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.
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. One benefit of not requiring to be partially ordered is that it simplifies defining equalizers in terms of inverse limits.
If is directed (respectively, partially ordered, countable) then the system is said to be directed (respectively, partially ordered, countable).
If no symbol is assigned to the preorder then unless indicated otherwise, it should automatically be assumed to be denoted by
Elements of are called indices of the system and a statement such as "for all that satisfy " may instead be more simply stated as: "for all indices in ". When the indexing set is understood then its mention may be omitted.
The homogeneous relation is identified with the set so that for all if and only if If then will denote the restriction of to (see this footnote[note 1] for the definition); however, notation is usually abused by writing where should technically be written instead.
is a family of objects, meaning that is an object in the category for every . For example, if is the category of sets (respectively, topological spaces, groups, etc.) then "object" means set (respectively, topological space, group, etc.)
is a morphism in the category for all indices . For example, if is the category of sets (resp. topological spaces, groups) then "morphism" means function (resp. continuous function, group homomorphism). These morphisms are called the bonding, connecting, transition, or linkingmaps/morphisms of the system or simply, the system's morphisms. Some authors reserve the term "bonding map/morphism" only for inverse systems that are indexed by the natural numbers.
Whenever is written then unless indicated otherwise, it should be assumed that and are indices satisfying .
By treating as a set, the notation is consequently a family of morphisms indexed by the set so this family can be denoted by although the notation or may be used instead. If the indexing set is understood then it may be omitted. If then the -indexed family may be written as or
The following compatibility conditionof inverse systems holds:
where if the arrows above are reversed or inverted (as is appropriate for inverse systems), then this condition for can be written as:
and almost[note 3] always, inverse systems are also required to satisfy the following additional condition:
is the identity morphism on for every
The tuple may also be written as as as if is understood, or even as if is understood.
If the connecting morphisms are understood or if there is no need to assign them symbols (e.g. as in the statements of some theorems) then the connecting 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 4]
Projective systems
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 connecting 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). For example, a projective system in the category of smooth manifolds (whose morphisms are smooth maps) is often defined as an inverse system in the category of smooth manifolds that is indexed by the natural numbers and all of whose bonding maps are surjectivesmoothsubmersions. Regardless of how "projective system" is defined, a projective limit means an inverse limit of a projective system.
Examples of inverse systems
Given any object in any category and given a preordered set the constant or trivial system over (that is constantly ) is the system
where all objects are and every connecting morphism is the identity morphism on which is denoted by That is, where for every index and for all indices the connecting morphism is the identity morphism
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
For example, where the right hand side is the following composition of morphisms:
This canonical system (whose preorder is the usual integer comparison ) is necessarily an inverse system. Although the natural numbers were used, the indexing set may, for example, alternatively be some other subset of the integers
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 a cofinal subset of then is called a cofinal subsystem of
The following generalization is of a subsystem is sometimes useful when dealing with limits of in the category Set. Given an index and subsets 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 If is directed and is cofinal in then the limit of this new system in Set can be identified as subset of 's limit.
Relationship with direct systems
Notations for connecting morphisms
For the inverse system and for any indices this article denotes the connecting morphism by However, 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 connecting 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.
One-to-one correspondence between inverse and direct systems
Given a preordered set its converse or transpose is the preordered set where by definition, which is also denoted by [note 5] Explicitly, 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 (not to be confused with a directed 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 connecting 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 which says exactly that is larger than with respect to ); however, this latter condition is in general not true of with respect to which is why in general, inverse systems like are not also direct systems. 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 the tuple (i.e. the equality for ) is exactly the same as the compatibility condition of inverse systems applied to the tuple (i.e. the equality for ).
Cone into an inverse system
A collection of morphisms from an object in is said to be compatible or consistent[3] with the system if for every index the morphism has prototype and if also for all indices the following compatibility condition is satisfied:
which happens if and only if the following diagram commutes:
If this is the case then the pair is called a cone from into the system the object is called the vertex of the cone and each is called the (th) morphism or projectionof the cone (into ).
