Table of contents |
2 Inverse limits in other categories 3 Examples 4 Related concepts and generalizations 5 See also |
Consider a partially ordered set I, and assume that for every i in I we are given a group A_{i}, and for every pair of elements i, j with i > j, we are given a group homomorphism f_{i,j}: A_{i} -> A_{j}. These homomorphisms are assumed to be compatible in the following sense: whenever i > j > k, then f_{i,k} = f_{j,k} o f_{i,j}. We define the inverse limit, A, as the set of all families {a_{i}}, where i ranges over I, we have a_{i} in A_{i} for all i, and such that for every i > j, f_{i,j}(a_{i}) = a_{j}. These families can be multiplied componentwise, and A is itself a group.
The inverse limit A together with the homomorphisms p_{i}({a_{k}}) = a_{i} (the natural projections) has the following universal property: For every group B and every set of homomorphisms g_{i}: B -> A_{i} such that for every i > j, g_{j} = f_{i,j} o g_{i}, there exists a unique homomorphism g: B -> A such that for every i, g_{i} = p_{i} o g.
The inverse limit A of the system (A_{i}, f_{i,j}) is denoted by
This same construction may be carried out if the A_{i} are sets, rings, algebras, fields, modules over the same ground ring or vector spaces over the same ground field, amongst others. The morphisms have to be morphisms in the corresponding category, and the inverse limit will then also belong to that category. The universal property mentioned above still holds in all these scenarios; in fact, this universal property can be used to define inverse limits abstractly in every category. In this way, inverse limits of topological spaces can also be defined. However, unlike in the categories mentioned, in some categories inverse limits do not always exist.
If every structure A_{i} is a topological space, then A is a subspace of the product topology and hence also a topological space. The universal property will still be satisfied if we require all maps to be continuous. If all A_{i} are compact and Hausdorff, then the inverse limit A will also be compact Hausdorff, since the rules describing an inverse limit are closed in this case.
The categorical dual of an inverse limit is a direct limit. More general concepts are the limits and colimits of category theory. The terminology is somewhat confusing: inverse limits are limits, while direct limits are colimits.