Direct limit

In mathematics, a direct limit (also called inductive limit) is a colimit of a "directed family 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 objects are understood to be sets with a given algebraic structure such as groups, rings, modules (over a fixed ring), algebras (over a fixed field), etc. With this in mind, homomorphisms are understood in the corresponding setting (group homomorphisms, etc.).

Start with the definition of a direct system of objects and homomorphisms. Let $\langle I,\le\rangle$ be a directed set. Let $\{A_i : i\in I\}$ be a family of objects indexed by $I\,$ and $f_{ij}: A_i \rightarrow A_j$ be a homomorphism for all $i \le j$ with the following properties:

1. $f_{ii}\,$ is the identity of $A_i\,$, and
2. $f_{ik}= f_{jk}\circ f_{ij}$ for all $i\le j\le k$.

Then the pair $\langle A_i,f_{ij}\rangle$ is called a direct system over $I\,$.

The underlying set of the direct limit, $A\,$, of the direct system $\langle A_i,f_{ij}\rangle$ is defined as the disjoint union of the $A_i\,$'s modulo a certain equivalence relation $\sim\,$:

$\varinjlim A_i = \bigsqcup_i A_i\bigg/\sim.$

Here, if $x_i\in A_i$ and $x_j\in A_j$, $x_i\sim\, x_j$ 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. An equivalent formulation that highlights the duality to the inverse limit is that an element is equivalent to all its images under the maps of the directed system, i.e. $x_i\sim\, f_{ik}(x_i)$.

One naturally obtains from this definition canonical morphisms $\phi_i: A_i\rightarrow A$ sending each element to its equivalence class. The algebraic operations on $A\,$ are defined via these maps in the obvious manner.

An important property is that taking direct limits in the category of modules is an exact functor.

Direct limit over a direct system in a category

The direct limit can be defined in an arbitrary category $\mathcal{C}$ by means of a universal property. Let $\langle X_i, f_{ij}\rangle$ be a direct system of objects and morphisms in $\mathcal{C}$ (same definition as above). The direct limit of this system is an object $X\,$ in $\mathcal{C}$ together with morphisms $\phi_i: X_i\rightarrow X$ satisfying $\phi_i =\phi_j \circ f_{ij}$. The pair $\langle X, \phi_i\rangle$ must be universal in the sense that for any other such pair $\langle Y, \psi_i\rangle$ there exists a unique morphism $u:X\rightarrow Y$ making the diagram

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

$X = \varinjlim X_i$

with the direct system $\langle X_i, f_{ij}\rangle$ 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 $\mathcal{C}$ admits an alternative description in terms of functors. Any directed poset $\langle I,\le \rangle$ can be considered as a small category $\mathcal{I}$ where the morphisms consist of arrows $i\rightarrow j$ if and only if $i\le j$. A direct system is then just a covariant functor $\mathcal{I}\rightarrow \mathcal{C}$.

General Definition

Let $\mathcal I$ and $\mathcal C$ categories. Let $c_X: \mathcal I\rightarrow \mathcal C$ be a constant functor with fixed object $X\in \mathcal C$. Define for every functor $F: \mathcal I\rightarrow \mathcal C$ the functor

$\lim_{\longrightarrow} F: \mathcal C \rightarrow \mathbf{Set}$

which assigns to each $X\in \mathcal C$ the set Hom(F,c_X) of natural transformations from F to c_X. If $\lim_{\longrightarrow} F$ is representable, the representing object in $\mathcal C$ is called the direct limit of F and is also denoted by $\lim_{\longrightarrow }F$.

If $\mathcal C$ is an abelian category where arbitrary (also infinite) direct sums of objects exists (this is Grothedieck's Axiom AB3). Then $\lim_{\longrightarrow} F$ is representable for every functor $F: \mathcal I\rightarrow \mathcal C$ and

$\lim_{\longrightarrow}: \mathrm{Hom}(\mathcal I, \mathcal C)\rightarrow \mathcal C, F\mapsto \lim_{\longrightarrow} F$

is a right-exact additive functor of abelian categories.

Examples

• A collection of subsets M_i of a set M can be partially ordered by inclusion. If the collection is directed, its direct limit is the union $\bigcup M_i$.
• 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 roots of unity of order some power of p, and is called the Prüfer group Z(p^∞).
• 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. For each neighborhood U of x, the canonical morphism F(U) → F_x associates to a section s of F over U an element s_x of the stalk F_x called the germ of s at x.
• 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

$\mathrm{Hom} (\varinjlim X_i, Y) = \varprojlim \mathrm{Hom} (X_i, Y).$

• 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.

References

• Bourbaki, Nicolas (1968), Elements of mathematics. Theory of sets, Translated from the French, Paris: Hermann, MR0237342 .
• Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics, 5 (2nd ed.), Springer-Verlag .