View source for Duality (mathematics)

{{Short description|General concept and operation in mathematics}}
{{For|the property of optimization problems|Duality (optimization)}}

In [[mathematics]], a '''duality''' translates concepts, [[theorem]]s or [[mathematical structure]]s into other concepts, theorems or structures in a [[Injective function|one-to-one]] fashion, often (but not always) by means of an [[Involution (mathematics)|involution]] operation: if the dual of {{mvar|A}} is {{mvar|B}}, then the dual of {{mvar|B}} is {{mvar|A}}. In other cases the dual of the dual – the double dual or bidual – is not necessarily identical to the original (also called ''primal''). Such involutions sometimes have [[fixed point (mathematics)|fixed points]], so that the dual of {{mvar|A}} is {{mvar|A}} itself. For example, [[Desargues' theorem]] is '''self-dual'''<!--boldface per WP:R#PLA--> in this sense under the ''standard [[duality (projective geometry)|duality]] in [[projective geometry]]''.

In mathematical contexts, ''duality'' has numerous meanings.<ref>{{harvnb|Atiyah|2007|page=1}}</ref> It has been described as "a very pervasive and important concept in (modern) mathematics"<ref>{{harvnb|Kostrikin|2001|loc=This quote is the first sentence of the final section named comments in this single-paged-document}}</ref> and "an important general theme that has manifestations in almost every area of mathematics".<ref name="PCM187L">{{harvnb|Gowers|2008|loc=p.&nbsp;187, col.&nbsp;1}}</ref>

Many mathematical dualities between objects of two types correspond to [[pairing]]s, [[bilinear function]]s from an object of one type and another object of the second type to some family of scalars. For instance, ''linear algebra duality'' corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the ''duality between [[distribution (mathematics)|distributions]] and the associated [[test function]]s'' corresponds to the pairing in which one integrates a distribution against a test function, and ''[[Poincaré duality]]'' corresponds similarly to [[intersection number]], viewed as a pairing between submanifolds of a given manifold.<ref name="PCM189R">{{harvnb|Gowers|2008|loc=p.&nbsp;189, col.&nbsp;2}}</ref>

From a [[category theory]] viewpoint, duality can also be seen as a [[functor]], at least in the realm of vector spaces. This functor assigns to each space its dual space, and the [[pullback (category theory)|pullback]] construction assigns to each arrow {{math|''f'': ''V'' → ''W''}} its dual {{math|''f''{{sup|∗}}: ''W''{{sup|∗}} → ''V''{{sup|∗}}}}.

== Introductory examples ==
In the words of [[Michael Atiyah]],
{{quote|Duality in mathematics is not a theorem, but a "principle".<ref>{{harvnb|Atiyah|2007|page=1}}</ref>}}

The following list of examples shows the common features of many dualities, but also indicates that the precise meaning of duality may vary from case to case.

=== Complement of a subset ===
A simple duality arises from considering [[subset]]s of a fixed set {{mvar|S}}. To any subset {{math|{{var|A}} ⊆ {{var|S}}}}, the [[complement (set theory)|complement]] {{math|{{var|A}}{{sup|∁}}}}<ref>The complement is also denoted as {{math|{{var|S}} \ {{var|A}}}}.</ref> consists of all those elements in {{mvar|S}} that are not contained in {{mvar|A}}. It is again a subset of {{mvar|S}}. Taking the complement has the following properties:
* Applying it twice gives back the original set, i.e., {{math|1=({{var|A}}{{sup|∁}}){{sup|∁}} = {{var|A}}}}. This is referred to by saying that the operation of taking the complement is an ''[[involution (mathematics)|involution]]''.
* An inclusion of sets {{math|{{var|A}} ⊆ {{var|B}}}} is turned into an inclusion in the ''opposite'' direction {{math|{{var|B}}{{sup|∁}} ⊆ {{var|A}}{{sup|∁}}}}.
* Given two subsets {{mvar|A}} and {{mvar|B}} of {{mvar|S}}, {{mvar|A}} is contained in {{math|{{var|B}}{{sup|∁}}}} [[if and only if]] {{mvar|B}} is contained in {{math|{{var|A}}{{sup|∁}}}}.

This duality appears in [[topology]] as a duality between [[open set|open]] and [[closed set|closed subset]]s of some fixed topological space {{mvar|X}}: a subset {{mvar|U}} of {{mvar|X}} is closed if and only if its complement in {{mvar|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.<ref>{{harvtxt|Rudin|1976|p=34}}</ref>  The [[interior (topology)|interior]] of a set is the largest open set contained in it, and the [[closure (topology)|closure]] of the set is the smallest closed set that contains it.  Because of the duality, the complement of the interior of any set {{mvar|U}} is equal to the closure of the complement of {{mvar|U}}.