Examples of cones
The empty cone and cones in the category of sets
In the category Set {\displaystyle \operatorname {Set} } of sets, the empty cone is the pair which has the empty set as its vertex and all of whose morphisms are the empty map, which is also denoted by The empty cone is a cone into every inverse system in category and for certain inverse systems, the empty cone might even be the only cone going into it. This is true, for example, if any is the empty set. If is an inverse system in then cones into it can be characterized in terms of fibers (which are inverse images of singleton sets) as follows: if are maps, each of which has prototype then is a cone into if and only if for all indices the map is constant on each fiber of and for every if the fiber is not empty then 's value on it must be
Restricting and extending cones
Suppose that is an inverse system and is a non-empty subset. If is a cone into then is called this cone's restriction to (or to ) and it is guaranteed to be a cone into the subsystem which was defined above. Similarly, any cone into that restricts to a given cone is called an extension to (or to ).
Given a cone into the subsystem to construct an extension to assume that is a cofinal subset of which means that for every there is some such that If is a directed set then the cone can be extended to a unique cone in where for each the morphism is for any/every that satisfies (see this footnote[proof 1] for a proof that these maps are well-defined).
If is not a directed set then an extension of a cone from a cofinal subsystem is not guaranteed to exist, as the following example demonstrates. Let be distinct objects, let
and partially order by declaring that if and only if or Let be the identity morphism for every and let and be the natural projections; that is,
and
This defines an inverse system . Let which is a cofinal subset of let be a singleton set, and define and by
and
Then is a cone into the subsystem but there does not exist any such that is a cone into because the compatibility condition would simultaneously require and
Inverse limit of an inverse system
An inverse limit can be defined abstractly in an arbitrary category as a cone that possesses a certain universal property. Throughout, let be an inverse system of objects and morphisms in a category (same definition as above).
An inverse limit or limit[3][4]of the inverse system is a cone into (so must consist of morphisms of the form and these morphisms must satisfy for all meaning that this diagram:
must commute) for which the following condition holds:
Universal property of ( inverse) 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 ); that is, the following diagram must commute for every index :
This unique morphism is called the (inverse) limitof the coneinto and it may also be denoted by , , or .
If this is the case then for all indices , the above diagrams can be combined to produce the following commutative diagram:
Said more succinctly and without indices, an inverse limitof an inverse system is a cone into such that for any cone into this system, there exists a unique morphism that satisfies This can be expressed by writing
where if the system's connecting morphisms and indexing set are understood then this may be written more simply as or However, despite the equals sign being used, limits are in general not unique although they are unique up to isomorphism. In the category Set, it is sufficient (and necessary) to check the universal property for all cones whose vertex is a singleton set (where such a cone into exists if and only if the vertex of 's limit is not the empty set; if the limit is empty, then the only cone into is the empty cone). In the category of groups, this is true if the object is instead
Each morphism of a limit is called the projection from to (the word "onto" is purposefully not used). Despite the name "projection", it should not be assumed that these maps are epimorphisms (e.g. surjections) because in general, this may not be true (examples of this may even be found in the category of sets).
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.
Basic properties of limits of cones
Correspondence between cones and morphisms into a limit
Suppose is a limit of in some category. If is any morphism and if then is a cone into whose limit into is the original morphism ; in symbols, for every morphism In particular, this shows that every morphism into (the vertex of) a limit arises as the limit of some cone. Every limit cone[note 6] thus establishes a one-to-one correspondence between morphisms into and cones into
The limit of a cone into is the identity morphism if and only if and (that is, for every index ). In particular, the limit of the cone into itself (that is, into ) is the identity morphism ; in symbols,
Equality of elements in the limit
Suppose is a limit of in Set. If given then if and only if for every index ;[proof 2] this characterization remains true if the index instead varies over some cofinal subset of (even if is not a directed set).
