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Direct limit

In , a direct limit (also known as an inductive limit) is a colimit taken over a directed , consisting of a directed poset I indexing objects A_i in a category \mathcal{C} together with morphisms \phi_{ij}: A_i \to A_j for i \leq j that satisfy compatibility conditions, such that the direct limit \varinjlim_{i \in I} A_i is an object equipped with morphisms \psi_i: A_i \to \varinjlim A_i forming a cocone—meaning it is among all such cocones over the . This construction generalizes intuitive notions like unions of nested sets or localization of rings, providing a way to "glue" compatible structures along a directed . The direct limit is dual to the (or projective limit), which instead involves cones over systems; while often correspond to "intersections," direct limits emphasize "unions" or extensions. In the , the direct limit of a directed system \{X_i, f_{ij}\} is explicitly constructed as the quotient of the \bigsqcup_{i \in I} X_i by the generated by identifying x \in X_i with f_{ij}(x) \in X_j for all i \leq j, yielding a set with inclusion maps from each X_i. For example, the rational numbers \mathbb{Q} arise as the direct limit of the localizations \mathbb{Z}[1/m] over all positive integers m, ordered by divisibility (with transition maps the natural into larger localizations), under localization maps, and the integers localized at powers of a prime p form \varinjlim \mathbb{Z}[p^{-n}]. Direct limits are ubiquitous in algebra and beyond, enabling the study of infinite constructions such as profinite groups (via limits, but with direct limits in quotients) and schemes in , where they facilitate gluing of affine schemes along covers. They exist in many categories like abelian groups, modules over a ring, and topological spaces (as inductive limits of embeddings), though their computation often relies on the category's completeness or cocompleteness properties. In filtered categories—where every pair of objects has a common upper bound—direct limits coincide with filtered colimits, a notion central to proving exactness properties in , such as the preservation of exact sequences under certain direct limits.

Background concepts

Directed sets

A directed set, also known as a directed poset, is a (I, \leq) in which every finite nonempty subset has an upper bound in I. That is, for any finite collection i_1, \dots, i_n \in I, there exists some k \in I such that i_j \leq k for all j = 1, \dots, n. This structure generalizes linearly ordered index sets like the natural numbers, providing a flexible framework for indexing families of mathematical objects while ensuring a form of coherence across finite collections. Classic examples include the set of natural numbers \mathbb{N} equipped with the usual order \leq, where the upper bound for any finite subset \{m_1, \dots, m_n\} is simply \max\{m_1, \dots, m_n\}. Another is the collection of all finite subsets of an arbitrary set X, ordered by inclusion (\subseteq), since the union of any finite number of finite subsets remains finite and serves as an upper bound. The directedness condition is essential for enabling "eventual" compatibility in indexed systems, such as direct systems of algebraic structures or topological spaces, where it guarantees that for any of indices, there is a common "later" index through which all relevant transition maps can factor, allowing the system to stabilize or cohere beyond any finite stage. In , directed sets are foundational to concepts like and . A filter in a poset is an upward-directed (nonempty, closed under finite meets, and upward closed), while an ideal is the order dual: a downward-directed lower set (nonempty, closed under finite joins, and downward closed). This duality highlights how directed sets underpin closure properties in and theory.

