Duality (mathematics)

Order-reversing dualities

Given a poset P = (X, ≤) (short for partially ordered set; i.e., a set that has a notion of ordering but in which two elements cannot necessarily be placed in order relative to each other), the dual poset P^d = (X, ≥) comprises the same ground set but the converse relation. Familiar examples of dual partial orders include

• the subset and superset relations ⊂ and ⊃ on any collection of sets, such as the subsets of a fixed set S. This gives rise to the first example of a duality mentioned above.
• the divides and multiple-of relations on the integers.
• the descendant-of and ancestor-of relations on the set of humans.

A duality transform is an involutive antiautomorphism f of a partially ordered set S, that is, an order-reversing involution f : S → S. In several important cases these simple properties determine the transform uniquely up to some simple symmetries. For example, if f₁, f₂ are two duality transforms then their composition is an order automorphism of S; thus, any two duality transforms differ only by an order automorphism. For example, all order automorphisms of a power set S = 2^R are induced by permutations of R.

A concept defined for a partial order P will correspond to a dual concept on the dual poset P^d. For instance, a minimal element of P will be a maximal element of P^d: minimality and maximality are dual concepts in order theory. Other pairs of dual concepts are upper and lower bounds, lower sets and upper sets, and ideals and filters.

In topology, open sets and closed sets are dual concepts: the complement of an open set is closed, and vice versa. In matroid theory, the family of sets complementary to the independent sets of a given matroid themselves form another matroid, called the dual matroid.

Dimension-reversing dualities

There are many distinct but interrelated dualities in which geometric or topological objects correspond to other objects of the same type, but with a reversal of the dimensions of the features of the objects. A classical example (known to Euclid) of this is the duality of the Platonic solids, in which the cube and the octahedron form a dual pair, the dodecahedron and the icosahedron form a dual pair, and the tetrahedron is self-dual. The dual polyhedron of any of these polyhedra may be formed as the convex hull of the center points of each face of the primal polyhedron, so the vertices of the dual correspond one-for-one with the faces of the primal. Similarly, each edge of the dual corresponds to an edge of the primal, and each face of the dual corresponds to a vertex of the primal. These correspondences are incidence-preserving: if two parts of the primal polyhedron touch each other, so do the corresponding two parts of the dual polyhedron. More generally, using the concept of polar reciprocation, any convex polyhedron, or more generally any convex polytope, corresponds to a dual polyhedron or dual polytope, with an i-dimensional feature of an n-dimensional polytope corresponding to an (n − i − 1)-dimensional feature of the dual polytope. The incidence-preserving nature of the duality is reflected in the fact that the face lattices of the primal and dual polyhedra or polytopes are themselves order-theoretic duals. Duality of polytopes and order-theoretic duality are both involutions: the dual polytope of the dual polytope of any polytope is the original polytope, and reversing all order-relations twice returns to the original order. Choosing a different center of polarity leads to geometrically different dual polytopes, but all have the same combinatorial structure.

From any convex three-dimensional polyhedron, one can form a planar graph, the graph of its vertices and edges. The dual polyhedron has a dual graph, a graph with one vertex for each face of the polyhedron and with one edge for every two adjacent faces. The same concept of planar graph duality may be generalized to graphs that are drawn in the plane but that do not come from a three-dimensional polyhedron, or more generally to graph embeddings on surfaces of higher genus: one may draw a dual graph by placing one vertex within each region bounded by a cycle of edges in the embedding, and drawing an edge connecting any two regions that share a boundary edge. An important example of this type comes from computational geometry: the duality for any finite set S of points in the plane between the Delaunay triangulation of S and the Voronoi diagram of S. As with dual polyhedra and dual polytopes, the duality of graphs on surfaces is a dimension-reversing involution: each vertex in the primal embedded graph corresponds to a region of the dual embedding, each edge in the primal is crossed by an edge in the dual, and each region of the primal corresponds to a vertex of the dual. The dual graph depends on how the primal graph is embedded: different planar embeddings of a single graph may lead to different dual graphs. Matroid duality is an algebraic extension of planar graph duality, in the sense that the dual matroid of the graphic matroid of a planar graph is isomorphic to the graphic matroid of the dual graph.

