Duality (mathematics)

A simple, maybe the most simple, duality arises from considering subsets of a fixed set S. To any subset A ⊆ S, the complement A^c[6] consists of all those elements in S that are not contained in A. It is again a subset of S. Taking the complement has the following properties:

• Applying it twice gives back the original set, i.e., (A^c)^c = A. This is referred to by saying that the operation of taking the complement is an involution.
• An inclusion of sets A ⊆ B is turned into an inclusion in the opposite direction B^c ⊆ A^c.
• Given two subsets A and B of S, A is contained in B^c if and only if B is contained in A^c.

This duality appears in topology as a duality between open and closed subsets of some fixed topological space X: a subset U of X is closed if and only if its complement in X is open. Because of this, many theorems about closed sets are dual to theorems about open sets. For example, any union of open sets is open, so dually, any intersection of closed sets is closed. The interior of a set is the largest open set contained in it, and the closure of the set is the smallest closed set that contains it. Because of the duality, the complement of the interior of any set U is equal to the closure of the complement of U.

A set C (blue) and its dual cone C^* (red).

A duality in geometry is provided by the dual cone construction. Given a set ${\displaystyle C}$ of points in the plane ${\displaystyle \mathbb {R} ^{2}}$ (or more generally points in ${\displaystyle \mathbb {R} ^{n}}$), the dual cone is defined as the set ${\displaystyle C^{*}\subseteq \mathbb {R} ^{2}}$ consisting of those points ${\displaystyle (x_{1},x_{2})}$ satisfying

$${\displaystyle x_{1}c_{1}+x_{2}c_{2}\geq 0}$$

for all points ${\displaystyle (c_{1},c_{2})}$ in ${\displaystyle C}$, as illustrated in the diagram. Unlike for the complement of sets mentioned above, it is not in general true that applying the dual cone construction twice gives back the original set ${\displaystyle C}$. Instead, ${\displaystyle C^{**}}$ is the smallest cone[7] containing ${\displaystyle C}$ which may be bigger than ${\displaystyle C}$. Therefore this duality is weaker than the one above, in that

• Applying the operation twice gives back a possibly bigger set: for all ${\displaystyle C}$, ${\displaystyle C}$ is contained in ${\displaystyle C^{**}}$. (For some ${\displaystyle C}$, namely the cones, the two are actually equal.)

The other two properties carry over without change:

• It is still true that an inclusion ${\displaystyle C\subseteq D}$ is turned into an inclusion in the opposite direction (${\displaystyle D^{*}\subseteq C^{*}}$).
• Given two subsets ${\displaystyle C}$ and ${\displaystyle D}$ of the plane, ${\displaystyle C}$ is contained in ${\displaystyle D^{*}}$ if and only if ${\displaystyle D}$ is contained in ${\displaystyle C^{*}}$.

A very important example of a duality arises in linear algebra by associating to any vector space V its dual vector space V^*. Its elements are the linear functionals ${\displaystyle \varphi :V\to K}$, where K is the field over which V is defined. The three properties of the dual cone carry over to this type of duality by replacing subsets of ${\displaystyle \mathbb {R} ^{2}}$ by vector space and inclusions of such subsets by linear maps. That is:

• Applying the operation of taking the dual vector space twice gives another vector space V^**. There is always a map V → V^**. For some V, namely precisely the finite-dimensional vector spaces, this map is an isomorphism.
• A linear map V → W gives rise to a map in the opposite direction (W^* → V^*).
• Given two vector spaces V and W, the maps from V to W^* correspond to the maps from W to V^*.

A particular feature of this duality is that V and V^* are isomorphic for certain objects, namely finite-dimensional vector spaces. However, this is in a sense a lucky coincidence, for giving such an isomorphism requires a certain choice, for example the choice of a basis of V. This is also true in the case if V is a Hilbert space, via the Riesz representation theorem.

