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 poset. 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(x) = x for all x ∈ 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 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:

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 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" identites true; i.e. the diagram

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

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 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 desription in terms of functors. Any directed poset I can be considered as a small category where the morphisms consist of arrows i → j iff i ≤ j. A direct system is then just a covariant functor I → C.

Examples

• Let I be any directed poset 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 iff 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.

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.