Inverse limit - Citizendia

In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Inverse limits can be defined in any category. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships

1 Formal definition
- 1.1 Algebraic objects
- 1.2 General definition
2 Examples
3 Related concepts and generalizations
4 References

Formal definition

Algebraic objects

We start with the definition of an inverse (or projective) system of groups and homomorphisms. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function Let (I, ≤) be a directed poset (not all authors require I to be directed). In Mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement Let (A_i)_i∈I be a family 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 is the identity in A_i,
f_ik = f_ij o f_jk for all i ≤ j ≤ k.

Then the set of pairs (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:

$\varprojlim A_i = \Big\{(a_i) \in \prod_{i\in I}A_i \;\Big|\; a_i = f_{ij}(a_j) \mbox{ for all } i \leq j\Big\}.$

The inverse limit, A, comes equipped with natural projections π_i: A → A_i which pick out the ith component of the direct product. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, one can often define a direct product of objectsalready known giving a new one The inverse limit and the natural projections satisfy a universal property described in the next section. In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism

This same construction may be carried out if the A_i's are sets, rings, modules (over a fixed ring), algebras (over a fixed field), etc. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with , and the homomorphisms are homomorphisms in the corresponding category. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets 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. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism Let (X_i, f_ij) be an inverse system of objects and morphisms in a category C (same definition as above). In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and 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 commute for all i, j. In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also 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 might not exist in an arbitrary category. If it does, however, it is unique in a strong sense: given any other inverse limit X′ there exists a unique isomorphism X′ → X commuting with the projection maps. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

We note that an inverse system in a category C admits an alternative description in terms of functors. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories Any partially ordered set I can be considered as a small category where the morphisms consist of arrows i → j iff i ≤ j. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships ↔ An inverse system is then just a contravariant functor I → C. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories

Examples

The ring of p-adic integers is the inverse limit of the rings Z/pⁿZ (see modular arithmetic) with the index set being the natural numbers with the usual order, and the morphisms being "take remainder". In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an The natural topology on the p-adic integers is the same as the one described here.
Pro-finite groups are defined as inverse limits of (discrete) finite groups. In Mathematics, profinite groups are Topological groups that are in a certain sense assembled from Finite groups they share many properties with their finite
Let the index set I of an inverse system (X_i, f_ij) have a greatest element m. In Mathematics, especially in Order theory, the greatest element of a subset S of a Partially ordered set (poset is an element of S 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. In Mathematics, the category of topological spaces, often denoted Top, is the category whose objects are Topological spaces and whose In General topology and related areas of Mathematics, the initial topology ( projective topology or weak topology) on a set X This is known as the limit topology.
- The set of infinite strings is the inverse limit of the set of finite strings, and is thus endowed with the limit topology. In Computer programming and some branches of Mathematics, a string is an ordered Sequence of Symbols. As the original spaces are discrete, the limit space is totally disconnected. In Topology, a discrete space is a particularly simple example of a Topological space or similar structure one in which the points are " isolated " In Topology and related branches of Mathematics, a totally disconnected space is a Topological space which is maximally disconnected in the sense that This is one way of realizing the p-adic numbers and the Cantor set (as infinite strings). In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 In Mathematics, the Cantor set, introduced by German Mathematician Georg Cantor in 1883 (but discovered in 1875 by Henry John Stephen Smith
Let (I, =) be the trivial order (not directed). The inverse limit of any corresponding inverse system is just the product. In Category theory, the product of two (or more objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as
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. In Category theory, a branch of Mathematics, a pullback (also called a fibered product or Cartesian square) is the limit of a

Related concepts and generalizations

The categorical dual of an inverse limit is a direct limit (or inductive limit). In Category theory, an abstract branch of Mathematics, the dual category or opposite category C op of a category C is the In Mathematics, the direct limit (also called the inductive limit) is a general method of taking colimits of "directed families of objects" More general concepts are the limits and colimits of category theory. In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts The terminology is somewhat confusing: inverse limits are limits, while direct limits are colimits.

References

Bourbaki, Nicolas (1989), Algebra I, Springer, ISBN 978-3540642435
Bourbaki, Nicolas (1989), General topology: Chapters 1-4, Springer, ISBN 978-3540642411
Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed. Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition Saunders Mac Lane ( 4 August 1909, Taftville, Connecticut – 14 April 2005, San Francisco) was an American Categories for the Working Mathematician is a textbook in Category theory written by American Mathematician Saunders Mac Lane, who ), Springer, ISBN 0-387-98403-8

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