Direct systems

A direct system, also known as an inductive system, consists of a collection of objects and s in a arranged according to a directed poset, providing the foundational structure for constructing direct limits. Formally, let I be a directed poset, viewed as a small where objects are elements of I and there is a unique i \to j i \leq j. A direct system indexed by I in a \mathcal{C} is a covariant F: I \to \mathcal{C}. Equivalently, it comprises objects A_i = F(i) in \mathcal{C} for each i \in I, together with s f_{ij}: A_i \to A_j in \mathcal{C} whenever i \leq j, satisfying the compatibility conditions: f_{ii} = \mathrm{id}_{A_i} for all i \in I, and f_{jk} \circ f_{ij} = f_{ik} whenever i \leq j \leq k. This second formulation emphasizes the diagram-theoretic perspective, where the morphisms form a compatible family over the poset order. The compatibility condition ensures transitivity along the directed order, which can be visualized via the following commutative diagram for i \leq j \leq k: \begin{CD} A_i @>f_{ij}>> A_j @>f_{jk}>> A_k \\ @V{f_{ik}}VV @. @| \\ A_i @= A_i @>>f_{ik}> A_k \end{CD} Here, the upper path f_{jk} \circ f_{ij} equals the direct morphism f_{ik}, guaranteeing consistency in the system's structure. Verification involves checking that the identities hold for the indexing poset and that composition in \mathcal{C} preserves the order relations. A example arises in the category of groups \mathbf{Grp}, where an increasing of subgroups H_0 \leq H_1 \leq H_2 \leq \cdots of a fixed group G, indexed by the natural numbers under the usual , forms a system via the morphisms f_{mn}: H_m \hookrightarrow H_n for m \leq n. This setup illustrates how systems capture ascending unions in algebraic structures. Direct systems indexed by different but related posets may yield equivalent structures. Specifically, if J \subseteq I is a cofinal —meaning that for every i \in I there exists j \in J with i \leq j—then the subsystem induced by J produces a direct system isomorphic to the original one, in the sense that their colimits coincide up to unique . This allows simplification by passing to cofinal subposets without altering the resulting object.

Formal definition

Direct limits of algebraic structures

In the category of abelian groups, the direct limit of a direct system (A_i, f_{ij})_{i \in I}, where I is a and each f_{ij}: A_i \to A_j for i \leq j is a satisfying the compatibility conditions f_{ii} = \mathrm{id}_{A_i} and f_{jk} \circ f_{ij} = f_{ik} for i \leq j \leq k, is constructed explicitly as a quotient group. Consider the direct sum \bigoplus_{i \in I} A_i, which consists of all families (a_i)_{i \in I} with a_i \in A_i and only finitely many a_i \neq 0. Let N be the subgroup generated by all elements of the form a_i - f_{ij}(a_i) for i \leq j and a_i \in A_i. The direct limit is then the quotient group \lim_{\to} A_i = \left( \bigoplus_{i \in I} A_i \right) / N, where the equivalence class of (a_i)_{i \in I} is denoted [(a_i)_{i \in I}]. The canonical maps \iota_i: A_i \to \lim_{\to} A_i are induced by the inclusions into the direct sum followed by the quotient map, satisfying \iota_j \circ f_{ij} = \iota_i for i \leq j, which verifies compatibility with the direct system. This construction extends naturally to modules over a fixed ring R. For a direct system of R-modules (M_i, \phi_{ij})_{i \in I}, the direct limit \lim_{\to} M_i is the quotient of the direct sum \bigoplus_{i \in I} M_i by the R-submodule K generated by elements m_i - \phi_{ij}(m_i) for i \leq j and m_i \in M_i, with R-linearity preserved in the quotient. The induced maps \psi_i: M_i \to \lim_{\to} M_i are R-module homomorphisms compatible with the system transitions \phi_{ij}. For rings, the direct limit of a direct system of rings (R_i, g_{ij})_{i \in I} (where each g_{ij}: R_i \to R_j is a ring homomorphism) is formed first as the direct limit in the category of abelian groups under addition, yielding an abelian group L = \lim_{\to} R_i. Multiplication on L is defined by _i _j = [g_{i k}(x) g_{j k}(y)]_k for some k \geq i, j, which is well-defined and associative due to the system compatibilities, making L into a ring with induced unital ring homomorphisms from each R_i. A concrete example arises from the direct system of cyclic groups \mathbb{Z}/n! \mathbb{Z} for n \in \mathbb{N}, with transition maps f_{m n}: \mathbb{Z}/m! \mathbb{Z} \to \mathbb{Z}/n! \mathbb{Z} for m \leq n given by multiplication by (n!)/ (m!), which is an . The direct limit is isomorphic to the Prüfer quasi-cyclic group \mathbb{Q}/\mathbb{Z}, as every torsion element in \mathbb{Q}/\mathbb{Z} has order dividing some n!, and the maps embed these subgroups compatibly. In , direct limits of rings appear prominently in localizations. For a R and multiplicative subset S \subseteq R, the localization S^{-1} R is the direct limit of the system indexed by finite products of elements from S, with maps sending r \in R to (r, s)/s in formal fractions, yielding the standard fraction ring structure.