A kind of geometric duality also occurs in optimization theory, but not one that reverses dimensions. A linear program may be specified by a system of real variables (the coordinates for a point in Euclidean space ${\displaystyle {\mathbb{R}}^n}$), a system of linear constraints (specifying that the point lie in a halfspace; the intersection of these halfspaces is a convex polytope, the feasible region of the program), and a linear function (what to optimize). Every linear program has a dual problem with the same optimal solution, but the variables in the dual problem correspond to constraints in the primal problem and vice versa.

Duality in logic and set theory

In logic, functions or relations A and B are considered dual if A(¬x) = ¬B(x), where ¬ is logical negation. The basic duality of this type is the duality of the ∃ and ∀ quantifiers in classical logic. These are dual because ∃x.¬P(x) and ¬∀x.P(x) are equivalent for all predicates P in classical logic: if there exists an x for which P fails to hold, then it is false that P holds for all x (but the converse does not hold constructively). From this fundamental logical duality follow several others:

• A formula is said to be satisfiable in a certain model if there are assignments to its free variables that render it true; it is valid if every assignment to its free variables makes it true. Satisfiability and validity are dual because the invalid formulas are precisely those whose negations are satisfiable, and the unsatisfiable formulas are those whose negations are valid. This can be viewed as a special case of the previous item, with the quantifiers ranging over interpretations.
• In classical logic, the ∧ and ∨ operators are dual in this sense, because (¬x ∧ ¬y) and ¬(x ∨ y) are equivalent. This means that for every theorem of classical logic there is an equivalent dual theorem. De Morgan's laws are examples. More generally, ∧ (¬ x_i) = ¬∨ x_i. The left side is true if and only if ∀i.¬x_i, and the right side if and only if ¬∃i.x_i.
• In modal logic, □p means that the proposition p is "necessarily" true, and ◊p that p is "possibly" true. Most interpretations of modal logic assign dual meanings to these two operators. For example, in Kripke semantics, "p is possibly true" means "there exists some world W such that p is true in W", while "p is necessarily true" means "for all worlds W, p is true in W". The duality of □ and ◊ then follows from the analogous duality of ∀ and ∃. Other dual modal operators behave similarly. For example, temporal logic has operators denoting "will be true at some time in the future" and "will be true at all times in the future" which are similarly dual.

Other analogous dualities follow from these:

• Set-theoretic union and intersection are dual under the set complement operator ^∁. That is, A^∁ ∩ B^∁ = (A ∪ B)^∁, and more generally, ⋂ A_α^∁ = (⋃ A_α)^∁. This follows from the duality of ∀ and ∃: an element x is a member of ⋂ A_α^∁ if and only if ∀α.¬x ∈ A_α, and is a member of (⋂ A_α)^∁ if and only if ¬∃α. x ∈ A_α.

Bidual

The dual of the dual, called the bidual or double dual, depending on context, is often identical to the original (also called primal), and duality is an involution. In this case the bidual is not usually distinguished, and instead one only refers to the primal and dual. For example, the dual poset of a dual poset is exactly the original poset, since the converse relation is defined by an involution.

In other cases, the bidual is not identical with the primal, though there is often a close connection. For example, the dual cone of the dual cone of a set contains the primal set (it is the smallest cone containing the primal set), and is equal if and only if the primal set is a cone.

An important case is for vector spaces, where there is a map from the primal space to the double dual, V → V^**, known as the "canonical evaluation map". For finite-dimensional vector spaces this is an isomorphism, but these are not identical spaces: they are different sets. In category theory, this is generalized by § Dual objects, and a "natural transformation" from the identity functor to the double dual functor. For vector spaces (considered algebraically), this is always an injection; see Dual space § Injection into the double-dual. This can be generalized algebraically to a dual module. There is still a canonical evaluation map, but it is not always injective; if it is, this is known as a torsionless module; if it is an isomorphism, the module is called reflexive.

