Inverse limit

In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to "glue together" several related objects, the precise matter of the gluing process being specified by morphisms between the objects. Inverse limits can be defined in any category (mathematics), but we will initially only consider inverse limits of group (mathematics). == Formal definition == === Algebraic objects === We start with the definition of an inverse system of group (mathematics) and group homomorphism. Let (''I'', ≤) be a directed set poset (not all authors require ''I'' to be directed). Let (''A''_''i'')_{''i''∈''I''} be a family (mathematics) of groups and suppose we have a family of homomorphisms ''f''_''ij'' : ''A''_''j'' → ''A''_''i'' for all ''i'' ≤ ''j'' (note the order) with the following properties: # ''f''_''ii''(''x'') = ''x'' for all ''x'' ∈ ''A''_''i'', # ''f''_''ik'' = ''f''_''ij'' O ''f''_''jk'' for all ''i'' ≤ ''j'' ≤ ''k''. Then the pair (''A''_''i'', ''f''_''ij'') is called an inverse system of groups and morphisms over ''I''. We define the inverse limit of the inverse system (''A''_''i'', ''f''_''ij'') as a particular subgroup of the direct product of the ''A''_''i'''s: :

\varprojlim A_i = \left\{(a_i) \in \prod_{i\in I}A_i \;\bigg|\; a_i = f_{ij}(a_j) \mbox{ for all } i \leq j\right\}

The inverse limit, ''A'', comes equipped with ''natural projections'' π_''i'' : ''A'' → ''A''_''i'' which pick out the ''i''th component of the direct product. 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 ''A''_''i'''s are sets, ring (mathematics), module (mathematics) (over a fixed ring), algebra over a field (over a fixed field), etc., and the homomorphisms are homomorphisms in the corresponding category theory. The inverse limit will also belong to that category. === General definition === The inverse limit can be defined abstractly in an arbitrary category (mathematics) by means of a universal property. Let (''X''_''i'', ''f''_''ij'') be an inverse system of objects and morphisms in a category ''C'' (same definition as above). The inverse limit of this system is an object ''X'' in ''C'' together with morphisms π_''i'' : ''X'' → ''X''_''i'' (called ''projections'') satisfying π_''i'' = ''f''_''ij'' O π_''j'' . The pair (''X'', π_''i'') must be universal in the sense that for any other such pair (''Y'', ψ_''i'') there exists a unique morphism ''u'' : ''Y'' → ''X'' making all the "obvious" identities true; i.e. the diagram.

must commutative diagram for all ''i'', ''j''. The inverse limit is often denoted :

X = \varprojlim X_i

with the inverse system (''X''_''i'', ''f''_''ij'') being understood. Unlike for algebraic objects, the inverse limit may not exist in an arbitrary category. If it does, however, it is unique in a strong sense: given any another inverse limit ''X''′ there exists is a ''unique'' isomorphism ''X''′ → ''X'' commuting with the projection maps. We note that an inverse system in category ''C'' admits an alternative description in terms of functors. Any partially ordered set ''I'' can be considered as a small category where the morphisms consist of arrows ''i'' → ''j'' iff ''i'' ≤ ''j''. An inverse system is then just a contravariant functor ''I'' → ''C''. == Examples == * The ring of p-adic numbers is the inverse limit of the rings Z/''p''^''n''Z (see modular arithmetic) with the index set being the natural numbers with the usual order, and the morphisms being "take remainder". The natural topology on the ''p''-adic integers is the same as the one described here. * Pro-finite group are defined as inverse limits of finite groups. * Let the index set ''I'' of an inverse system (''X''_''i'', ''f''_''ij'') have a greatest element ''m''. Then the natural projection π_''m'' : ''X'' → ''X''_''m'' is an isomorphism. * Inverse limits in the category of topological spaces are given by placing the initial topology on the underlying set-theoretic inverse limit. * Let (''I'', =) be the trivial order (not directed). The inverse limit of any corresponding inverse system is just the product (category theory). * Let ''I'' consist of three elements ''i'', ''j'', and ''k'' with ''i'' ≤ ''j'' and ''i'' ≤ ''k'' (not directed). The inverse limit of any corresponding inverse system is the pullback (category theory). == Related concepts and generalizations == The dual (category theory) of an inverse limit is a direct limit (or inductive limit). More general concepts are the limit (category theory) of category theory. The terminology is somewhat confusing: inverse limits are limits, while direct limits are colimits. Category theory Abstract algebra

Inverse limit

There should be some mention of the derived functor,

{\varprojlim}^{1}

. Likewise for the direct limit, if somebody happens to know how that works.

See other meanings of words starting from letter:

I

IA | IB | IC | ID | IE | IF | IG | IH | IJ | IK | IL | IM | IN | IO | IP | IR | IS | IT | IU | IW | IX | IY | IZ |

Words begining with Inverse_limit:

Inverse_limit
Inverse_limit