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Inclusion map

In mathematics, an inclusion map, also known as an inclusion mapping, is an injective function that embeds a subset B of a set A into A by mapping each element b \in B to itself, denoted as f: B \to A where f(b) = b.^[1]^[2] This natural embedding preserves the identity of elements from the subset, effectively treating B as a part of A without altering its structure.^[2] The inclusion map is fundamentally an injection, ensuring that distinct elements in the domain remain distinct in the codomain, and it is often symbolized by the hookrightarrow notation \hookrightarrow to distinguish it from general functions.^[2] In set theory, it formalizes the subset relation, allowing seamless integration of smaller structures into larger ones, such as the canonical inclusion of the natural numbers into the integers.^[1] Key properties include its continuity in topological contexts and preservation of algebraic or geometric structures when applicable.^[3] Inclusion maps play a crucial role across mathematical disciplines, including topology, where they embed subspaces like the n-sphere S^n into \mathbb{R}^{n+1}; functional analysis, where they appear as bounded or compact operators between spaces such as Sobolev embeddings; and differential geometry, facilitating the study of submanifolds via smooth embeddings.^[2]^[3] In category theory, they represent monomorphisms, enabling commutative diagrams that express relationships between objects.^[3] These maps are essential for constructing chains of embeddings and analyzing properties like compactness or differentiability in advanced settings.^[3]

Definition

In Set Theory

In set theory, the inclusion map, also known as the canonical injection or embedding, is a function \iota: A \to B defined whenever A \subseteq B, such that \iota(x) = x for every x \in A.^[2]^[1] This map simply identifies each element of the subset A with itself in the larger set B, effectively treating A as embedded within B without altering its elements.^[4] The inclusion map acts as the identity function restricted to A, preserving the natural membership relation between the sets while establishing a one-to-one correspondence between A and its image in B.^[2] It is inherently injective, as distinct elements in A map to distinct elements in B, but it is generally not surjective unless A = B.^[1] This foundational construction underpins subset relationships in pure set theory, where no additional operations or structures are imposed on the sets. A classic example is the inclusion map \iota: \mathbb{N} \to \mathbb{Z}, where \mathbb{N} denotes the natural numbers (e.g., {0, 1, 2, \dots}) and \mathbb{Z} the integers; here, \iota(n) = n embeds the non-negative integers into the full ring of integers.^[4] Another simple case is the inclusion of even integers into all integers, \iota: 2\mathbb{Z} \to \mathbb{Z}, with \iota(2k) = 2k for k \in \mathbb{Z}, illustrating how the map respects the subset structure without introducing new elements.^[2] The inclusion map is often denoted using the hooked arrow notation \iota: A \hookrightarrow B to emphasize its injective nature and the embedding aspect.^[1] This concept extends naturally to settings with additional structure, such as ordered sets or topological spaces, but in pure set theory, it remains a basic tool for analyzing subset inclusions.^[2]