Uniqueness and isomorphisms of inverse limits
In some categories, there are certain inverse systems for which an inverse limit does not exist. However, if an inverse limit does exist then it is unique up to isomorphism in a strong sense: any two limits of an inverse system differ from each other by at most a unique isomorphism that commutes with the projection morphisms, as is now described in detail.
Assume throughout that is a limit of If is an isomorphism from some object and if then the cone is also an inverse limit of in this category and moreover, as mentioned above, the limit of the cone into is Conversely, assuming now that is any limit of then the limit of the cone into which is denoted by is necessarily an isomorphism that satisfies (that is, for every index ) and consequently, . Furthermore, if denotes the limit of the cone into then in addition to satisfying this morphism also satisfies and This also shows that an arbitrary given cone into is (also) an inverse limit of this system if and only if there exists an isomorphism such that (or said differently, such that is equal to the original limit ); moreover, this isomorphism will necessarily be unique.
Cofinal subsystems of directed systems
Suppose that is a directed inverse system and that is a cofinal subset of the directed set If is a limit of then this cone's restriction to is a limit of the subsystem Conversely, if is a limit of the subsystem then this cone's unique extension to (described above) will be a limit of the original system Specifically, for each the morphism was defined by where is any index satisfying ; however, if is not a directed set then there might not even exist a cone into that extends Using the method just described, it is thus possible to pass between a limit of a directed system and a limit of a cofinal subsystem by doing nothing more than removing or adding (uniquely defined) projections from/to the limit cone; in particular, the cone's vertex does not need to be changed.
Vertex of a limit as the limit of a totally ordered system
It is now demonstrated how the vertex of a limit of a partially ordered inverse system in a category is also the vertex of a limit of a totally ordered inverse system in if the limits of the subsystems that are defined below all exist (which happens, in particular, if inverse limits always exist in ). Partially order by set inclusion the set of all non-empty ideals in which by definition are non-empty subsets with the property that whenever satisfies for some then necessarily Let be a totally ordered subset of such that (in particular, the possibility is allowed although might alternatively (and more usefully) consist entirely of proper ideals) and for all assume that is a limit in of the -indexed subsystem For all with each of and is a cone into so denote these cones' respective limits into by and Then is a totally ordered inverse system and it has as an inverse limit in Note in particular that the vertex of this new limit cone is the same as that of the original limit
For example of this construction in the category of sets, suppose that is partially ordered by declaring if and only if and let and be the identity map for every index Then the vertex of a limit of the inverse system is the space of real sequences Let consist of all subsets of the form (as ranges over ) so that is order isomorphic to the usual total order Then is also the limit of the totally ordered inverse system where every is the canonical projection (as described in an example below).
Interpretations
For every time let denote some state space at time where for ease of discussion, it will be assumed that time is measured in seconds. If is some state at time then let denote its subsequent state at time so that this assignment defines a function Assume that when then is the identity map, which can be interpreted as: there is no change in state if there is no change in time. Given times assume also that ; this can interpreted as: the state that results after waiting seconds is the same state that results from waiting seconds followed immediately by waiting another seconds. In other words, this assumes that the future state of depends only on the current state and not on some combination of both the current state and some other state(s) belonging to the strict past (i.e. belonging to some at some time ).
With respect to the usual order on the index of the codomain of is greater than (or equal to) the index of the domain, so the tuple
might be expected to form a direct system (and not an inverse system), and indeed the assumption made above guarantee this (because those assumptions are precisely the required compatibility conditions). Similarly, with respect to the transpose (defined above) of the above system
might be expected to form an inverse system (and not a direct system), and indeed the assumption made above are precisely what guarantee this.
