Inverse limit

In mathematics, the inverse limit (also called the projective limit) is a construction which 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, but we will initially only consider inverse limits of groups.

Formal definition

Algebraic objects

We start with the definition of an inverse (or projective) system of groups and homomorphisms. Let (I, ≤) be a directed poset (not all authors require I to be directed). Let (A_i)_i∈I be a family of groups and suppose we have a family of homomorphisms f_ij : A_j → A_i for all i ≤ j (note the order) with the following properties:

1. f_ii is the identity in A_i,
2. f_ik = f_ij O f_jk for all i ≤ j ≤ k.

Then the set of pairs (A_i, f_ij) is called an inverse system of groups and morphisms over I.

We define the inverse limit of the inverse system (A_i, f_ij) as a particular subgroup of the direct product of the A_i's:

$\varprojlim A_{i}={\Big \{}(a_{i})\in \prod _{i\in I}A_{i}\;{\Big |}\;a_{i}=f_{ij}(a_{j}){\mbox{ for all }}i\leq j{\Big \}}$

The inverse limit, A, comes equipped with natural projections π_i : A → A_i which pick out the ith component of the direct product. The inverse limit and the natural projections satisfy a universal property described in the next section.

This same construction may be carried out if the A_i's are sets, rings, modules (over a fixed ring), algebras (over a fixed field), etc., and the homomorphisms are homomorphisms in the corresponding category. The inverse limit will also belong to that category.

General definition

The inverse limit can be defined abstractly in an arbitrary category by means of a universal property. Let (X_i, f_ij) be an inverse system of objects and morphisms in a category C (same definition as above). The inverse limit of this system is an object X in C together with morphisms π_i : X → X_i (called projections) satisfying π_i = f_ij O π_j . The pair (X, π_i) must be universal in the sense that for any other such pair (Y, ψ_i) there exists a unique morphism u : Y → X making all the "obvious" identities true; i.e. the diagram.

must commute for all i, j. The inverse limit is often denoted

$X=\varprojlim X_{i}$

with the inverse system (X_i, f_ij) being understood.

Unlike for algebraic objects, the inverse limit may not exist in an arbitrary category. If it does, however, it is unique in a strong sense: given any another inverse limit X′ there exists is a unique isomorphism X′ → X commuting with the projection maps.

We note that an inverse system in category C admits an alternative description in terms of functors. Any partially ordered set I can be considered as a small category where the morphisms consist of arrows i → j iff i ≤ j. An inverse system is then just a contravariant functor I → C.

Examples

• The ring of p-adic integers is the inverse limit of the rings Z/pⁿZ (see modular arithmetic) with the index 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 (discrete) finite groups.
• Let the index set I of an inverse system (X_i, f_ij) have a greatest element m. Then the natural projection π_m : X → X_m is an isomorphism.
• Inverse limits in the category of topological spaces are given by placing the initial topology on the underlying set-theoretic inverse limit. This is known as the limit topology.
• Let (I, =) be the trivial order (not directed). The inverse limit of any corresponding inverse system is just the product.
• Let I consist of three elements i, j, and k with i ≤ j and i ≤ k (not directed). The inverse limit of any corresponding inverse system is the pullback.

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 limits, while direct limits are colimits.