In Structured Sets

In mathematical structures, the inclusion map generalizes the set-theoretic inclusion by ensuring preservation of the defining operations, relations, or axioms. Given an algebraic structure B with universe |B| and a substructure A whose universe is a subset of |B|, the inclusion map \iota: A \to B is defined by \iota(x) = x for all x \in A. This map is a homomorphism, meaning it respects the structure: for any n-ary operation f in the signature, \iota(f_A(x_1, \dots, x_n)) = f_B(\iota(x_1), \dots, \iota(x_n)) for all x_1, \dots, x_n \in A.^[5] Similarly, for relations R, if (x_1, \dots, x_n) \in R_A, then (\iota(x_1), \dots, \iota(x_n)) \in R_B. This preservation ensures A inherits the structure from B via restriction of operations and relations to A.^[5] The requirement that \iota is a homomorphism positions it within the category of the relevant structures, where objects are algebras or relational structures and morphisms are structure-preserving maps. A subset A \subseteq |B| qualifies as a substructure precisely if it is closed under all operations (i.e., f_B(a_1, \dots, a_n) \in A for a_i \in A) and, for relational structures, if relations on A match those induced from B. The inclusion map then serves as the canonical embedding, confirming A's status as a substructure without additional mapping.^[5] This builds on the pure set inclusion as the underlying function, but adds the structural fidelity.^[5] A representative example occurs in vector spaces over a field K. If W is a subspace of a vector space V, the inclusion map \iota: W \to V given by \iota(w) = w is a linear transformation, preserving vector addition and scalar multiplication: \iota(w_1 + w_2) = w_1 + w_2 = \iota(w_1) + \iota(w_2) and \iota(c w) = c w = c \iota(w) for w_1, w_2 \in W and c \in K. This linearity follows directly from the subspace axioms, ensuring linear combinations in W remain unchanged in V.^[6] Unlike arbitrary embeddings, which are injective homomorphisms that may relabel elements via composition with an isomorphism, the inclusion map uses the identity function on the shared universe, providing a direct identification of elements without renaming or permutation. This naturalness makes it the standard choice for substructures, distinguishing it from more general structure-preserving injections.^[5]

Properties

Injectivity and Monomorphisms

An inclusion map \iota: A \to X, where A \subseteq X, is defined by \iota(a) = a for all a \in A. This map is injective because if \iota(a) = \iota(b), then a = b by the identity nature of the mapping on A.^[7] In category theory, a monomorphism is a morphism f: X \to Y that is left-cancellative, meaning that for any object Z and any pair of morphisms g_1, g_2: Z \to X, if f \circ g_1 = f \circ g_2, then g_1 = g_2.^[8]^[9] Inclusion maps are always monomorphisms in standard categories such as \mathbf{Set}, \mathbf{Grp}, and \mathbf{Top}. In \mathbf{Set}, every inclusion is an injective function, and monomorphisms coincide precisely with injective functions.^[8]^[9] In \mathbf{Grp}, monomorphisms are injective group homomorphisms, and inclusions of subgroups satisfy this condition.^[8]^[10] In \mathbf{Top}, monomorphisms are injective continuous maps, with subspace inclusions serving as regular and strong monomorphisms when equipped with the subspace topology.^[8] For example, in \mathbf{Set}, the injectivity of an inclusion map directly implies it is a monomorphism, as left-cancellativity follows from the one-to-one correspondence of elements.^[9] This property aligns with the broader universal property of inclusions but emphasizes their cancellative behavior in compositions.^[8]

Universal Property

In category theory, the inclusion map \iota: A \to B often satisfies a universal property when B is constructed as a free or induced object generated by A, such as in algebraic categories. Specifically, for any object C in the category and any morphism f: A \to C that respects the relevant structure (e.g., a set map to a group when B is free), there exists a unique morphism \bar{f}: B \to C such that f = \bar{f} \circ \iota.^[11] This property characterizes the inclusion up to isomorphism as the canonical morphism from A to the universal object B that "freely" completes A under the category's operations. This universal property manifests as the initial object in the comma category (A \downarrow \mathcal{C}), whose objects are morphisms from A to other objects in \mathcal{C} and whose morphisms are commuting triangles over A. The pair (B, \iota) is initial, ensuring unique factorizations through \iota for compatible maps from A.^[11] For instance, in the category of groups, if A is a set and B is the free group on A with \iota including the generators, any group homomorphism f: A \to G (treating A as a discrete group) extends uniquely to a group homomorphism \bar{f}: B \to G.^[12] The inclusion map plays a key role in forming induced maps and restrictions across categories. Composition with \iota induces a natural transformation on hom-sets, \operatorname{Hom}(B, C) \to \operatorname{Hom}(A, C) given by g \mapsto g \circ \iota, which restricts functions or homomorphisms from B to A. This is essential in constructing colimits, such as pushouts where inclusions serve as legs of diagrams, ensuring compatible extensions or gluings.^[12] In the category of sets, the inclusion \iota: A \hookrightarrow B of a subset allows extending maps from A to any set C by arbitrarily defining values on B \setminus A, though uniqueness fails unless B = A. This reflects the coproduct structure B \cong A \sqcup (B \setminus A), where \iota is one coproduct inclusion, facilitating constructions like disjoint unions.^[13] More generally, such inclusions connect to initial objects in slice categories: the pair (B, \iota) is initial in the coslice category A / \mathcal{C} (or equivalently the comma category above), underscoring their role in universal approximations and free completions without delving into specific variances.^[11]