Direct limits in categories

In , the direct limit of a direct system is formalized as a colimit. Specifically, given a directed poset I viewed as a small (with objects the elements of I and morphisms the order relations i \leq j), and a F: I \to \mathcal{C} from I to an arbitrary category \mathcal{C}, the direct limit \varinjlim F is defined to be the colimit \varinjlim_I F of the diagram F. The colimit \varinjlim F is an object L in \mathcal{C} equipped with a family of morphisms \psi_i: F(i) \to L for each i \in I, forming a cocone over the diagram F, such that the following compatibility condition holds: for every morphism f: i \to j in I (i.e., i \leq j), the diagram \begin{tikzcd} F(i) \arrow[r, "F(f)"] \arrow[d, "\psi_i"] & F(j) \arrow[d, "\psi_j"] \\ L \arrow[rr, equal] & & L \end{tikzcd} commutes, meaning \psi_j \circ F(f) = \psi_i. This cocone is universal in the sense that for any other object L' in \mathcal{C} with morphisms \psi'_i: F(i) \to L' satisfying the same compatibility, there exists a unique morphism \phi: L \to L' such that \psi'_i = \phi \circ \psi_i for all i \in I. The existence of direct limits is not guaranteed in every \mathcal{C}; it depends on the properties of \mathcal{C} and the directed poset I. However, direct limits exist in many commonly studied categories, such as the \mathbf{Set} and the category of abelian groups \mathbf{Ab}, where they can be constructed explicitly.

Construction and universal property

Explicit constructions

In the category of sets, the direct limit of a direct system (A_i, f_{ij})_{i \in I} over a I is constructed explicitly as the of the \coprod_{i \in I} A_i by the smallest \sim such that a \sim f_{ij}(a) for all a \in A_i and i \leq j. The canonical maps \iota_i: A_i \to \lim_{\to} A_i send each element to its equivalence class, and this satisfies the universal property of the direct limit. This construction extends to the category of modules over a ring R, where the direct limit is the of the direct sum \bigoplus_{i \in I} M_i by the submodule generated by elements of the form \iota_i(m) - \iota_j(f_{ij}(m)) for m \in M_i and i \leq j, with \iota_i the canonical inclusions. For topological spaces, the direct limit of a direct system (X_i, f_{ij}) is formed by equipping the set-theoretic direct limit \lim_{\to} X_i with the finest (also known as the direct limit topology or inductive limit topology) such that each \iota_i: X_i \to \lim_{\to} X_i is continuous. A set U \subseteq \lim_{\to} X_i is open if and only if \iota_i^{-1}(U) is open in X_i for every i \in I. This topology ensures that the direct limit satisfies the universal property in the . In the category of schemes, particularly for direct systems of affine schemes with affine transition morphisms, the direct limit is constructed as the spectrum of the direct limit of the corresponding rings of sections. Specifically, if (S_i \to S, f_{ij}) is a direct system of affine schemes over a base scheme S indexed by a I, with S_i = \Spec R_i, then the direct limit S = \lim_{\to} S_i is affine and isomorphic to \Spec(\colim_{i \in I} R_i), where the colimit is taken in the . The structure sheaf on S is induced accordingly, ensuring compatibility with the transition maps. Direct limits do not exist in every . For instance, in the category of finite sets, consider a direct system given by strictly increasing inclusions of finite sets A_1 \subset A_2 \subset \cdots whose union is countably infinite; the set-theoretic direct limit would be infinite and thus not an object in the category, so no direct limit exists. To compute the direct limit of a direct system of chain complexes (C^i_*, f^i_{j*})_{i \in I} in the category of chain complexes over an (such as abelian groups), proceed degreewise as follows: for each degree n, form the direct limit \lim_{\to} C^i_n using the restrictions of the transition maps f^i_{j,n}: C^i_n \to C^j_n; this yields the nth term of the limiting complex C_* = \lim_{\to} C^i_*. Define the differential d_n: C_n \to C_{n-1} on the equivalence classes by _n \mapsto [d^i_n(c)]_{n-1} for a representative c \in C^i_n, which is well-defined because the direct system maps commute with the differentials d^i_* and direct limits in abelian categories preserve the necessary commutativity. Verify that d_{n-1} \circ d_n = 0 in the limit, as it holds in each C^i_* and passes to the quotient. This componentwise construction yields the direct limit complex, which satisfies the universal property in the category of chain complexes.

