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Concept
Inverse limit
Summary
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 between the objects. Thus, inverse limits can be defined in any although their existence depends on the category that is considered. They are a special case of the concept of in category theory.
By working in the , that is by reverting the arrows, an inverse limit becomes a direct limit or inductive limit, and a limit becomes a colimit.
We start with the definition of an inverse system (or projective system) of groups and homomorphisms. Let be a directed poset (not all authors require I to be directed). Let (Ai)i∈I be a family 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 of the direct product of the 's:
The inverse limit comes equipped with natural projections pii: A → Ai which pick out the ith component of the direct product for each in . 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 's are sets, semigroups, topological spaces, rings, modules (over a fixed ring), algebras (over a fixed ring), etc., and the homomorphisms are morphisms in the corresponding . The inverse limit will also belong to that category.
The inverse limit can be defined abstractly in an arbitrary by means of a universal property. Let 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 pii: X → Xi (called projections) satisfying pii = ∘ pij for all i ≤ j.
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