Now pick some arbitrary time to interpret as being "now" or "the current time", where for concreteness it will be assumed that Because is a directed set and is a cofinal subset of the direct limit of the direct system is, up to a unique isomorphism, the same as the direct limit of the subsystem which results from restricting the indexing set to Similarly, because is a directed set and is a cofinal subset of the inverse limit of is, up to a unique isomorphism, the same as the inverse limit of the subsystem In words, this means that the inverse limit of this system depends only on the past (that is, only on the times ) while the direct limit of this system depends only on the future (that is, only on the times ).
The inverse limit of this (sub)system, say can be interpreted as all the set of all possible (infinite) "timelines" that "go back eternally", where some given "infinite timeline" is said to "pass through" some given state at time if and only if Inspecting the definition of the canonical inverse limit (defined below) makes this interpretation clearer. It may also help to use as the indexing set instead of the continua or ; doing this is also technically sound.[note 7] The meaning and importance of the words "infinite" and "eternally" are now clarified. If is some state at time and if there exists any time such that then the compatibility condition makes it impossible for there to exist an "infinite timeline" passing through the state ; this is true even if there exist some time between and (i.e. ) such that because this just means that there is a "finite timeline" passing through which not the same as an "infinite/eternal timeline". Importantly, knowing only that for every real is not sufficient to guarantee the existence of an "infinite timeline" passing through (that is, satisfying );[note 8]; additional information or assumptions are needed to guarantee this.
In contrast, by definition of the canonical direct limit, given some state the states in the set [note 9] (consisting of the current state and all of its future states) are identified together as one under a canonical equivalence relation (that is described in the article on direct limits). Note that this aforementioned set is a subset of the equivalence class containing (which will be denoted by ) but it might be a proper subset of ; however, these two sets will be equal to each other if, for instance, all of the maps are injective for all real
Regardless of whether this set equality holds, every state is associated with exactly one element in this system's direct limit (where ). In this system's inverse limit however, there is might not exist any single "infinite timeline" passing through (although this is a possibility[note 10]). In fact, depending on the connecting morphisms there might not exist any "infinite timeline" passing through there might exist only finitely many, or there might even exist infinitely many. See also this footnote[note 11] for a related, but more accessible, interpretation.
The canonical limit
The canonical (inverse) limit of the inverse system in the category Set is the cone into consisting of the following subset of the Cartesian product of the 's:
together with the associated projections that are defined to be the restrictions to of the Cartesian product'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 into consisting of and the maps satisfies the universal property of limits described above and so is an inverse limit of in the category Set. This definition of the canonical limit thus proves that inverse limits always exist in Set. Importantly, however, it is possible for the canonical limit to 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 in Set). And moreover, it is possible for a projection to not be surjective (that is, to not be an epimorphism in Set); this is possible even if is not empty.
If is a cone into then for any subset
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.
The existence of limits in the category Set can be used to help prove the existence of limits in other concrete categories.
Limits exist in the category of topological spaces
In the category of topological spaces, becomes a limit in this category when is endowed with the initial topology, also known as the weak topology or the limit topology, induced on it by the projections Consequently, inverse limits always exist the category of topological spaces.[5] If every is endowed with a uniformity and if every is uniformly continuous then becomes a limit of in the category of uniform spaces (where morphisms are uniformly continuous maps) if is endowed with the relative uniformity induced on by the product uniformity on
Limits exist in the category of groups
In the category of groups, is necessarily a subgroup of the product group and the projections will necessarily be group homomorphisms. These observations, together with the fact that is a limit in the category of sets, readily imply that inverse limits always exist in the category of groups.[5] The subset is not empty because it contains the identity element of
If is any limit of in the category of groups and if then because there exists some element in which will be denoted by such that
Limits in other algebra related categories
This same construction of canonical limits may be carried out in the categories of semigroups,[5] rings, modules (over a fixed ring), algebras (over a fixed ring), etc., where the morphisms are homomorphisms of the corresponding algebraic structure. The construction of these canonical limits thus proves that inverse limit always exist in these categories. For example, an inverse system of rings and ring homomorphisms will be a ring together with ring homomorphisms.