Universal property

The direct limit of a direct system \{A_i, f_{ij}\}_{i \in I} in a \mathcal{C}, denoted \lim_{\to} A_i with canonical morphisms \psi_i: A_i \to \lim_{\to} A_i, satisfies the following : for any object B in \mathcal{C} and any family of morphisms \phi_i: A_i \to B compatible with the direct system (i.e., \phi_j \circ f_{ij} = \phi_i whenever i \leq j), there exists a unique morphism \lambda: \lim_{\to} A_i \to B such that the diagrams commute, meaning \phi_i = \lambda \circ \psi_i for all i \in I. To see that this property holds, assume an explicit construction of the direct limit as a object (such as the \coprod_{i \in I} A_i modulo the generated by the transition maps f_{ij}), with \psi_i as the induced quotient maps; then \lambda is constructed by defining it on representatives from each A_i via the \phi_i and extending by with the relations, which is well-defined due to the directedness ensuring any two representatives can be connected through a common upper bound. Uniqueness follows because any two such maps \lambda, \lambda' must agree on the images of all \psi_i (by the \phi_i = \lambda \circ \psi_i = \lambda' \circ \psi_i), and the \psi_i jointly generate \lim_{\to} A_i as a colimit. This universal property is illustrated by the commuting triangles for each i \in I: \begin{CD} A_i @>{\psi_i}>> \lim_{\to} A_i \\ @V{\phi_i}VV @VV{\lambda}V \\ B \end{CD} along with the compatibility squares for transition maps: \begin{CD} A_i @>{f_{ij}}>> A_j @>{\psi_j}>> \lim_{\to} A_i \\ @V{\phi_i}VV @VV{\phi_j}V @VV{\lambda}V \\ B @= B \end{CD} where \phi_j \circ f_{ij} = \phi_i and \lambda \circ \psi_j = \phi_j, ensuring the entire diagram over the I commutes uniquely through \lambda. As a consequence of this , any two direct limits of the same direct system are via a unique compatible with the canonical morphisms, establishing that the direct limit is unique up to unique in the \mathcal{C}. Direct limits generalize finite colimits such as pushouts, which arise as direct limits over finite directed sets (e.g., the two-element poset for parallel arrows).