In Algebraic Structures

Subgroups and Homomorphisms

In group theory, given a group G and a subgroup H \leq G, the inclusion map \iota: H \to G is defined by \iota(h) = h for all h \in H. This map is a group homomorphism because the binary operation on H is the restriction of the operation on G, ensuring that \iota(h_1 h_2) = h_1 h_2 = \iota(h_1) \iota(h_2) for all h_1, h_2 \in H.^[14] The inclusion map \iota is injective, since \iota(h_1) = \iota(h_2) implies h_1 = h_2, and its image \iota(H) = H exactly, providing an embedding that identifies the subgroup H with its isomorphic copy within G.^[15] A representative example is the inclusion \iota: 2\mathbb{Z} \to \mathbb{Z} of the cyclic subgroup of even integers into the additive group of all integers, where \iota(2k) = 2k for k \in \mathbb{Z}; this preserves addition as \iota(2k_1 + 2k_2) = 2(k_1 + k_2) = 2k_1 + 2k_2 = \iota(2k_1) + \iota(2k_2). In the context of quotient groups, the inclusion map aids in characterizing normal subgroups through kernels of homomorphisms. For a normal subgroup N \trianglelefteq G, the inclusion \iota: N \to G has trivial kernel, and pairs with the canonical projection \pi: G \to G/N (whose kernel is N) to form the short exact sequence $1 \to N \xrightarrow{\iota} G \xrightarrow{\pi} G/N \to 1, illustrating how normality enables the quotient structure while the inclusion embeds N faithfully.^[16]

Subrings and Ideals

In ring theory, an inclusion map arises naturally when one ring is a subring of another. If R is a subring of a ring S, the inclusion map \iota: R \to S is defined by \iota(r) = r for all r \in R. This map is a ring homomorphism because it preserves the ring operations inherited from S: \iota(r_1 + r_2) = r_1 + r_2 = \iota(r_1) + \iota(r_2) and \iota(r_1 r_2) = r_1 r_2 = \iota(r_1) \iota(r_2). Since subrings share the multiplicative identity of the ambient ring, \iota(1_R) = 1_S, ensuring the homomorphism respects the unit.^[17]^[18] A classic example is the inclusion of the ring of integers \mathbb{Z} into the field of rational numbers \mathbb{Q}, where \mathbb{Z} serves as a subring under the standard addition and multiplication. Here, \iota: \mathbb{Z} \to \mathbb{Q} maps each integer to itself, preserving all operations and the identity 1. This inclusion highlights how integer arithmetic embeds into the broader structure of rational numbers, facilitating extensions in algebraic number theory.^[17] While ideals I \trianglelefteq R are not typically subrings (as they often lack the multiplicative unit unless I = R), inclusions involving ideals connect to quotient rings through homomorphisms. Specifically, if J \subseteq I are ideals of R, the inclusion J \to I (viewed in the context of the quotient map R \to R/J) induces a surjective ring homomorphism R/J \to R/I with kernel I/J, reflecting the lattice structure of ideals and their quotients. This relationship underscores the role of inclusion maps in the correspondence theorem for rings.^[19]