Limits in Set and Top
Throughout, will be a limit of a partially ordered inverse system in the category of sets and will be a cone into whose limit into will be denoted by If is non-empty and for every then
so that in particular, for every and every subset To non-trivially relate a subset with its images under and the s, the set
is necessarily a subset of
For any given index the map is injective if and only if this is true of for every index in which case the limit map will necessarily also be injective,
will hold on
and
will hold on
If for some index then is the identity morphism on if and only if for every index
Assume that is an inverse system in the category Top {\displaystyle \operatorname {Top} } of topological spaces and that is a limit in Then the topology on is equal to the weak topology or the limit topology, induced on it by the projections which by definition is the weakest topology on making every continuous. A subbase for this topology consists of all subsets of the form as ranges over the open subsets of and ranges over ; if is a directed set then these subbasic open subsets actually form a basis for the limit topology (rather than merely just a subbasis).
The system is said to be a uniform system if every is endowed with a uniformity that is consistent with 's given topology (that is, the topology that induces on is equal to 's given topology) and if every is uniformly continuous. A uniform system is nothing more than an inverse system in the category of uniform spaces, where morphisms are uniformly continuous maps. In this case, becomes an inverse limit of in the category of uniform spaces if is endowed with the weakest uniformity making every uniformly continuous. The topology induced on by the uniformity is equal to the weak topology on induced by the If in addition every is metrizable with as a metric and if the indexing set is then is also metrizable and a metric compatible with its topology is given by the Fréchet combination
for all
Examples of limits
Limits of a constant system
Given any object in any category and given a preordered set the limit of the constant system is the cone If is a set then the vertex of the canonical limit of this constant system consists of all elements of the product that are constant; that is, all for which there exists some such that for
Greatest index
Suppose is an inverse system in some category and is a partially ordered set with a greatest element which means that for every Then is a limit of in this category. Moreover, if is any limit of then its projection is necessarily an isomorphism.
Pullbacks as limits
A pullback of any two given morphisms and having the same codomain object is an inverse limit of the following inverse system:
where the set of three distinct indices[note 12] is partially ordered by declaring that for any if and only if or ; this makes the least element of where this partially ordered set is not (upward) directed. In terms of the notation this article has been using, for every the th object is and the connecting morphism is (as usual) the identity map; the only remaining connecting morphisms are and because and A limit of this inverse system is a cone into where by the definition of a limit, these morphisms satisfy
and
and thus also
The vertex is typically denoted by and the morphism is usually not mentioned because whenever it is needed, it can be reconstructed from the morphisms
and
via either of the compatibility conditions or The universal property of limits that satisfies can be restated as:
Universal property of pullbacks: If and are any morphisms that satisfy then there exists a unique morphism such that and
This unique morphism is the limit of the cone into where is defined by
Products as limits
Arbitrary products
Let be a family of sets indexed by For declare that if and only if This makes a partially ordered set but it is a directed set if and only if is a singleton set. For every let be the identity map and let denote the canonical projection onto the coordinate. Then is an inverse system in the category Set and is a limit of in this category, where If the are topological spaces then is an inverse system in the category of topological spaces and moreover, endowing with the product topology (which is equal to the weak topology on induced by the projections ) makes a limit of this system in
Several limits of finite powers of a set
Let be any non-empty set (for example, ) and for every natural number let denote the canonical projection defined by As described above, these maps induce a canonical inverse system
where is the map This system has as a limit in Set, where is the canonical projection defined by (the symbols were used above to help define the canonical limit's projections and by using each symbol will continue to denote the map that picks out exactly one component − its th component − and ignores all others; in particular, for this system, if ). In contrast to this limit, if has at least two distinct elements then the vertex of this system's canonical limit (defined above) is a proper subset of even though this product can be canonically identified with via
The elements of the canonical limit are exactly those that are of the form So for instance, if are distinct then under this canonical identification, belongs to but not to the canonical limit (because ). Because it is simpler than the canonical limit, the limit is typically preferred over the canonical limit for this particular system.