Examples

Sequence limits

In , a direct limit of a arises from a direct system (A_n, f_n)_{n \in \mathbb{N}} in a category \mathcal{C}, where each A_n is an object and f_n: A_n \to A_{n+1} is a satisfying the conditions for a indexed by the natural numbers. The direct limit \varinjlim A_n, also known as the colimit of this system, is an object A equipped with s \phi_n: A_n \to A such that \phi_{n+1} \circ f_n = \phi_n for all n, and it is with respect to this property: any other object B with compatible s \psi_n: A_n \to B factors uniquely through A. This construction generalizes the notion of a by incorporating identifications dictated by the transition maps f_n, effectively quotienting the \coprod_{n \in \mathbb{N}} A_n by the where elements x \in A_m and y \in A_n (with m \leq n) are identified if there exists k \geq n such that the composite map sends x to y in A_k. A prominent example in field theory is the algebraic closure \overline{\mathbb{Q}} of the rational numbers \mathbb{Q}, which can be realized as the direct limit of the directed system consisting of all finite extensions of \mathbb{Q} (ordered by , forming a since any two finite extensions are contained in a common larger finite extension). The transition maps are the natural morphisms, and the direct limit imposes the necessary identifications to form a where every non-constant over \mathbb{Q} splits completely, while remaining algebraic over \mathbb{Q}. This presentation highlights the direct limit's role as a that adjoins roots of polynomials in a controlled, inductive manner. In the category of abelian groups, the Prüfer p-group \mathbb{Z}(p^\infty) (for a prime p) exemplifies a computation: it is the direct limit of the (\mathbb{Z}/p^n \mathbb{Z})_{n \in \mathbb{N}}, where the transition map f_n: \mathbb{Z}/p^n \mathbb{Z} \to \mathbb{Z}/p^{n+1} \mathbb{Z} sends x \pmod{p^n} to p x \pmod{p^{n+1}}. The resulting group consists of elements representable as fractions a/p^k \pmod{1} for integers a, k with p \nmid a, and it is divisible and torsion, serving as the p-primary component of \mathbb{Q}/\mathbb{Z}. In contrast, the p-adic integers \mathbb{Z}_p arise as the of the same rings \mathbb{Z}/p^n \mathbb{Z} but with projection maps, illustrating how direct limits expand structures by "adding infinities" outward, while inverse limits complete them inward by refining approximations. Direct limits of sequences play a key role in applications like persistent homology in topological data analysis, where a filtered simplicial complex K_\bullet (indexed by a sequence of parameters, such as distances in a point cloud) gives rise to persistent homology groups via the direct limit of the induced system on homology groups \varinjlim H_*(K_n), capturing topological features that persist across scales in the filtration. This framework quantifies multi-scale invariants, such as holes or connected components, by tracking their birth and death in the persistent homology construction, enabling robust analysis of noisy data in dimensions beyond zero.

Union of subgroups

A fundamental example of a direct limit arises from an increasing of subgroups in a group G. Consider a I and a direct system (H_i, \iota_{ij}), where each H_i is a of G and the transition maps \iota_{ij}: H_i \to H_j for i \le j are the maps H_i \subset H_j. The direct limit \lim_{\to} H_i is the H = \bigcup_{i \in I} H_i \subset G, equipped with the induced group structure from G, and the canonical maps H_i \to H are the inclusions. This is the prototypical "union of subgroups," illustrating how direct limits "glue" compatible algebraic structures along a directed poset. A related example involves normal subgroups and their quotients. Suppose (N_i)_{i \in I} is a direct system of normal subgroups of G with N_i \supset N_j for i \le j, inducing natural projection maps G/N_i \to G/N_j. The direct limit \lim_{\to} G/N_i is isomorphic to the quotient group G / \bigcap_{i \in I} N_i, where the canonical maps G/N_i \to G / \bigcap N_k are the natural projections. This identifies the colimit as the "universal quotient" by the intersection kernel, preserving the group structure through the directed system. If the intersection \bigcap N_i = \{e\}, the limit is isomorphic to G; otherwise, it is a proper quotient. The analogous construction holds for rings and ideals. Let (R, I_i)_{i \in I} be a commutative ring with a direct system of ideals I_i \supset I_j for i \le j, inducing surjective ring homomorphisms R/I_i \to R/I_j. The direct limit \lim_{\to} R/I_i is the R / \bigcap_{i \in I} I_i, with canonical maps R/I_i \to R / \bigcap I_k given by the natural projections. This yields the "universal quotient" by the intersection ideal, a key tool in commutative algebra for localizing or completing structures. In , this translates to affine schemes. Consider the direct system of affine schemes \Spec(R/I_i) with transition maps \Spec(R/I_j) \to \Spec(R/I_i) induced by the ring projections R/I_i \to R/I_j for i \le j. The colimit in the category of schemes is \Spec(R / \bigcap I_i), representing the "gluing" of the schemes along the directed system. More generally, colimits of varieties glued along open sets—such as inductive systems of varieties with affine open covers compatible via inclusions—yield the union variety, with the structure sheaf determined by direct limits of the corresponding rings on affines. This construction is essential for building global objects from local data in scheme theory. Not every direct limit of subgroups yields the full ambient group. For instance, consider the additive group \mathbb{Q} of rational numbers and the directed system of subgroups H_n = \mathbb{Z}[1/2^n] for n \in \mathbb{N}, ordered by inclusion. The direct limit is the union \bigcup H_n, the dyadic rationals \{ a/2^b \mid a \in \mathbb{Z}, b \ge 0 \}, which is a proper dense subgroup of \mathbb{Q}. This illustrates how the colimit can remain proper even when the subgroups are dense in a topological sense, contrasting with cases where finite-index chains exhaust the group.