For topological vector spaces (including normed vector spaces), there is a separate notion of a topological dual, denoted ⁠${\displaystyle V^{\prime }}$⁠ to distinguish from the algebraic dual V^*, with different possible topologies on the dual, each of which defines a different bidual space ⁠${\displaystyle V^{″}}$⁠. In these cases the canonical evaluation map ⁠${\displaystyle V\to V^{″}}$⁠ is not in general an isomorphism. If it is, this is known (for certain locally convex vector spaces with the strong dual space topology) as a reflexive space.

In other cases, showing a relation between the primal and bidual is a significant result, as in Pontryagin duality (a locally compact abelian group is naturally isomorphic to its bidual).

Dual objects

A group of dualities can be described by endowing, for any mathematical object X, the set of morphisms Hom (X, D) into some fixed object D, with a structure similar to that of X. This is sometimes called internal Hom. In general, this yields a true duality only for specific choices of D, in which case X^* = Hom (X, D) is referred to as the dual of X. There is always a map from X to the bidual, that is to say, the dual of the dual, $${\displaystyle X\to X^{\ast \ast }:=(X^{\ast }{)}^{\ast }=\mathrm{Hom}(\mathrm{Hom}(X,D),D).}$$ It assigns to some x ∈ X the map that associates to any map f : X → D (i.e., an element in Hom(X, D)) the value f(x). Depending on the concrete duality considered and also depending on the object X, this map may or may not be an isomorphism.

Dual vector spaces revisited

The construction of the dual vector space $${\displaystyle V^{\ast }=\mathrm{Hom}(V,K)}$$ mentioned in the introduction is an example of such a duality. Indeed, the set of morphisms, i.e., linear maps, forms a vector space in its own right. The map V → V^** mentioned above is always injective. It is surjective, and therefore an isomorphism, if and only if the dimension of V is finite. This fact characterizes finite-dimensional vector spaces without referring to a basis.

Isomorphisms of V and V∗ and inner product spaces

A vector space V is isomorphic to V^∗ precisely if V is finite-dimensional. In this case, such an isomorphism is equivalent to a non-degenerate bilinear form $${\displaystyle \phi :V\times V\to K}$$ In this case V is called an inner product space. For example, if K is the field of real or complex numbers, any positive definite bilinear form gives rise to such an isomorphism. In Riemannian geometry, V is taken to be the tangent space of a manifold and such positive bilinear forms are called Riemannian metrics. Their purpose is to measure angles and distances. Thus, duality is a foundational basis of this branch of geometry. Another application of inner product spaces is the Hodge star which provides a correspondence between the elements of the exterior algebra. For an n-dimensional vector space, the Hodge star operator maps k-forms to (n − k)-forms. This can be used to formulate Maxwell's equations. In this guise, the duality inherent in the inner product space exchanges the role of magnetic and electric fields.

Duality in projective geometry

In some projective planes, it is possible to find geometric transformations that map each point of the projective plane to a line, and each line of the projective plane to a point, in an incidence-preserving way. For such planes there arises a general principle of duality in projective planes: given any theorem in such a plane projective geometry, exchanging the terms "point" and "line" everywhere results in a new, equally valid theorem. A simple example is that the statement "two points determine a unique line, the line passing through these points" has the dual statement that "two lines determine a unique point, the intersection point of these two lines". For further examples, see Dual theorems.

A conceptual explanation of this phenomenon in some planes (notably field planes) is offered by the dual vector space. In fact, the points in the projective plane ${\displaystyle {\mathbb{R}\mathbb{P}}^2}$ correspond to one-dimensional subvector spaces ${\displaystyle V\subset {\mathbb{R}}^3}$ while the lines in the projective plane correspond to subvector spaces ${\displaystyle W}$ of dimension 2. The duality in such projective geometries stems from assigning to a one-dimensional ${\displaystyle V}$ the subspace of ${\displaystyle ({\mathbb{R}}^3{)}^{\ast }}$ consisting of those linear maps ${\displaystyle f:{\mathbb{R}}^3\to \mathbb{R}}$ which satisfy ${\displaystyle f(V)=0}$. As a consequence of the dimension formula of linear algebra, this space is two-dimensional, i.e., it corresponds to a line in the projective plane associated to ${\displaystyle ({\mathbb{R}}^3{)}^{\ast }}$.

