Mitchell, Barry (1972), "Rings with several objects", Advances in Mathematics, 8: 1–161, doi:10.1016/0001-8708(72)90002-3, MR 0294454
Neeman, Amnon (2002), "A counterexample to a 1961 "theorem" in homological algebra (with appendix by Pierre Deligne)", Inventiones Mathematicae, 148 (2): 397–420, doi:10.1007/s002220100197, MR 1906154, S2CID 121186299
Roos, Jan-Erik (1961), "Sur les foncteurs dérivés de lim. Applications", C. R. Acad. Sci. Paris, 252: 3702–3704, MR 0132091
Roos, Jan-Erik (2006), "Derived functors of inverse limits revisited", J. London Math. Soc., Series 2, 73 (1): 65–83, doi:10.1112/S0024610705022416, MR 2197371
Section 3.5 of Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Some authors (e.g. Dugundji) do note require that $f_{ii}$ be the identity map. The definition of the inverse limit of such a generalized system is identical to the definition given below.

[2] This is abuse of notation and terminology since calling $A_{\bullet }$ an inverse system is technically incorrect.

[same-construction-3] John Rhodes & Benjamin Steinberg. The q-theory of Finite Semigroups. p. 133. ISBN 978-0-387-09780-0.

[FOOTNOTEMac_Lane199868-69-4] Mac Lane 1998, pp. 68-69.

[FOOTNOTEDugundji1966427-435-5] Dugundji 1966, pp. 427-435.

[6] Dugger, Daniel. "A Primer on Homotopy Colimits" (PDF). Archived (PDF) from the original on 3 Dec 2020.