Editing Inverse limit (section)

=== Algebraic objects ===

We start with the definition of an '''inverse system''' (or projective system) of [[group (mathematics)|groups]] and [[group homomorphism|homomorphisms]]. Let <math>(I, \leq)</math> be a [[directed set|directed]] [[poset]] (not all authors require ''I'' to be directed). Let (''A''<sub>''i''</sub>)<sub>''i''∈''I''</sub> be a [[indexed family|family]] of groups and suppose we have a family of homomorphisms <math>f_{ij}: A_j \to A_i</math> for all <math>i \leq j</math> (note the order) with the following properties:
# <math>f_{ii}</math> is the identity on <math>A_i</math>,
# <math>f_{ik} = f_{ij} \circ f_{jk} \quad \text{for all } i \leq j \leq k.</math>
Then the pair <math>((A_i)_{i\in I}, (f_{ij})_{i\leq j\in I})</math> is called an inverse system of groups and morphisms over <math>I</math>, and the morphisms <math>f_{ij}</math> are called the transition morphisms of the system.

We define the '''inverse limit''' of the inverse system <math>((A_i)_{i\in I}, (f_{ij})_{i\leq j\in I})</math> as a particular [[subgroup]] of the [[direct product]] of the ''<math>A_i</math>''<nowiki/>'s:

:<math>A = \varprojlim_{i\in I}{A_i} = \left\{\left.\vec a \in \prod_{i\in I}A_i \;\right|\; a_i = f_{ij}(a_j) \text{ for all } i \leq j \text{ in } I\right\}.</math>

The inverse limit <math>A</math> comes equipped with ''natural projections'' {{math|{{pi}}<sub>''i''</sub>: ''A'' → ''A''<sub>''i''</sub>}} which pick out the {{math|''i''}}th component of the direct product for each <math>i</math> in <math>I</math>. 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 <math>A_i</math>'s are [[Set (mathematics)|sets]],<ref name="same-construction">John Rhodes & Benjamin Steinberg. The q-theory of Finite Semigroups. p. 133. {{ISBN|978-0-387-09780-0}}.</ref> [[semigroup]]s,<ref name="same-construction"/> [[topological space]]s,<ref name="same-construction"/> [[ring (mathematics)|rings]], [[module (mathematics)|modules]] (over a fixed ring), [[algebra over a field|algebras]] (over a fixed ring), etc., and the [[homomorphism]]s are morphisms in the corresponding [[category theory|category]]. The inverse limit will also belong to that category.