In all the dualities discussed before, the dual of an object is of the same kind as the object itself. For example, the dual of a vector space is again a vector space. Many duality statements are not of this kind. Instead, such dualities reveal a close relation between objects of seemingly different nature. One example of such a more general duality is from Galois theory. For a fixed Galois extension K / F, one may associate the Galois group Gal(K/E) to any intermediate field E (i.e., F ⊆ E ⊆ K). This group is a subgroup of the Galois group G = Gal(K/F). Conversely, to any such subgroup H ⊆ G there is the fixed field K^H consisting of elements fixed by the elements in H.

Compared to the above, this duality has the following features:

• An extension F ⊆ F′ of intermediate fields gives rise to an inclusion of Galois groups in the opposite direction: Gal(K/F′) ⊆ Gal(K/F).
• Associating Gal(K/E) to E and K^H to H are inverse to each other. This is the content of the fundamental theorem of Galois theory.

The construction of the dual vector space

$${\displaystyle V^{*}=\operatorname {Hom} (V,K)}$$

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.

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

$${\displaystyle \varphi :V\times V\to K}$$

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.

The complete quadrangle, a configuration of four points and six lines in the projective plane (left) and its dual configuration, the complete quadrilateral, with four lines and six points (right).

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.[10] 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.[11] 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 ${\displaystyle \mathbb {RP} ^{2}}$ correspond to one-dimensional subvector spaces ${\displaystyle V\subset \mathbb {R} ^{3}}$[12] 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 ${\displaystyle (\mathbb {R} ^{3})^{*}}$ 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 ${\displaystyle (\mathbb {R} ^{3})^{*}}$.

The (positive definite) bilinear form

$${\displaystyle \langle \cdot ,\cdot \rangle$$ :\mathbb {R} ^{3}\times \mathbb {R} ^{3}\to \mathbb {R} ,\langle x,y\rangle =\sum _{i=1}^{3}x_{i}y_{i}}

yields an identification of this projective plane with the ${\displaystyle \mathbb {RP} ^{2}}$. Concretely, the duality assigns to ${\displaystyle V\subset \mathbb {R} ^{3}}$ its orthogonal ${\displaystyle \left\{w\in \mathbb {R} ^{3},\langle v,w\rangle =0{\text{ for all }}v\in V\right\}}$. The explicit formulas in duality in projective geometry arise by means of this identification.

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
  $${\displaystyle H\to H^{*},v\mapsto (w\mapsto \langle w,v\rangle ),}$$

  
  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.[13] All the standard spaces of distributions — ${\displaystyle {\mathcal {D}}'(U)}$, ${\displaystyle {\mathcal {S}}'(\mathbb {R} ^{n})}$, ${\displaystyle {\mathcal {C}}^{\infty }(U)'}$ — are reflexive locally convex spaces.[14]

The dual lattice of a lattice L is given by

$${\displaystyle \operatorname {Hom} (L,\mathbf {Z} ),}$$

which is used in the construction of toric varieties.[15] The Pontryagin dual of locally compact topological groups G is given by

$${\displaystyle \operatorname {Hom} (G,S^{1}),}$$

continuous group homomorphisms with values in the circle (with multiplication of complex numbers as group operation).

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:

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

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.[16] 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.[17]

A category that is equivalent to its dual is called self-dual. An example of self-dual category is the category of Hilbert spaces.[18]

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

and

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,[19] fibrations and cofibrations in topology and more generally model categories.[20]

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

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

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,

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.[21]

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)[22]

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.[23]

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.[24]

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 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)).[25]

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.

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.[26] 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 an 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.

The same duality pattern holds for a smooth projective variety over a separably closed field, using l-adic cohomology with Q_ℓ-coefficients instead.[27] This is further generalized to possibly singular varieties, using intersection cohomology instead, a duality called Verdier duality.[28] Serre duality or coherent duality are similar to the statements above, but applies to cohomology of coherent sheaves instead.[29]

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 ${\displaystyle {\widehat {\mathbf {Z} }}}$, 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[30]

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).[31]

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