In Topology

Continuous Maps

In topology, the inclusion map \iota: A \to X, where A \subseteq X is a subset of a topological space (X, \mathcal{T}), is defined by \iota(a) = a for all a \in A, and A is equipped with the subspace topology \mathcal{T}_A = \{ U \cap A \mid U \in \mathcal{T} \}.^[20] This map is continuous by construction, as it ensures that the topological structure of A is compatible with that of X.^[21] To verify continuity, consider an arbitrary open set U \in \mathcal{T} in X. The preimage under \iota is \iota^{-1}(U) = \{ a \in A \mid \iota(a) \in U \} = U \cap A, which is open in the subspace topology \mathcal{T}_A by definition.^[21] Since every open set in X has a preimage that is open in A, \iota satisfies the continuity condition. The subspace topology, also known as the relative topology, thus inherits openness from intersections with open sets in X, preserving the local properties of the ambient space on A.^[20] A representative example is the inclusion \iota: (0,1) \to \mathbb{R}, where \mathbb{R} has the standard topology generated by open intervals. Open sets in (0,1) are of the form (a,b) \cap (0,1) for $0 \leq a < b \leq 1 and open intervals (a,b) in \mathbb{R}, such as (0.2, 0.8), which remains open in the subspace topology.^[20] This illustrates how the inclusion map maintains continuity while inducing the expected Euclidean structure on the subspace.^[22]

Embeddings of Spaces

In topology, an inclusion map \iota: A \to X, where A is a subset of a topological space X equipped with the subspace topology, is a topological embedding if it is a homeomorphism onto its image \iota(A) endowed with the subspace topology induced from X.^[23] This means \iota is continuous, injective, and the inverse map from \iota(A) to A is also continuous with respect to the subspace topology.^[24] By construction, every such inclusion map satisfies this property, as the subspace topology on A is defined precisely to make \iota a homeomorphism onto its image.^[24] The inclusion map \iota is continuous for any subset A \subseteq X, but additional properties like being an open or closed map depend on the nature of A. Specifically, \iota is an open map if and only if A is an open subset of X, meaning that the image of every open set in A (under the subspace topology) is open in X.^[25] Conversely, \iota is a closed map if A is a closed subset of X. In general, for \iota to behave as an open map onto saturated sets or under other conditions, A must satisfy corresponding topological criteria, such as being saturated with respect to certain open covers, ensuring the embedding preserves openness in the image.^[23] A classic example is the inclusion of the unit circle S^1 = \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1 \} into \mathbb{R}^2, where S^1 carries the subspace topology. This map \iota: S^1 \to \mathbb{R}^2 is a topological embedding, as it is a homeomorphism onto its image, the unit circle itself, preserving the topological structure of S^1.^[23] Unlike an immersion, which requires only a local homeomorphism onto the image (typically in the context of differentiable manifolds), a topological embedding demands a global homeomorphism to the image, ensuring no topological distortions occur across the entire space.^[26]

Applications

In Homotopy Theory

In homotopy theory, the inclusion map \iota: A \hookrightarrow X of a subspace A into a topological space X induces group homomorphisms \iota_*: \pi_n(A, a_0) \to \pi_n(X, x_0) on the nth homotopy groups for each n \geq 1, where basepoints are preserved under the inclusion. These induced maps capture how homotopy classes in A extend or relate to those in X. Analogously, on singular homology, the inclusion induces chain maps that yield homomorphisms H_n(\iota): H_n(A; \mathbb{Z}) \to H_n(X; \mathbb{Z}) for each n \geq 0.^[27] A fundamental result states that if A is a deformation retract of X, then \iota is a homotopy equivalence, so \iota_* is an isomorphism on every homotopy group \pi_n for n \geq 0. This holds because a deformation retract provides a homotopy H: X \times I \to X such that H_0 = \mathrm{id}_X, H_1(X) \subseteq A, and H_t|_A = \mathrm{id}_A for all t, implying the retraction r: X \to A composed with \iota is homotopic to the identity on X, and vice versa relative to A. The same applies to homology, where \iota induces isomorphisms H_n(\iota) under these conditions.^[27] For instance, the inclusion of the equator S^1 \hookrightarrow S^2 induces the zero homomorphism on \pi_1(S^1) \cong \mathbb{Z} \to \pi_1(S^2) = 0, as loops on the equator bound hemispheres in S^2. On higher homotopy groups \pi_n for n > 1, \pi_n(S^1) = 0, so \iota_* yields the unique homomorphism from the trivial group to \pi_n(S^2), which is an isomorphism if and only if \pi_n(S^2) = 0 (though \pi_n(S^2) \neq 0 for n \geq 2). This example illustrates how inclusions can detect non-trivial extensions in higher dimensions.^[27] Inclusion maps feature prominently in the long exact sequence of the homotopy pair (X, A):

