Inverse limit

In abstract algebra, the inverse limit (also called projective limit) is a construction which allows to "glue together" several related objects; the precise matter of the gluing 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.

Table of contents

1 Definition for groups and universal property
2 Inverse limits in other categories
3 Examples
4 Related concepts and generalizations
5 See also

Definition for groups and 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.

The inverse limit A together with the homomorphisms p_i({a_k}) = a_i (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.

The inverse limit A of the system (A_i, f_i,j) is denoted by

This notation is somewhat unfortunate, since it suppresses the partial order on I, the crucially important connecting maps f_i,j, and the natural projections from the limit to the A_i.

Inverse limits in other categories

This same construction may be carried out if the A_i are sets, rings, algebras, fields, modules over the same ground ring or vector spaces over the same ground field, amongst others. The morphisms have to be morphisms in the corresponding category, and the inverse limit will then also belong to that category. The universal property mentioned above still holds in all these scenarios; in fact, this universal property can be used to define inverse limits abstractly in every category. In this way, inverse limits of topological spaces can also be defined. However, unlike in the categories mentioned, in some categories inverse limits do not always exist.

If every structure A_i is a topological space, then A is a subspace of the product topology and hence also a topological space. The universal property will still be satisfied if we require all maps to be continuous. If all A_i are compact and Hausdorff, then the inverse limit A will also be compact Hausdorff, since the rules describing an inverse limit are closed 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 are defined as inverse limits of finite groups.

Related concepts and generalizations

The categorical dual of an inverse limit is a direct limit. More general concepts are the limits and colimits of category theory. The terminology is somewhat confusing: inverse limits are limits, while direct limits are colimits.