In
mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related
objects, the precise gluing process being specified by
morphisms
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphism ...
between the objects. Thus, inverse limits can be defined in any
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
although their existence depends on the category that is considered. They are a special case of the concept of
limit
Limit or Limits may refer to:
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* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
in category theory.
By working in the
dual category
In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields ...
, that is by reverting the arrows, an inverse limit becomes a
direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cat ...
or ''inductive limit'', and a ''limit'' becomes a
colimit
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions su ...
.
Formal definition
Algebraic objects
We start with the definition of an inverse system (or projective system) of
groups and
homomorphisms
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
. Let
be a
directed
Director may refer to:
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Music
* Director (band), an Irish rock band
* ''D ...
poset
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
(not all authors require ''I'' to be directed). Let (''A''
''i'')
''i''∈''I'' be a
family
Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Idea ...
of groups and suppose we have a family of homomorphisms
for all
(note the order) with the following properties:
#
is the identity on
,
#
Then the pair
is called an inverse system of groups and morphisms over
, and the morphisms
are called the transition morphisms of the system.
We define the inverse limit of the inverse system
as a particular
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
of the
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
of the ''
''
's:
:
The inverse limit
comes equipped with ''natural projections'' which pick out the th component of the direct product for each
in
. The inverse limit and the natural projections satisfy a
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently ...
described in the next section.
This same construction may be carried out if the
's are
sets,
[John Rhodes & Benjamin Steinberg. The q-theory of Finite Semigroups. p. 133. .] semigroups,
topological spaces,
rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
,
modules (over a fixed ring),
algebras
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and additio ...
(over a fixed ring), etc., and the
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...
s are morphisms in the corresponding
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
. The inverse limit will also belong to that category.
General definition
The inverse limit can be defined abstractly in an arbitrary
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
by means of a
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently ...
. Let
be an inverse system of objects and
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphis ...
s in a category ''C'' (same definition as above). The inverse limit of this system is an object ''X'' in ''C'' together with morphisms
''i'': ''X'' → ''X''
''i'' (called ''projections'') satisfying
''i'' =
∘
''j'' for all ''i'' ≤ ''j''. The pair (''X'',
''i'') must be universal in the sense that for any other such pair (''Y'', ψ
''i'') there exists a unique morphism ''u'': ''Y'' → ''X'' such that the diagram
commutes for all ''i'' ≤ ''j''. The inverse limit is often denoted
:
with the inverse system
being understood.
In some categories, the inverse limit of certain inverse systems does not exist. If it does, however, it is unique in a strong sense: given any two inverse limits ''X'' and ''X of an inverse system, there exists a ''unique''
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
''X''′ → ''X'' commuting with the projection maps.
Inverse systems and inverse limits in a category ''C'' admit an alternative description in terms of
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...
s. Any partially ordered set ''I'' can be considered as a
small category
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows as ...
where the morphisms consist of arrows ''i'' → ''j''
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
''i'' ≤ ''j''. An inverse system is then just a
contravariant functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
''I'' → ''C''. Let
be the category of these functors (with
natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a na ...
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
The direct limit, if it exists, is defined as a
right adjoint
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kn ...
of this trivial functor.
Examples
* The ring of
''p''-adic integers is the inverse limit of the rings
(see
modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his bo ...
) with the index set being the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s with the usual order, and the morphisms being "take remainder". That is, one considers sequences of integers
such that each element of the sequence "projects" down to the previous ones, namely, that
whenever