The (positive definite) bilinear form $${\displaystyle ⟨\cdot ,\cdot ⟩:{\mathbb{R}}^3\times {\mathbb{R}}^3\to \mathbb{R},⟨x,y⟩=\sum _{i=1}^3{x}_i{y}_i}$$ yields an identification of this projective plane with the ${\displaystyle {\mathbb{R}\mathbb{P}}^2}$. Concretely, the duality assigns to ${\displaystyle V\subset {\mathbb{R}}^3}$ its orthogonal ${\displaystyle \left\{w\in {\mathbb{R}}^3,⟨v,w⟩=0\text{ for all }v\in V\right\}}$. The explicit formulas in duality in projective geometry arise by means of this identification.

Topological vector spaces and Hilbert spaces

In the realm of topological vector spaces, a similar construction exists, replacing the dual by the topological dual vector space. There are several notions of topological dual space, and each of them gives rise to a certain concept of duality. A topological vector space ${\displaystyle X}$ that is canonically isomorphic to its bidual ${\displaystyle X^{″}}$ is called a reflexive space: $${\displaystyle X\cong X^{″}.}$$

Examples:

• As in the finite-dimensional case, on each Hilbert space H its inner product ⟨⋅, ⋅⟩ defines a map $${\displaystyle H\to H^{\ast },v\mapsto (w\mapsto ⟨w,v⟩),}$$ which is a bijection due to the Riesz representation theorem. As a corollary, every Hilbert space is a reflexive Banach space.
• The dual normed space of an L^p-space is L^q where 1/p + 1/q = 1 provided that 1 ≤ p < ∞, but the dual of L^∞ is bigger than L¹. Hence L¹ is not reflexive.
• Distributions are linear functionals on appropriate spaces of functions. They are an important technical means in the theory of partial differential equations (PDE): instead of solving a PDE directly, it may be easier to first solve the PDE in the "weak sense", i.e., find a distribution that satisfies the PDE and, second, to show that the solution must, in fact, be a function. All the standard spaces of distributions — ${\displaystyle {\mathcal{D}}^{\prime }(U)}$, ${\displaystyle {\mathcal{S}}^{\prime }({\mathbb{R}}^n)}$, ${\displaystyle {\mathcal{C}}^{\mathrm{\infty }}(U{)}^{\prime }}$ — are reflexive locally convex spaces.

Further dual objects

The dual lattice of a lattice L is given by $${\displaystyle \mathrm{Hom}(L,\mathbb{Z}),}$$ the set of linear functions on the real vector space containing the lattice that map the points of the lattice to the integers ${\displaystyle \mathbb{Z}}$. This is used in the construction of toric varieties. The Pontryagin dual of locally compact topological groups G is given by $${\displaystyle \mathrm{Hom}(G,S^1),}$$ continuous group homomorphisms with values in the circle (with multiplication of complex numbers as group operation).

Dual categories

Opposite category and adjoint functors

In another group of dualities, the objects of one theory are translated into objects of another theory and the maps between objects in the first theory are translated into morphisms in the second theory, but with direction reversed. Using the parlance of category theory, this amounts to a contravariant functor between two categories C and D:

F: C → D

which for any two objects X and Y of C gives a map

Hom_C(X, Y) → Hom_D(F(Y), F(X))

That functor may or may not be an equivalence of categories. There are various situations, where such a functor is an equivalence between the opposite category C^op of C, and D. Using a duality of this type, every statement in the first theory can be translated into a "dual" statement in the second theory, where the direction of all arrows has to be reversed. Therefore, any duality between categories C and D is formally the same as an equivalence between C and D^op (C^op and D). However, in many circumstances the opposite categories have no inherent meaning, which makes duality an additional, separate concept.