For every define a map by
and for every
where only the first two coordinates are out of their usual order.
Then is also a limit of in Set and it has the same vertex as the previous limit ; the only difference is in the projection morphisms. To see that this difference in the projection morphisms is important, set let denote any singleton set, and for every let be defined by which makes a cone into The limit of the cone into is the map defined by whereas the limit of into is the map defined by where only the first two coordinates are out of their natural order. Moreover, the limit of the cone into is the isomorphism of sets (i.e. a bijection) defined by This map also happens to be the limit of the cone into
Pro-finite groups and algebraic examples
Pro-finite groups, p-adic integers, and p-adic solenoids
Pro-finite groups are defined as inverse limits of (discrete) finite groups.
The ring of p {\displaystyle p}
-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 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 {\displaystyle p}
-adic numbers and the Cantor set (as infinite strings).
The p {\displaystyle p}
-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
Formal power series
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.
Intersections as limits
Intersections and unions
Let be a non-empty family of sets and define
and
This example will turn into an inverse system and then later, also into a direct system. Given any sets and such that let
denote the inclusion map (defined by ). Let and for every let so that Thus is the systems' indexing set, and the systems' objects are the elements of The superset inclusion relation and the subset inclusion relation each define a partial order on the set For example, is a partially ordered set because if is denoted by (that is, if we define if and only if so that subsets are considered "greater" while supersets are considered "lesser") then is an antisymmetric preorder on The index of the codomain of with respect to is less than or equal to the index of its domain (because which is if the notation above is used). So with respect to the partial order these inclusions maps might be expected to form an inverse system (and not a direct system), and indeed the tuple
does form a partially ordered inverse system because if are indices then If is a directed set and if then is an inverse limit of in the category Set. The indexing set has a greatest element if and only if in which case is this (unique) greatest element.
Consider now which is the transpose of The index of the codomain of with respect to is greater than or equal to the index of its domain (because ). So with respect to the partial order these inclusions maps might be expected to form a direct system (and not an inverse system), and indeed this is the case because as mentioned above, the transpose
of the inverse system is necessarily a direct system. If is a directed set and if then is a direct limit in the category Set of the direct system The indexing set has a greatest element if and only if in which case is this (unique) greatest element.
This example demonstrates why in general, intersections are considered to be "inverse limit constructions" while unions (which are the "dual" of intersections) are considered to be "direct limit constructions." This duality extends further. For instance, Cartesian products (which are products in the category of sets) are generally considered to be inverse limit constructions while disjoint unions (which are coproducts in the category of sets and are the "dual" to products) are considered direct limit constructions.
Returning to inverse limits, when is not a directed set, which will now be assumed, then constructing a limit of that is easier to work with than the canonical limit becomes more technically challenging. One issue with the canonical limit of this system is that it may be difficult to determine its basic properties, such as for instance, whether or not the its vertex is the empty set. Indeed, it is even possible for this limit to be a Cartesian product of sets. One special case is now considered. For all let and let If there exists some such that is the empty set then the limit of in Set is necessarily the empty cone.
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 it is meant the continuous function defined on (if then is an empty list). For the map is defined in the usual way (e.g. etc.). Each is an injective map (a fact that can be verified by using Taylor's theorem, for example) so that the same is true of every bonding map Consequently, the intersection of the images is a limit of this inverse system (where this limit's projection into is the restriction to of the inverse of the bijection ).
This limit is now shown to be the set of smooth functions on Suppose Then if and only if for some real and some which by Taylor's theorem happens if and only if is continuously differentiable (i.e. ). 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 possibly also 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
Inverse system morphisms
Throughout, and will be inverse systems in some category An inverse system morphism (ISM) from to is a pair consisting of
an order-preserving map ; that is, whenever satisfy and
an -indexed collection of morphisms where for every is a morphism in
with the property that for all satisfying
That is, the following diagram must commute:
This situation may be summarized by writing that is a morphism (of inverse systems).
