In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
between the objects. Thus, inverse limits can be defined in any
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
although their existence depends on the category that is considered. They are a special case of the concept of
limit in category theory.
By working in the
dual category
In category theory, a branch of mathematics, the opposite category or dual category C^ of a given Category (mathematics), category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal ...
, that is by reversing 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 cate ...
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 s ...
.
Formal definition
Algebraic objects
We start with the definition of an inverse system (or projective system) of
groups and
homomorphisms. Let
be a
directed
Direct may refer to:
Mathematics
* Directed set, in order theory
* Direct limit of (pre), sheaves
* Direct sum of modules, a construction in abstract algebra which combines several vector spaces
Computing
* Direct access (disambiguation), a ...
poset
In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
(not all authors require ''I'' to be directed). Let (''A''
''i'')
''i''∈''I'' be a
family
Family (from ) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). It forms the basis for social order. Ideally, families offer predictabili ...
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, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation  ...
of the
direct product
In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abs ...
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 fro ...
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. .] semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
s,
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s,
rings,
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 addition ...
(over a fixed ring), etc., and the
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
s are morphisms in the corresponding
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
. 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:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
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 fro ...
. Let
be an inverse system of objects and
morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
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
and the canonical projections
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 or morphism 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 the ...
''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 Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
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 asso ...
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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
''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 natur ...
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 inverse 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 k ...
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 operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to mo ...
) with the index set being the
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
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