A category that is equivalent to its dual is called self-dual. An example of self-dual category is the category of Hilbert spaces.

Many category-theoretic notions come in pairs in the sense that they correspond to each other while considering the opposite category. For example, Cartesian products Y₁ × Y₂ and disjoint unions Y₁ ⊔ Y₂ of sets are dual to each other in the sense that

Hom (X, Y₁ × Y₂) = Hom (X, Y₁) × Hom (X, Y₂)

and

Hom (Y₁ ⊔ Y₂, X) = Hom (Y₁, X) × Hom (Y₂, X)

for any set X. This is a particular case of a more general duality phenomenon, under which limits in a category C correspond to colimits in the opposite category C^op; further concrete examples of this are epimorphisms vs. monomorphism, in particular factor modules (or groups etc.) vs. submodules, direct products vs. direct sums (also called coproducts to emphasize the duality aspect). Therefore, in some cases, proofs of certain statements can be halved, using such a duality phenomenon. Further notions displaying related by such a categorical duality are projective and injective modules in homological algebra, fibrations and cofibrations in topology and more generally model categories.

Two functors F: C → D and G: D → C are adjoint if for all objects c in C and d in D

Hom_D(F(c), d) ≅ Hom_C(c, G(d)),

in a natural way. Actually, the correspondence of limits and colimits is an example of adjoints, since there is an adjunction

colim: C^I ↔ C: Δ

between the colimit functor that assigns to any diagram in C indexed by some category I its colimit and the diagonal functor that maps any object c of C to the constant diagram which has c at all places. Dually,

Δ: C ↔ C^I: lim.

Spaces and functions

Gelfand duality is a duality between commutative C*-algebras A and compact Hausdorff spaces X is the same: it assigns to X the space of continuous functions (which vanish at infinity) from X to C, the complex numbers. Conversely, the space X can be reconstructed from A as the spectrum of A. Both Gelfand and Pontryagin duality can be deduced in a largely formal, category-theoretic way.

In a similar vein there is a duality in algebraic geometry between commutative rings and affine schemes: to every commutative ring A there is an affine spectrum, Spec A. Conversely, given an affine scheme S, one gets back a ring by taking global sections of the structure sheaf O_S. In addition, ring homomorphisms are in one-to-one correspondence with morphisms of affine schemes, thereby there is an equivalence

(Commutative rings)^op ≅ (affine schemes)

Affine schemes are the local building blocks of schemes. The previous result therefore tells that the local theory of schemes is the same as commutative algebra, the study of commutative rings.

Noncommutative geometry draws inspiration from Gelfand duality and studies noncommutative C*-algebras as if they were functions on some imagined space. Tannaka–Krein duality is a non-commutative analogue of Pontryagin duality.

Galois connections

In a number of situations, the two categories which are dual to each other are actually arising from partially ordered sets, i.e., there is some notion of an object "being smaller" than another one. A duality that respects the orderings in question is known as a Galois connection. An example is the standard duality in Galois theory mentioned in the introduction: a bigger field extension corresponds—under the mapping that assigns to any extension L ⊃ K (inside some fixed bigger field Ω) the Galois group Gal (Ω / L) —to a smaller group.

The collection of all open subsets of a topological space X forms a complete Heyting algebra. There is a duality, known as Stone duality, connecting sober spaces and spatial locales.

• Birkhoff's representation theorem relating distributive lattices and partial orders

Pontryagin duality

Pontryagin duality gives a duality on the category of locally compact abelian groups: given any such group G, the character group

χ(G) = Hom (G, S¹)

given by continuous group homomorphisms from G to the circle group S¹ can be endowed with the compact-open topology. Pontryagin duality states that the character group is again locally compact abelian and that

G ≅ χ(χ(G)).

Moreover, discrete groups correspond to compact abelian groups; finite groups correspond to finite groups. On the one hand, Pontryagin is a special case of Gelfand duality. On the other hand, it is the conceptual reason of Fourier analysis, see below.

