Inverse limit

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In abstract algebra, the inverse limit is a construction which allows to "glue together" several related objects; the precise matter of the glueing process being specified by morphisms between the objects. Inverse limits can be defined in any category, but we will initially only consider inverse limits of groups.

Consider a partially ordered set I, and assume that for every i in I we are given a group Ai, and for every pair of elements i, j with i > j, we are given a group homomorphism fi,j: Ai -> Aj. These homomorphisms are assumed to be compatible in the following sense: whenever i > j > k, then fi,k = fj,k o fi,j. We define the inverse limit, A, as the set of all families {ai}, where i ranges over I, we have ai in Ai for all i, and such that for every i > j, fi,j(ai) = aj. These families can be multiplied componentwise, and A is itself a group.

The inverse limit A together with the functions pj({ai}) = aj (the natural projections) has the following universal property: For every group B and every set of homomorphisms gi: B -> Ai such that for every i > j, gj = fi,j o gi there exists a unique homomorphism g: B -> A such that for every i, gi = pi o g.

This same construction may be carried out if the Ai are rings, algebras, fields, groups, modules or vector spaces, amongst other. The morphisms have to be morphisms in the corresponding category, and the inverse limit will then also belong to that category. The universal property mentionen above still holds; in fact, this universal property can be used to define inverse limits in every category. However, unlike in the categories mentioned above, in some categories inverse limits do not always exist.

If every structure Ai is finite, we can give A the product topology of discrete spaces. Since the rules describing an inverse limit are closed, A will be compact and Hausdorff in this case.


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