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.
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 (Ai)i∈I be a family of groups and suppose we have a family of homomorphisms fij: Aj → Ai for all i ≤ j (note the order) with the following properties:
Then the pair ((Ai)i∈I, (fij)i≤ j∈I) is called an inverse system of groups and morphisms over I, and the morphisms fij are called the transition morphisms of the system.
We define the inverse limit of the inverse system ((Ai)i∈I, (fij)i≤ j∈I) as a particular subgroup of the direct product of the Ai's:
The inverse limit, A, comes equipped with natural projections πi: A → Ai which pick out the ith component of the direct product for each i in I. 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 Ai'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.
The inverse limit can be defined abstractly in an arbitrary category by means of a universal property. Let (Xi, fij) 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 πi: X → Xi (called projections) satisfying πi = fij o πj. The pair (X, πi) must be universal in the sense that for any other such pair (Y, ψi) there exists a unique morphism u: Y → X making all the "obvious" identities true; i.e., the diagram
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