Properties

Preservation by functors

In , left exact functors preserve finite limits but do not in general preserve colimits, including direct limits. Right exact functors preserve finite colimits but typically fail to preserve infinite colimits such as direct limits. In abelian categories, the direct limit functor turns short exact sequences of directed systems into short exact sequences. For example, in the category of modules over a R, the functor M \otimes_R (-) preserves direct limits for any fixed R-module M, as it is a left adjoint and hence commutes with all colimits. As a counterexample, the contravariant Hom functor \Hom_R(-, N) preserves limits but does not preserve colimits, including direct limits. In the category of sets, direct limits (as filtered colimits) commute with finite products, so \varinjlim_I (\prod_{j=1}^n X_{i_j}) \cong \prod_{j=1}^n (\varinjlim_I X_{i_j}) for a directed system (X_i)_{i \in I}.

Exactness

In homological algebra, the direct limit functor preserves exact sequences in certain abelian categories. Specifically, in the category of modules over a ring R, the direct limit (a type of filtered colimit) is an exact functor: if (L_i \to M_i \to N_i)_{i \in I} is a directed system of exact sequences of R-modules, then the induced sequence \varinjlim L_i \to \varinjlim M_i \to \varinjlim N_i is exact. This holds for any directed system in the category of R-modules. In contrast to inverse limits, which generally require the Mittag-Leffler condition on the to ensure exactness (e.g., that the images of maps stabilize appropriately), direct limits in module categories are exact without additional hypotheses. For instance, given a short of R-modules $0 \to A_i \to B_i \to C_i \to 0 compatible with a directed , the induced sequence $0 \to \varinjlim A_i \to \varinjlim B_i \to \varinjlim C_i \to 0 remains short exact. However, this exactness fails in non-abelian settings, such as the category of groups, where the notion of relies on subgroups and the direct limit does not generally preserve kernels or the exactness condition \operatorname{im} f = \ker g. In such categories, the lack of an abelian structure prevents the direct limit from acting as an in the homological sense. An important application arises in sheaf : in the category of sheaves of abelian groups (or modules) on a , filtered colimits, including direct limits, are exact, ensuring that direct limits of exact sequences of sheaves yield exact sequences of direct limit sheaves. This property underpins computations in sheaf , where direct limits often appear in stalks or inductive limits of coherent sheaves.

