Editing Inverse limit (section)

== Formal definition ==

=== 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.

=== General definition ===

The inverse limit can be defined abstractly in an arbitrary [[category (mathematics)|category]] by means of a [[universal property]]. Let <math display=inline> (X_i, f_{ij})</math> be an inverse system of objects and [[morphism]]s  in a category ''C'' (same definition as above). The '''inverse limit''' of this system is an object ''X'' in ''C'' together with morphisms {{pi}}<sub>''i''</sub>: ''X'' → ''X''<sub>''i''</sub> (called ''projections'') satisfying {{pi}}<sub>''i''</sub> = <math>f_{ij}</math> ∘ {{pi}}<sub>''j''</sub> for all ''i'' ≤ ''j''. The pair (''X'', {{pi}}<sub>''i''</sub>)  must be universal in the sense that for any other such pair (''Y'', ψ<sub>''i''</sub>) there exists a unique morphism ''u'': ''Y'' → ''X'' such that the diagram

<div style="text-align: center;">[[File:InverseLimit-01.svg|175px|class=skin-invert]]</div>

[[commutative diagram|commutes]] for all ''i'' ≤ ''j''. The inverse limit is often denoted
:<math>X = \varprojlim X_i</math>
with the inverse system <math display=inline>(X_i, f_{ij})</math> and the canonical projections <math>\pi_i</math> being understood.

In some categories, the inverse limit of certain inverse systems does not exist. If it does, however, it is unique in a strong sense: given any two inverse limits ''X'' and ''X''' of an inverse system, there exists a ''unique'' [[isomorphism]] ''X''&prime; → ''X'' commuting with the projection maps.

Inverse systems and inverse limits in a category ''C'' admit an alternative description in terms of [[functor]]s. Any partially ordered set ''I'' can be considered as a [[small category]] where the morphisms consist of arrows ''i'' → ''j'' [[if and only if]] ''i'' ≤ ''j''. An inverse system is then just a [[contravariant functor]] ''I'' → ''C''. Let <math>C^{I^\mathrm{op}}</math> be the category of these functors (with [[natural transformation]]s as morphisms). An object ''X'' of ''C'' can be considered a trivial inverse system, where all objects are equal to ''X'' and all arrow are the identity of ''X''. This defines a "trivial functor" from ''C'' to <math>C^{I^\mathrm{op}}.</math> The inverse limit, if it exists, is defined as a [[right adjoint]] of this trivial functor.