Wikipedia

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

Inverse systems

An inverse limit is intimately tied to an associated a tuple called an inverse system. Inverse limits depend on the category in which they are taken. In many concrete categories such as the category of topological spaces or the category of groups, a part of any inverse limit consists of an inverse limit in the category of sets, where the canonical inverse limit in the category of sets provides an explicit example.

An inverse system in a category over is a tuple

which may also be written as if is understood, that has all of the following properties:

  1. The indexing set is a preordered set where many authors also require that be a directed set and/or a partially ordered set.
    • Some authors, such as Bourbaki,[1] require that be a partially ordered set but not necessarily a directed set while others, such as Dugundji,[2] require that be a directed set but not necessarily a partially ordered set.
    • If is directed (resp. partially ordered, countable) then the system is said to be directed (resp. partially ordered, countable). Elements of are called indices of the system and the relation is identified with the set
  2. is a family of objects, meaning that is an object in the category for every . For example, if is the category of sets (resp. topological spaces, groups) then "object" means set (resp. topological space, group), etc.
  3. is a morphism in the category for every that satisfy For example, if is the category of sets (resp. topological spaces, groups) then "morphism" means function (resp. continuous function, group homomorphism), etc. These morphism are called the bonding, connecting, transition, or linking maps/morphisms of the system.
    • Whenever is written then unless indicated otherwise, it should be assumed that and are indices satisfying
    • The system is said to be monomorphic (resp. epimorphic, injective, surjective, etc.) if this is true of all bonding morphisms.
  4. The following compatibility condition of inverse systems holds:
    [note 1] for all indices ;
    that is,

and usually, but not always,[note 2] inverse systems are also required to satisfy the following additional condition:

  1. is the identity morphism on for every

If the bonding morphisms are understood or if there is no need to assign them symbols (e.g. as in the statements of some theorems) then the bonding morphisms will often be omitted (i.e. not written); for this reason it is common to see statements such as "let be an inverse system."[note 3]

The term "projective system" is sometimes used as a synonym for "inverse system" although some author use the term "projective system" to refer to a specific type of inverse system. In particular, some authors use projective system to refer to inverse systems directed by the natural numbers , a surjective/epimorphic inverse system, and/or an inverse system whose bonding morphisms are all projections (assuming that morphisms called "projections" are defined in the category as they are in the category of topological vector spaces for example). In this case, projective limit then means an inverse limit of a projective system.

Relationship with direct systems and notations

Given a preordered set its converse or transpose is the preordered set where by definition, ; that is, for all declare that holds if and only if holds. A tuple is an inverse system if and only if its transpose or opposite, which is the tuple is a direct system. This characterization may be used to define direct systems in terms of inverse systems, or to define inverse systems in terms of direct systems.

To illustrate how inverse systems differ from direct systems, the reason why is a direct system whenever is an inverse system is now explained in detail. One of the most prominent features distinguishing a direct system from an inverse system is that in an inverse system like if then the morphisms are of the form where the index of the bonding morphism's codomain is smaller (i.e. less than or equal to) with respect to than the index of its domain whereas in a direct system, the codomain of a morphism would instead have a larger index, which is true of with respect to (because implies so is larger than with respect to ) although it is in general not true of with respect to (which is why in general, is not also a direct system). Similarly, by its very definition, the compatibility condition of direct systems is satisfied by if and only if holds for all indices satisfying (or equivalently ), which can also be expressed by stating that the composition:

where as before, the index of the codomain is larger (with respect to ) than the index of the domain. But this compatibility condition of direct systems applied to (i.e. the equality for ) is exactly the same compatibility condition that the inverse system is required to satisfy (i.e. the equality for ).

For the inverse system and for any indices this article denotes the morphism by but some authors may instead denote this same morphism by (with the positions of and swapped) while others may denote it by or While the notation used to denote an inverse system's morphisms may vary, what does not vary is that in an inverse system, the index of the codomain is always smaller than (i.e. less than or equal to) the index of the domain (while for direct systems, it is the opposite). Focusing on this invariant of the morphisms instead of the specific index convention/notation used by a particular author, may help when switching from reading one author's work to another's.

Inverse limits and cones

The inverse limit can be defined abstractly in an arbitrary category by means of a universal property. Let be an inverse system of objects and morphisms in a category (same definition as above).

A collection of morphisms from an object in is said to be compatible or consistent[3] with the system if for all indices the morphism has prototype and the following compatibility condition is satisfied:

in which case the pair is called a cone from into the system. The object is called the vertex of the cone

An inverse limit[3][4] of this inverse system is a cone into (so these morphisms must satisfy ) for which the following condition holds:

Universal property of limits: If is any cone into this system (so by assumption these morphisms satisfy ) then there exists a unique morphism such that for every index (this may be abbreviated as );
  • In this case, the following diagram will commute for all indices
InverseLimit-01.png
  • This unique morphism is called the limit of the cone and it may also be denoted by or

Said more succinctly and without indices, an inverse limit of an inverse system is a cone into such that for any cone into this system, there exists a unique morphism such that

Each morphism is called the projection from to although these projections are not guaranteed to be surjections. An inverse limit is often denoted

when the bonding morphisms and indexing set of the system are 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 and of an inverse system, there exists a unique isomorphism commuting with the projection morphisms.

An inverse system in a category admits an alternative description in terms of functors. Specifically, any partially ordered set can be considered as a small category where the morphisms consist of arrows if and only if An inverse system is then just a contravariant functor and the inverse limit functor is a covariant functor.

