Direct limit

From Academic Kids

In mathematics, the direct limit (also called the inductive limit) is a general method of taking limits of "directed families of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.

Contents

Formal definition

Algebraic objects

In this section we will understand objects to be sets with a given algebraic structure such as groups, rings, modules (over a fixed ring), algebras (over a fixed field), etc. We will also understand homomorphisms in the corresponding setting (group homomorphisms, etc.).

We start with the definition of a direct system of objects and homomorphisms. Let (I, ≤) be a directed poset. Let {Ai | iI} be a family of objects indexed by I and suppose we have a family of homomorphisms fij : AiAj for all ij with the following properties:

  1. fii(x) = x for all xAi,
  2. fik = fjk O fij for all ijk.

Then the pair (Ai, fij) is called a direct system over I.

The direct limit, A, of the direct system (Ai, fij) is defined as the disjoint union of the Ai's modulo a certain equivalence relation:

<math>\varinjlim A_i = \left(\coprod_i A_i\right)\bigg/\{x_i \sim x_j \mid \mbox{there exists } k\in I \mbox{ such that } f_{ik}(x_i) = f_{jk}(x_j)\}<math>

Heuristically, two elements in the disjoint union are equivalent iff they "eventually become equal" in the direct system. One naturally obtains from this definition canonical morphisms φi : AiA sending each element to its equivalence class. The algebraic operations on A are defined via these maps in the obvious manner.

General definition

The direct limit can be defined abstractly in an arbitrary category by means of a universal property. Let (Xi, fij) be a direct system of objects and morphisms in a category C (same definition as above). The direct limit of this system is an object X in C together with morphisms φi : XiX satisfying φi = φj O fij. The pair (X, φi) must be universal in the sense that for any other such pair (Y, ψi) there exists a unique morphism u : XY making all the "obvious" identities true; i.e. the diagram.

Missing image
DirectLimit-01.png
Image:DirectLimit-01.png

must commute for all i, j. The direct limit is often denoted

<math>X = \varinjlim X_i<math>

with the direct system (Xi, fij) being understood.

Unlike for algebraic objects, the direct limit may not exist in an arbitrary category. If it does, however, it is unique in a strong sense: given any another direct limit X' there exists is a unique isomorphism X' → X commuting with the canonical morphisms.

We note that a direct system in category C admits an alternative description in terms of functors. Any directed poset I can be considered as a small category where the morphisms consist of arrows ij iff ij. A direct system is then just a covariant functor IC.

Examples

  • Let I be any directed poset with a greatest element m. The direct limit of any corresponding direct system is isomorphic to Xm and the canonical morphism φm : XmX is an isomorphism.
  • Let p be a prime number. Consider the direct system composed of the groups Z/pnZ and the homomorphisms Z/pnZZ/pn+1Z which are induced by multiplication by p. The direct limit of this system consists of all the pnth roots of unity, and is called the p-group.
  • Let F be a C-valued sheaf on a topological space X. Fix a point x in X. The open neighborhoods of x form a directed poset ordered by inclusion (UV iff U contains V). The corresponding direct system is (F(U), rU,V) where r is the restriction map. The direct limit of this system is called the stalk of F at x, denoted Fx. The canonical morphisms F(U) → Fx are called germs.
  • Direct limits in the category of topological spaces are given by placing the final topology on the underlying set-theoretic direct limit.

Related constructions and generalizations

The categorical dual of the direct limit is called the inverse limit (or projective limit). More general concepts are the limits and colimits of category theory. The terminology is somewhat confusing: direct limits are colimits while inverse limits are limits.

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