Composition of inverse system morphisms
If is another inverse system and is an inverse system morphism (so is order-preserving and is a morphism for all ) then the composition
is defined to be the inverse system morphism
consisting of the order morphism and the -indexed morphisms By endowing inverse system morphisms with this definition of composition, the class of all inverse systems in a given category becomes a category.
Limit of an inverse system morphism
Assume that is an inverse limit of and that is an inverse limit of The canonical conefrominduced by (or associated with) the ISM is the cone
from into where the family may be denoted by The limitfrom into of the inverse system morphism is defined to be the limit of the canonical cone This limit is a morphism which may also be denoted by or This limit morphism is the unique morphism satisfying
for all
in which case the following diagram will commute for all indices satisfying :
It is in general not true that for given inverse systems, every morphism between inverse limits arises as the limit of an inverse system morphism. Specifically, in the category of sets, (together with the canonical projections ) is a limit of the surjective inverse system and there exists an isomorphism such that is not equal to the limit of any inverse system morphism of the form When the space of all real sequences is endowed with the usual product topology then this map is even a homeomorphism. Moreover, when is considered as a topological vector space (TVS) then both and are continuously differentiable with respect to the Fréchet derivative and for each, its Fréchet derivative at any point of its domain is an isomorphism of TVSs. Each coordinate of is smooth and the same is true of each coordinate of
Equivalence transformation
Two inverse systems over the same category are said to be equivalent (in the given category) if there exists an equivalence transformation between them. Two inverse system morphisms and are said to form an equivalence transformation (of inverse systems) if
for all
and also
for all
In this case, the following diagrams will commute for all in and all in :
and
from which it follows that in particular, the following diagram will also commute.
Equivalent systems have isomorphic limits.
Examples of inverse system morphisms and their limits
Identity morphisms
Suppose that is an inverse system indexed by that has a limit If is the identity morphism and if for every is also the identity morphism, then is an inverse system morphism whose limit from and into is the identity morphism
Series
Let be a subset of an additive group that contains the additive identity is closed under addition, and has the property that if is non-zero then its additive inverse does not belong to (alternatively, with implies ). For example, could be or or or even the complex plane's upper half .
For all natural numbers let and let denote the canonical projection defined by so that as described in an example above, these maps form an inverse system
that has as a limit in the category Set, where is the canonical projection defined by .
For every let and let be the map defined by
where in particular, Let
be the canonical inverse system that these maps induce (as defined above) and let be the canonical limit of in Set. Elements of the canonical limit's vertex are exactly those elements of that are of the form where for all integers both and belong to and Intuitively, elements of can be identified with infinite lists of equations of the form:
where all and belong to (This interpretation often generalizes and may help to explain why the limits of some inverse systems are empty: nothing satisfies that system's infinite list of requirements.) For example, the sequence of all s belongs to And given any and any the sequence (where is added to every ) will also belong to
Let be defined by For every let be the canonical projection as defined above. Then is an inverse system morphism because the following diagram commutes for all :
Let denote the limit of into If is as described above, then this element's image under is ; thus "forgets" every but remembers every Observe, in particular, that the map both the systems and and their limits and are not dependent on any topology.
Assume now that Then is neither injective nor surjective. And despite being completely independent of any topology, it is readily verified that an element belongs to 's image if and only if the series converges in with its usual Euclidean topology. In this way, inverse limits in the category Set of sets can be used to completely characterize those non-negative real sequences that are absolutely convergent in (with its usual topology). However, because is not injective, it is not possible to use to (non-arbitrarily) assign a distinguished "sum" to such a convergent series (that is, to each element in ); to do that, it is necessary to pick a topology on (such as the Euclidean topology, for example).
