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 direc...
more
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 set of pairs (Ai, fij) 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, fij) 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...
less
