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Inverse limit


In mathematics, the inverse limit (also called the projective limit or limit) is a construction that 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 system (or projective system) of groups and homomorphisms. Let (I, ≤) be a directed poset (not all authors require I to be directed). Let (Ai)iI be a family of groups and suppose we have a family of homomorphisms fij: AjAi for all ij (note the order), called bonding maps, with the following properties:

Then the pair ((Ai)iI, (fij)ijI) 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)iI, (fij)ijI) as a particular subgroup of the direct product of the Ai's:

The inverse limit, A, comes equipped with natural projections πi: AAi 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, semigroups, topological spaces,rings, modules (over a fixed ring), algebras (over a fixed ring), etc., and the homomorphisms are morphisms in the corresponding category. The inverse limit will also belong to that category.


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