Related constructions

Inverse limits

In category theory, an system in a category \mathcal{C} indexed by a directed I consists of a F: I^{\mathrm{op}} \to \mathcal{C}, where I^{\mathrm{op}} is the opposite category of I, assigning to each i \in I an object F(i) and to each morphism i \leq j in I a morphism F(j \to i): F(i) \to F(j) in \mathcal{C} satisfying compatibility conditions. This structure contrasts with a direct system, which uses the functor F: I \to \mathcal{C} and morphisms in the forward direction. The inverse limit of such a system, denoted \varprojlim_{i \in I} F(i), is the limit of the diagram F, comprising an object L in \mathcal{C} together with a universal cone: projections \pi_i: L \to F(i) for each i \in I such that for all i \leq j, the diagram \begin{CD} L @>\pi_j>> F(j) \\ @V\pi_iVV @VVF(j \to i)V \\ F(i) @= F(i) \end{CD} commutes, and this cone is universal in the sense that any other cone from an object X to the F(i) factors uniquely through L. This universal property ensures the inverse limit, if it exists, is unique up to unique isomorphism. The construction of inverse limits exhibits a profound duality with direct limits. In the opposite category \mathcal{C}^{\mathrm{op}}, an inverse limit in \mathcal{C} corresponds to a direct (co)limit in \mathcal{C}^{\mathrm{op}}, reflecting the arrow-reversing nature of categorical duality. More specifically, in abelian categories, the inverse limit functor \varprojlim: \mathrm{Fun}(I^{\mathrm{op}}, \mathcal{C}) \to \mathcal{C} is the direct limit functor applied in the opposite category under suitable conditions, such as when the index category admits finite products or the ambient category has exact limits. This duality highlights key differences: direct limits tend to "glue" objects along compatible maps to form larger structures, often preserving colimits, whereas inverse limits "project" onto consistent subsystems, typically preserving limits but not always exactness in non-abelian settings. A canonical example illustrating this contrast is provided by the rings \mathbb{Z}/p^n\mathbb{Z} for a prime p and n \in \mathbb{N}, with transition maps the natural projections \mathbb{Z}/p^{n+1}\mathbb{Z} \to \mathbb{Z}/p^n\mathbb{Z}. The inverse limit of this system yields the ring of p-adic integers \mathbb{Z}_p, consisting of coherent sequences (a_n)_{n \in \mathbb{N}} where a_n \in \mathbb{Z}/p^n\mathbb{Z} and a_{n+1} \equiv a_n \pmod{p^n}, equipped with componentwise addition and multiplication. In duality, the direct limit of the same rings but with forward inclusion maps \mathbb{Z}/p^n\mathbb{Z} \hookrightarrow \mathbb{Z}/p^{n+1}\mathbb{Z} (multiplication by p) produces the Prüfer p-group \mathbb{Z}(p^\infty), the p-primary component of \mathbb{Q}/\mathbb{Z}, which is divisible and torsion but countable, unlike the uncountable, integral domain \mathbb{Z}_p. Direct and inverse limits coincide in certain cases, such as when the directed set I is finite. For a finite directed poset, the direct limit of a direct system over I equals the inverse limit of the corresponding inverse system over I^{\mathrm{op}}, as the universal properties align due to the finiteness ensuring that colimits and limits compute similarly via finite coproducts and products in categories like abelian groups. This equivalence underscores the foundational role of finite diagrams in , where dual constructions overlap.

Filtered colimits

A filtered category is a small category \mathcal{I} such that every finite diagram in \mathcal{I} admits a cocone under it. Equivalently, \mathcal{I} is nonempty, any two objects in \mathcal{I} have morphisms to a common object, and any pair of parallel morphisms between objects can be coequalized by a morphism from their common . This structure ensures that colimits over \mathcal{I} behave in a controlled manner, generalizing the notion of directed sets to arbitrary categories. A filtered colimit is the colimit of a diagram F: \mathcal{I} \to \mathcal{C} where \mathcal{I} is filtered and \mathcal{C} is a cocomplete category. In the category of sets \mathbf{Set}, such a colimit is constructed as the quotient of the disjoint union \coprod_{i \in \mathcal{I}} F(i) by the equivalence relation generated by identifying x \in F(i) with F(f)(x) \in F(j) for every morphism f: i \to j in \mathcal{I}. Direct limits, as previously discussed, are precisely the filtered colimits indexed by directed posets, where the poset order provides the necessary cocones for finite subdiagrams. For example, consider the diagram in \mathbf{Set} consisting of finite sets A_n = \{1, 2, \dots, n\} for n \in \mathbb{N}, with inclusion maps A_m \hookrightarrow A_n for m \leq n. The index category is the poset \mathbb{N} under the usual order, which is filtered. The filtered colimit is the union \bigcup_n A_n = \mathbb{N}, where elements are identified via the inclusions. Filtered colimits commute with finite limits in \mathbf{Set}; that is, for a filtered diagram F: \mathcal{I} \to \mathbf{Set} and a finite limit diagram L: \mathcal{J} \to \mathbf{Set} with |\mathcal{J}| < \infty, we have \varinjlim_{\mathcal{I}} (\lim_{\mathcal{J}} (F \times L)) \cong \lim_{\mathcal{J}} (\varinjlim_{\mathcal{I}} (F \times L)). This property holds more generally in categories with finite limits and filtered colimits, but fails for non-filtered colimits, such as those over discrete categories. In enriched category theory, filtered colimits relate to left Kan extensions, where the colimit of a diagram can be expressed as a pointwise left Kan extension along the Yoneda embedding, preserving the enriched structure over a monoidal category \mathcal{V}. This connection facilitates the study of colimits in \mathcal{V}-enriched settings, such as abelian groups or topological spaces.

