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In mathematics, the inverse limit (also called the 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. Thus, 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 limit in category theory. By working in the dual category, that is by reverting 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 (Ai)i∈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 for all  i ≤ j ≤ k . {\displaystyle f_{ik}=f_{ij}\circ f_{jk}\quad {\text{for all }}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. We define 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})} as a particular subgroup of the direct product of the A i {\displaystyle A_{i}} 's: 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 inverse limit A {\displaystyle A} comes equipped with natural projections πi: A → Ai which pick out the ith component of the direct product for each i {\displaystyle i} in I {\displaystyle I} . 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 {\displaystyle A_{i}} 's are sets, semigroups, topological spaces, rings, modules (over a fixed ring), algebras (over a fixed ring), etc., and the homomorphisms are morphisms in the corresponding category. The inverse limit will also belong to that category. General definition The inverse limit can be defined abstractly in an arbitrary category by means of a universal property. Let ( X i , f i j ) {\textstyle (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 πi: X → Xi (called projections) satisfying πi = f i j {\displaystyle f_{ij}} ∘ πj for all i ≤ j. Th.... Discover the Jk Roos Jr popular books. Find the top 100 most popular Jk Roos Jr books.