Direct limit

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.

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 set. Let {A_i | i ∈ I} be a family of objects indexed by I and suppose we have a family of homomorphisms f_ij: A_i → A_j for all i ≤ j with the following properties:

1. f_ii is the identity in A_i,
2. f_ik = f_jk o f_ij for all i ≤ j ≤ k.

Then the pair (A_i, f_ij) is called a direct system over I.

The underlying set of the direct limit, A, of the direct system (A_i, f_ij) is defined as the disjoint union of the A_i's modulo a certain equivalence relation ~:

Failed to parse (Missing texvc executable; please see math/README to configure.): \varinjlim A_i = \coprod_i A_i\bigg/\sim.

Here, if x_i is in A_i and x_j is in A_j, x_i ~ x_j if there is some k in I such that f_ik (x_i) = f_jk (x_j).

Heuristically, two elements in the disjoint union are equivalent if and only if they "eventually become equal" in the direct system. One naturally obtains from this definition canonical morphisms φ_i: A_i → A 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 (X_i, f_ij) 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 φ_i: X_i → X satisfying φ_i = φ_j O f_ij. The pair (X, φ_i) must be universal in the sense that for any other such pair (Y, ψ_i) there exists a unique morphism u: X → Y making all the "obvious" identities true; i.e., the diagram

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

Failed to parse (Missing texvc executable; please see math/README to configure.): X = \varinjlim X_i

with the direct system (X_i, f_ij) 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 another direct limit X′ there exists a unique isomorphism X′ → X commuting with the canonical morphisms.

We note that a direct system in a 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 i → j if and only if i ≤ j. A direct system is then just a covariant functor I → C.

Examples

• A collection of subsets Failed to parse (Missing texvc executable; please see math/README to configure.): M_i

of a set Failed to parse (Missing texvc executable; please see math/README to configure.): M
can be partially ordered by inclusion. Its limit is the union Failed to parse (Missing texvc executable; please see math/README to configure.): \bigcup M_i

.

• Let I be any directed set with a greatest element m. The direct limit of any corresponding direct system is isomorphic to X_m and the canonical morphism φ_m: X_m → X is an isomorphism.
• Let p be a prime number. Consider the direct system composed of the groups Z/pⁿZ and the homomorphisms Z/pⁿZ → Z/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 (U ≤ V if and only if U contains V). The corresponding direct system is (F(U), r_U,V) where r is the restriction map. The direct limit of this system is called the stalk of F at x, denoted F_x. The canonical morphisms F(U) → F_x 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.
• Inductive limits are linked to projective ones via

Failed to parse (Missing texvc executable; please see math/README to configure.): Hom (\varinjlim X_i, Y) = \varprojlim Hom (X_i, Y).

• Consider a sequence {A_n, φ_n} where A_n is a C*-algebra and φ_n : A_n → A_{n + 1} is a *-homomorphism. The C*-analog of the direct limit construction gives a C*-algebra satisfying the universal property above.

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.fr:Limite inductive de:Direkter Limes