Analytic dualities

In analysis, problems are frequently solved by passing to the dual description of functions and operators.

Fourier transform switches between functions on a vector space and its dual: $${\displaystyle \hat{f}(\xi ):={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)e^{-2\pi ix\xi }dx,}$$ and conversely $${\displaystyle f(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\hat{f}(\xi )e^{2\pi ix\xi }d\xi .}$$ If f is an L²-function on R or R^N, say, then so is ${\displaystyle \hat{f}}$ and ${\displaystyle f(-x)=\hat{\hat{f}}(x)}$. Moreover, the transform interchanges operations of multiplication and convolution on the corresponding function spaces. A conceptual explanation of the Fourier transform is obtained by the aforementioned Pontryagin duality, applied to the locally compact groups R (or R^N etc.): any character of R is given by ξ ↦ e^−2πixξ. The dualizing character of Fourier transform has many other manifestations, for example, in alternative descriptions of quantum mechanical systems in terms of coordinate and momentum representations.

• Laplace transform is similar to Fourier transform and interchanges operators of multiplication by polynomials with constant coefficient linear differential operators.
• Legendre transformation is an important analytic duality which switches between velocities in Lagrangian mechanics and momenta in Hamiltonian mechanics.

Homology and cohomology

Theorems showing that certain objects of interest are the dual spaces (in the sense of linear algebra) of other objects of interest are often called dualities. Many of these dualities are given by a bilinear pairing of two K-vector spaces

A ⊗ B → K.

For perfect pairings, there is, therefore, an isomorphism of A to the dual of B.

Poincaré duality

Poincaré duality of a smooth compact complex manifold X is given by a pairing of singular cohomology with C-coefficients (equivalently, sheaf cohomology of the constant sheaf C)

Hⁱ(X) ⊗ H²ⁿ⁻ⁱ(X) → C,

where n is the (complex) dimension of X. Poincaré duality can also be expressed as a relation of singular homology and de Rham cohomology, by asserting that the map

${\displaystyle (\gamma ,\omega )\mapsto {\int }_{\gamma }\omega }$

(integrating a differential k-form over a (2n − k)-(real-)dimensional cycle) is a perfect pairing.

Poincaré duality also reverses dimensions; it corresponds to the fact that, if a topological manifold is represented as a cell complex, then the dual of the complex (a higher-dimensional generalization of the planar graph dual) represents the same manifold. In Poincaré duality, this homeomorphism is reflected in an isomorphism of the kth homology group and the (n − k)th cohomology group.

Duality in algebraic and arithmetic geometry

The same duality pattern holds for a smooth projective variety over a separably closed field, using l-adic cohomology with Q_ℓ-coefficients instead. This is further generalized to possibly singular varieties, using intersection cohomology instead, a duality called Verdier duality. Serre duality or coherent duality are similar to the statements above, but applies to cohomology of coherent sheaves instead.

With increasing level of generality, it turns out, an increasing amount of technical background is helpful or necessary to understand these theorems: the modern formulation of these dualities can be done using derived categories and certain direct and inverse image functors of sheaves (with respect to the classical analytical topology on manifolds for Poincaré duality, l-adic sheaves and the étale topology in the second case, and with respect to coherent sheaves for coherent duality).

Yet another group of similar duality statements is encountered in arithmetics: étale cohomology of finite, local and global fields (also known as Galois cohomology, since étale cohomology over a field is equivalent to group cohomology of the (absolute) Galois group of the field) admit similar pairings. The absolute Galois group G(F_q) of a finite field, for example, is isomorphic to ${\displaystyle \hat{\mathbf{Z}}}$, the profinite completion of Z, the integers. Therefore, the perfect pairing (for any G-module M)

Hⁿ(G, M) × H¹⁻ⁿ (G, Hom (M, Q/Z)) → Q/Z

is a direct consequence of Pontryagin duality of finite groups. For local and global fields, similar statements exist (local duality and global or Poitou–Tate duality).