Terminology and variants

Inductive limits

In category theory and algebra, the term "inductive limit" serves as a synonym for the direct limit, particularly in older literature and certain algebraic traditions. This usage emphasizes the construction's role in successively building larger objects from a directed system of smaller ones, akin to an inductive process. The direct limit, or inductive limit, is the colimit of a directed system, where canonical morphisms from each object in the system factor uniquely through the limit object. The nomenclature "inductive limit" is especially prevalent in French mathematical literature, where it is rendered as "limite inductive," as employed extensively by the Bourbaki group in works such as Algèbre commutative. This regional preference reflects the influence of the school on and topological vector spaces, where inductive limits describe topologies on unions of increasing sequences of spaces. In English-language texts, the term appears in mid-20th-century algebraic contexts but has largely yielded to "direct limit" in modern category-theoretic treatments. A specialized variant known as the "stationary limit" (or stationary inductive limit) arises in certain algebraic settings, such as limit algebras or operator algebras, where the directed system consists of an infinite sequence with the same fixed transition map (e.g., multiplication by a fixed positive ) at each step. This construction simplifies the direct limit to an without further identifications, often used to model infinite-dimensional structures as stabilized finite approximations. Direct limits admit variants distinguished by strictness: strict direct limits occur when the morphisms are monomorphisms (e.g., inclusions), preserving the internal structure without , as in countable directed systems of Lie groups or vector spaces. Non-strict direct limits, in contrast, involve canonical maps that may identify elements across the system, leading to a quotient construction in the limit object. Notation for inductive limits varies by tradition: the arrowed limit "\lim_{\to}" is standard for direct limits, symbolizing the "forward" direction of the system, while "ind-lim" or "lim ind" appears in texts emphasizing the inductive aspect, particularly in ind-category or ind-scheme contexts. The etymology of "inductive limit" derives from the notion of induction as progressively enlarging from base cases to a comprehensive whole, mirroring mathematical 's step-by-step generalization.

Historical notes

The concept of direct limits has roots in early 20th-century , particularly in Emmy Noether's foundational work on ideals during the . In her 1921 paper, Noether developed the theory of ideals in rings, where she implicitly employed unions of ascending chains of ideals to establish key results like the primary decomposition theorem, foreshadowing the structure of directed unions that characterize direct limits, though without a formal definition. The formalization of direct limits, often termed inductive limits in early literature, occurred through the efforts of the collective in the mid-20th century. In their multi-volume treatise , particularly the Algèbre chapters published starting in the late 1940s and 1950s, Bourbaki systematically introduced inductive limits as a general construction for algebraic structures like groups, rings, and modules, emphasizing their role in unifying various limit processes within a structuralist framework. A broader category-theoretic perspective on direct limits emerged shortly after , integrating them into the newly developed as colimits of directed systems. and Saunders Mac Lane's seminal 1945 paper on general algebra introduced the notions of limits and colimits, providing a diagrammatic framework that encompassed direct limits as universal cocones over directed diagrams, thus extending their algebraic origins to a wide array of mathematical contexts. This categorical viewpoint gained traction in subsequent works, such as Roger Godement's 1958 monograph Topologie algébrique et théorie des faisceaux, which applied direct limits extensively in the construction and of sheaves on topological spaces. By the 1960s, direct limits evolved significantly in , particularly through Alexander Grothendieck's innovations. In his development of abelian categories and derived functors, as detailed in works like the 1957 Tôhoku paper and subsequent seminars on sheaf theory, Grothendieck utilized direct limits to handle exactness properties and colimit preservation in categories of sheaves and modules, bridging algebraic constructions with geometric and topological applications.