\cdots \to \pi_n(A, a_0) \xrightarrow{\iota_*} \pi_n(X, x_0) \to \pi_n(X, A; x_0) \xrightarrow{\partial} \pi_{n-1}(A, a_0) \to \cdots,

which is exact, allowing computation of relative homotopy groups \pi_n(X, A) via the kernel and image of \iota_*. A parallel long exact sequence exists in homology:

\cdots \to H_n(A; \mathbb{Z}) \xrightarrow{H_n(\iota)} H_n(X; \mathbb{Z}) \to H_n(X, A; \mathbb{Z}) \to H_{n-1}(A; \mathbb{Z}) \to \cdots.

These sequences exploit inclusions to relate absolute and relative invariants, as in CW complexes where the inclusion of the n-skeleton induces isomorphisms on \pi_i for i < n.^[27]

In Scheme Theory

In scheme theory, inclusion maps manifest as closed immersions, which are morphisms f: X \to Y of schemes such that f induces a homeomorphism from X onto a closed subset of Y, and the induced map f^\#: \mathcal{O}_Y \to f_* \mathcal{O}_X on structure sheaves is surjective.^[28] For affine schemes, a closed immersion \iota: \Spec(A) \to \Spec(R) corresponds precisely to a surjective ring homomorphism R \twoheadrightarrow A, such as when A = R/I for some ideal I \subset R.^[28] This surjection defines the kernel ideal sheaf on \Spec(R), which is quasi-coherent, ensuring the embedding captures the structure of the closed subscheme.^[28] Closed immersions possess key properties that underscore their role as inclusions: they are monomorphisms in the category of schemes, meaning they are injective on points and faithfully reflect the scheme structure without isomorphisms beyond the identity.^[29] When defined via quotient rings, these immersions are affine morphisms, preserving the affine nature of the source scheme within the target.^[28] Moreover, the ideal sheaf associated to the immersion allows for a precise description of the closed subscheme as the zero set of that ideal, enabling local computations on affine opens.^[28] A representative example is the closed immersion \Spec(k/(x^2)) \to \Spec(k), induced by the surjection k \twoheadrightarrow k/(x^2).^[28] This embeds the spectrum of the dual numbers as a closed subscheme of the affine line \mathbb{A}^1_k, where the unique prime ideal (x) of k/(x^2) maps to the origin, incorporating a nilpotent element \overline{x} with \overline{x}^2 = 0; geometrically, it models an infinitesimal thickening of the point at the origin, often interpreted as a "double point" on the line.^[30] Closed immersions play a crucial role in defining closed subschemes and facilitating the construction of schemes through gluing. A closed subscheme of a scheme Y is given by a closed immersion i: Z \to Y, allowing the specification of subscheme structures via ideals.^[28] In gluing constructions, pushouts along closed immersions exist in the category of schemes—for instance, given a closed immersion Z \to X and a morphism Z \to Y, the resulting pushout X \cup_Z Y forms a scheme that effectively glues X and Y along the shared closed subscheme Z, preserving scheme-theoretic properties like separatedness.^[31] This mechanism is fundamental for building composite schemes from simpler components while respecting their closed substructures.^[28]

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