The canonical limit

The canonical inverse limit of the inverse system in the category of sets is the following subset of the Cartesian product of the 's:

where the associated morphisms are defined to be the restrictions to of the Cartesian project's natural projections. Explicitly, for every index the map is the restriction to of the natural projection which picks out the component of the Cartesian product; that is The canonical limit, which is the cone consisting of and the maps satisfies the universal property of limits described above and so is an inverse limit of in the category of sets. This definition of the canonical limit thus proves that inverse limits always exist in the category of sets. Importantly, however, the canonical limit may be the empty set even if all are non-empty (the limit being the empty set does not mean that the limit does not exist). And moreover, it is possible for a projection to not be surjective even if is not empty.

In the category of groups, is a subgroup of the group In the category of topological spaces, becomes a limit in this category when is endowed with the weak topology induced on it by the projections Consequently, inverse limits always exist in both the category of groups as well as in the category of topological spaces.

This same construction of canonical limits may be carried out if the 's are groups,[5] semigroups,[5] topological spaces,[5] 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. For example, an inverse system (or projective system) of groups and homomorphisms will be a group and homomorphisms.

Examples

  • Given any object in any category and given a preordered set the constant or trivial system over is the system where and is the identity map on for every index The limit of this system is the cone where
  • If is a sequence of object and if is a sequence of morphisms, each of which has prototype then these objects and morphism are automatically associated with the following induced or canonical system where for all satisfying the morphism is defined by while is defined to be the identity morphism
  • If is an inverse system and is a subset then this system's restriction to is the inverse system where any such system is known as a subsystem of .
  • If is an inverse system in the category of sets, and then this system's restriction to and is the inverse system where for every if then and otherwise If is not specified then it is to be assumed that
  • The ring of -adic integers is the inverse limit of the rings (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 such that each element of the sequence "projects" down to the previous ones, namely, that whenever The natural topology on the -adic integers is the one implied here, namely the product topology with cylinder sets as the open sets.
  • The -adic solenoid is the inverse limit of the compact Hausdorff 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
  • The ring of formal power series over a commutative ring can be thought of as the inverse limit of the rings indexed by the natural numbers as usually ordered, with the morphisms from to given by the natural projection.
  • Pro-finite groups are defined as inverse limits of (discrete) finite groups.
  • Let the index set of an inverse system have a greatest element Then the natural projection 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 -adic numbers and the Cantor set (as infinite strings).
  • Smooth functions defined via limits without differentiation: Let be a convex open subset of and fix Let be the algebra of continuous -valued functions on and let for all For each integer define the bonding map by
    where by we mean the continuous function defined on (if then is an empty list). For the mape is defined in the usual way (e.g. etc.). Using Taylor's theorem, it is easily seen that each is injective so consequently, is a limit of this system. We now identify this limit as the set of smooth functions. Suppose Then if and only if for some real and some which by Taylor's theorem happens if and only if is continuously differentiable. By induction, if and only if
    for some real and some which by Taylor's theorem is true if and only if (in this case, is the derivative of at ). Thus the limit of the above system is Note that this construction of the smooth functions on 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 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 ). This construction can be generalized to define smooth functions on a convex open subset of where

Derived functors of the inverse limit

For an abelian category the inverse limit functor

is left exact. If is ordered (not simply partially ordered) and countable, and is the category Ab of abelian groups, the Mittag-Leffler condition is a condition on the transition morphisms that ensures the exactness of Specifically, Eilenberg constructed a functor

(pronounced "lim one") such that if and are three inverse systems of abelian groups, and

is a short exact sequence of inverse systems, then

is an exact sequence in Ab.

Mittag-Leffler condition

If the ranges of the morphisms of an inverse system of abelian groups are stationary, that is, for every there exists such that for all  : 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 are surjective
  • a system of finite-dimensional vector spaces or finite abelian groups or modules of finite length or Artinian modules.

An example[6]pg 83 where is non-zero is obtained by taking to be the non-negative integers, letting Xi = piZ, Bi = Z, and Ci = Bi / Xi = Z/piZ. Then

where Zp denotes the p-adic integers.

Further results

More generally, if is an arbitrary abelian category that has enough injectives, then so does and the right derived functors of the inverse limit functor can thus be defined. The right derived functor is denoted

In the case where satisfies Grothendieck's axiom (AB4*), Jan-Erik Roos generalized the functor lim1 on AbI to series of functors limn such that

It was thought for almost 40 years that Roos had proved (in Sur les foncteurs dérivés de lim. Applications) that for an inverse system with surjective transition morphisms and 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 Roos has since shown (in "Derived functors of inverse limits revisited") that his result is correct if 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 has cardinality (the th infinite cardinal), then is zero for all This applies to the -indexed diagrams in the category of -modules, with 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 on diagrams indexed by a countable set, is nonzero for ).

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.

See also

Notes

  1. ^ Mnemonic: The two "inner"/"middle" indices in which is the right hand side of the compatibility condition should always be the same in order for the composition to be guaranteed to be valid/well-defined. For example, the composition is valid because both middle indices are whereas if then the composition might not even be well-defined (unless, for instance, it happens to be the case that ). Moreover, this common "inner"/"middle" index is missing from the "simplified" left hand side of this equality.
  2. ^ Some authors (e.g. Dugundji) do note require that be the identity map. The definition of the inverse limit of such a generalized system is identical to the definition given below.
  3. ^ This is abuse of notation and terminology since calling an inverse system is technically incorrect.

Citations

  1. ^ Bourbaki 1989.
  2. ^ Dugundji 1966.
  3. ^ a b Mac Lane 1998, pp. 68-69.
  4. ^ Dugundji 1966, pp. 427-435.
  5. ^ a b c John Rhodes & Benjamin Steinberg. The q-theory of Finite Semigroups. p. 133. ISBN 978-0-387-09780-0.
  6. ^ Dugger, Daniel. "A Primer on Homotopy Colimits" (PDF). Archived (PDF) from the original on 3 Dec 2020.

References

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