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 and Then
where 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
Cartesian closed category – Type of category in category theory
Direct limit
Equaliser (mathematics) – Set of arguments where two or more functions have the same value
Limit (category theory) – Terminology used in theoretical mathematics involving categories.
Product (category theory) – Generalized object in category theory
Protorus – Mathematical object
Notes
^The restriction of to is defined by This definition is used in the definition of a subsystem.
^Mnemonic: The two "inner"/"middle" indices in which is the left 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" right hand side of this equality.
^Some authors (e.g. Dugundji) do not require that 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 an inverse system is technically incorrect.
^Although is equal to this article will use the notation instead of because, for instance, the statement " is less than or equal to with respect to " (which unambiguously means ) is clearer and less ambiguous to non-specialists than the equivalent statement: " is less than or equal to with respect to " (also meaning but the latter is usually read as " is greater than or equal to ", which might cause confusion).
^It is emphasized that in general, this one-to-one correspondence depends on both the object and the morphisms
^Just like the set is a cofinal subset of the directed set Consequently, and its subsystems and all have the same limit (up to a unique isomorphism).
^For an example of how this might happen, consider an inverse system whose objects are all non-empty sets but whose limit in Set is the empty cone (so and for every ). Assuming that has a least element and that is a singleton set, say then for every index because every is non-empty, but there nevertheless does not exist any such that An alternative to assuming that has a least element and that is a singleton set is to instead extend and its limit cone by appending a new least element to the indexing set letting be any singleton set, and defining and each in the only possible way, so that each map is necessarily surjective and This new extended cone which is still an empty cone, is necessarily a limit of this new extended system. Thus despite the fact that for every index
^Here it is being assumed that all of the sets are pairwise disjoint; if they are not then they can be identified with (or replaced by) pairwise disjoint copies.
^A unique "infinite timeline" passing through exists if and only if the fiber is a singleton set for every If is injective for all real then for every there will exist at most one "infinite timeline" passing through ; under these injectivity assumptions, it will exist if and only if for every real (or equivalently, for every in some infinite unbounded subset of ).
^For a much less technical interpretation that is more accessible to the general public and grounded in the physical world, rather than interpreting an element as a state, consider instead as representing some particle as it exists at time and let instead represent this particle as it exists at the future time (Instead of a particle, it could also be a ship belonging to Theseus). Then the equivalence class that belongs to the resulting canonical direct limit may be interpreted as representing "the particle considered as a single entity existing throughout time" as opposed to the extreme opposite view that at any two distinct times the particle as it exists at time is distinct/different from its "future self" as it exists at time (this may also be described as being the assumption that all of the sets are pairwise disjoint, which if true, greatly simplifies the construction of the canonical direct limit by obviating the otherwise necessary step of constructing pairwise disjoint copies of these sets). The canonical direct limit's morphism can be interpreted as the bridge between (1) viewing the particle as infinitely many distinct instances (as time varies over ) and (2) viewing it as a single entity existing throughout time. Any element of the canonical inverse limit passing through (i.e. satisfying ) can then be interpreted as one particular "timeline" of particles that eventually became
^The objects and are treated as if they are distinct although if they are not, then 's elements may respectively be replaced by the distinct symbols and or alternatively, by any other three distinct objects or symbols such as and for instance.
Proofs
^Fix and let be such that and Proving will show that is well-defined. Pick any such that and and then pick such that Because the equality holds and so Similarly, follows because Thus
^Let be any singleton set and define by and The conclusion follows by applying the universal property of inverse limits.
Citations
^Bourbaki 1989.
^Dugundji 1966.
^ a bMac Lane 1998, pp. 68-69.
^Dugundji 1966, pp. 427-435.
^ a b cJohn Rhodes & Benjamin Steinberg. The q-theory of Finite Semigroups. p. 133. ISBN 978-0-387-09780-0.
^Dugger, Daniel. "A Primer on Homotopy Colimits" (PDF). Archived (PDF) from the original on 3 Dec 2020.
References
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