Wikipedia: Inverse limit

Difference (from prior author revision) (major diff, minor diff)

Changed: 1,26c1

Given a mapping A from a lattice to a set of algebraic structures
(sets, groups, algebras, rings etc.), and a mapping from each pair
of lattice elements i, j such that i>j to a morphism
f_i,j: A_i -> A_j such that
f_i,k = f_i,j o f_j,k, we define
the inverse limit, A, as the set of all mappings
a_i from the
lattice to the union of all A_j such that for every i
a_i is a member of A_i and such that for every
i > j, f_i,j(a<sub>i) = a_j.

The inverse limit together with the functions
f_i({a_j}) = a_i (the
projections) has the universal property that for every structure
B and a set of morphisms g_i: B -> A_i
such that for every i>j,
g_j = g_i o f_i,j'' there is a unique
morphism g: B -> A such that for every i,
g_i = g o f_i

Note that A is often the same kind of algebraic structure with the
operations defined element-wise: this holds for rings, algebras, fields,
groups and vector spaces, amongst other. If every structure is finite,
we can give A the product topology of discrete spaces. Since the
rules describing an inverse limit are closed, A will be compact and
Haussdorf in this case.

In abstract algebra, the inverse limit is a construction which allows to "glue together" several related objects; the precise matter of the glueing process being specified by morphisms between the objects. Inverse limits can be defined in any category, but we will initially only consider inverse limits of groups.

Changed: 28c3,5

Examples:

Consider a partially ordered set I, and assume that for every i in I we are given a group A_i, and for every pair of elements i, j with i > j, we are given a group homomorphism f_i,j: A_i -> A_j. These homomorphisms are assumed to be compatible in the following sense: whenever i > j > k, then f_i,k = f_j,k o f_i,j. We define
the inverse limit, A, as the set of all families {a_i}, where i ranges over I, we have a_i in A_i for all i, and such that for every
i > j, f_i,j(a_i) = a_j. These families can be multiplied componentwise, and A is itself a group.

Changed: 30c7,32

* Z_p, the p-adic numbers are an inverse limit of Z/pⁿ with the lattice being the natural numbers with the usual order, and the functions being "take reminder". The natural topology on the p-adic numbers is the same as the one described here.

The inverse limit A together with the functions
p_j({a_i}) = a_j (the
natural projections) has the following universal property: For every group
B and every set of homomorphisms g_i: B -> A_i
such that for every i > j,
g_j = f_i,j o g_i there exists a unique
homomorphism g: B -> A such that for every i,
g_i = p_i o g.

This same construction may be carried out if the A_i are rings, algebras, fields,
groups, modules or vector spaces, amongst other. The morphisms have to be morphisms in the corresponding category, and the inverse limit will then also belong to that category. The universal property mentionen above still holds; in fact, this universal property can be used to define inverse limits in every category. However, unlike in the categories mentioned above, in some categories inverse limits do not always exist.

If every structure A_i is finite,
we can give A the product topology of discrete spaces. Since the
rules describing an inverse limit are closed, A will be compact and
Hausdorff in this case.

Examples:

* The ring of p-adic integers is the inverse limit of the rings Z/pⁿ (see modular arithmetic) with the partially ordered set being the natural numbers with the usual order, and the morphisms being "take remainder". The natural topology on the p-adic integers is the same as the one described here.
* Pro-finite groups

See also:
* [Direct limit]?
* Category theory
* Universal property

Consider a partially ordered set I, and assume that for every i in I we are given a group A_i, and for every pair of elements i, j with i > j, we are given a group homomorphism f_i,j: A_i -> A_j. These homomorphisms are assumed to be compatible in the following sense: whenever i > j > k, then f_i,k = f_j,k o f_i,j. We define the inverse limit, A, as the set of all families {a_i}, where i ranges over I, we have a_i in A_i for all i, and such that for every i > j, f_i,j(a_i) = a_j. These families can be multiplied componentwise, and A is itself a group.

This same construction may be carried out if the A_i are rings, algebras, fields, groups, modules or vector spaces, amongst other. The morphisms have to be morphisms in the corresponding category, and the inverse limit will then also belong to that category. The universal property mentionen above still holds; in fact, this universal property can be used to define inverse limits in every category. However, unlike in the categories mentioned above, in some categories inverse limits do not always exist.

If every structure A_i is finite, we can give A the product topology of discrete spaces. Since the rules describing an inverse limit are closed, A will be compact and Hausdorff in this case.

Examples:

The ring of p-adic integers is the inverse limit of the rings Z/pⁿ (see modular arithmetic) with the partially ordered set being the natural numbers with the usual order, and the morphisms being "take remainder". The natural topology on the p-adic integers is the same as the one described here.
Pro-finite groups

See also: