In mathematics, an inverse limit (also called a projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Inverse limits can be defined in any category, although their existence depends on the category that is considered. They are a special case of the concept of a limit in category theory. By working in the dual category—that is, by reversing the arrows—an inverse limit becomes a direct limit or inductive limit, and a limit becomes a colimit.
Formal definition
Algebraic objects We start with the definition of an inverse system (or projective system) of groups and homomorphisms. Let ( I , ≤ ) {\displaystyle (I,\leq )} be a directed poset (not all authors require I to be directed). Let ( A i ) i ∈ I {\displaystyle (A_{i})_{i\in I}} be a family of groups and suppose we have a family of homomorphisms f i j : A j → A i {\displaystyle f_{ij}:A_{j}\to A_{i}} for all i ≤ j {\displaystyle i\leq j} (note the order) with the following properties:
f i i {\displaystyle f_{ii}} is the identity on A i {\displaystyle A_{i}} ,
f i k = f i j ∘ f j k {\displaystyle f_{ik}=f_{ij}\circ f_{jk}} for all i ≤ j ≤ k . {\displaystyle i\leq j\leq k.}
Then the pair ( ( A i ) i ∈ I , ( f i j ) i ≤ j ∈ I ) {\displaystyle ((A_{i})_{i\in I},(f_{ij})_{i\leq j\in I})} is called an inverse system of groups and morphisms over I {\displaystyle I} , and the morphisms f i j {\displaystyle f_{ij}} are called the transition morphisms of the system. The inverse limit of the inverse system ( ( A i ) i ∈ I , ( f i j ) i ≤ j ∈ I ) {\displaystyle ((A_{i})_{i\in I},(f_{ij})_{i\leq j\in I})} is the subgroup of the direct product of the A i {\displaystyle A_{i}} 's defined as
A = lim ← i ∈ I A i = { a → ∈ ∏ i ∈ I A i | a i = f i j ( a j ) for all i ≤ j in I } . {\displaystyle 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\}.}
The definition above of an inverse system implies, that A {\displaystyle A} is closed under pointwise multiplication, and therefore a group, since
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