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.
Formal definition
Algebraic objects
We start with the definition of an inverse system (or projective system) of groups and homomorphisms. Let (I, \leq) 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 f_{ij}: A_j \to A_i for all i \leq j (note the order) with the following properties:

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