=== Dual cone ===
[[File:Dual cone illustration.svg|right|thumb|A set {{mvar|C}} (blue) and its dual cone {{math|{{var|C}}{{sup|*}}}} (red).]]
A duality in [[geometry]] is provided by the [[dual cone]] construction. Given a set <math>C</math> of points in the plane <math>\mathbb R^2</math> (or more generally points in {{nowrap|<math>\mathbb R^n</math>),}} the dual cone is defined as the set <math>C^* \subseteq \mathbb R^2</math> consisting of those points <math>(x_1, x_2)</math> satisfying
<math display="block">x_1 c_1 + x_2 c_2 \ge 0</math>
for all points <math>(c_1, c_2)</math> in <math>C</math>, 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 <math>C</math>. Instead, <math>C^{**}</math> is the smallest cone<ref>More precisely, <math>C^{**}</math> is the smallest [[closed set|closed]] [[convex set|convex]] cone containing <math>C</math>.</ref> containing <math>C</math> which may be bigger than <math>C</math>. Therefore this duality is weaker than the one above, in that
* Applying the operation twice gives back a possibly bigger set: for all <math>C</math>, <math>C</math> is contained in <math>C^{**}</math>. (For some <math>C</math>, namely the cones, the two are actually equal.)
The other two properties carry over without change:
* It is still true that an inclusion <math>C \subseteq D</math> is turned into an inclusion in the opposite direction (<math>D^* \subseteq C^*</math>).
* Given two subsets <math>C</math> and <math>D</math> of the plane, <math>C</math> is contained in <math>D^*</math> if and only if <math>D</math> is contained in <math>C^*</math>.

=== Dual vector space ===
A very important example of a duality arises in [[linear algebra]] by associating to any [[vector space]] {{mvar|V}} its [[dual vector space]] {{math|{{var|V}}{{sup|*}}}}. Its elements are the [[linear functional]]s <math>\varphi: V \to K</math>, where {{mvar|K}} is the [[field (mathematics)|field]] over which {{mvar|V}} is defined.
The three properties of the dual cone carry over to this type of duality by replacing subsets of <math>\mathbb R^2</math> 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 {{math|{{var|V}}{{sup|**}}}}. There is always a map {{math|{{var|V}} → {{var|V}}{{sup|**}}}}. For some {{mvar|V}}, namely precisely the [[finite-dimensional vector space]]s, this map is an [[isomorphism]]. 
* A linear map {{math|{{var|V}} → {{var|W}}}} gives rise to a map in the opposite direction ({{math|{{var|W}}{{sup|*}} → {{var|V}}{{sup|*}}}}).
* Given two vector spaces {{mvar|V}} and {{mvar|W}}, the maps from {{mvar|V}} to {{math|{{var|W}}{{sup|*}}}} correspond to the maps from {{mvar|W}} to {{math|{{var|V}}{{sup|*}}}}.
A particular feature of this duality is that {{mvar|V}} and {{math|{{var|V}}{{sup|*}}}} 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 (linear algebra)|basis]] of {{mvar|V}}. This is also true in the case if {{mvar|V}} is a [[Hilbert space]], ''via'' the [[Riesz representation theorem]].

=== Galois theory ===

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]] {{math|{{var|K}} / {{var|F}}}}, one may associate the [[Galois group]] {{math|Gal({{var|K}}/{{var|E}})}} to any intermediate field {{mvar|E}} (i.e., {{math|{{var|F}} ⊆ {{var|E}} ⊆ {{var|K}}}}). This group is a subgroup of the Galois group {{math|1={{var|G}} = Gal({{var|K}}/{{var|F}})}}. Conversely, to any such subgroup {{math|{{var|H}} ⊆ {{var|G}}}} there is the fixed field {{math|{{var|K}}{{sup|{{var|H}}}}}} consisting of elements fixed by the elements in {{mvar|H}}.

Compared to the above, this duality has the following features:
* An extension {{math|{{var|F}} ⊆ {{var|F}}&prime;}} of intermediate fields gives rise to an inclusion of Galois groups in the opposite direction: {{math|Gal({{var|K}}/{{var|F}}&prime;) ⊆ Gal({{var|K}}/{{var|F}})}}.
* Associating {{math|Gal({{var|K}}/{{var|E}})}} to {{mvar|E}} and {{math|{{var|K}}{{sup|{{var|H}}}}}} to {{mvar|H}} are inverse to each other. This is the content of the [[fundamental theorem of Galois theory]].

== Order-reversing dualities ==
[[File:Upset.svg|thumb|[[Hasse diagram]] of the power set of {{math|{1,2,3,4}<nowiki/>}}, partially ordered by {{math|⊂}}. The dual poset, i.e. ordering by {{math|⊃}}, is obtained by turning the diagram upside-down. The green nodes form an [[upper set]] and a lower set in the original and the dual order, respectively.]]
{{main article|Duality (order theory)}}
Given a [[poset]] {{math|1={{var|P}} = ({{var|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 [[duality (order theory)|dual]] poset {{math|1={{var|P}}{{sup|{{var|d}}}} = ({{var|X}}, ≥)}} comprises the same ground set but the [[converse relation]]. Familiar examples of dual partial orders include
* the subset and superset relations {{math|⊂}} and {{math|⊃}} on any collection of sets, such as the subsets of a fixed set {{mvar|S}}. This gives rise to the first example of a duality mentioned [[#Complement of a subset|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]] {{mvar|f}} of a [[partially ordered set]] {{mvar|S}}, that is, an [[Order theory#Functions between orders|order-reversing]] involution {{math|{{var|f}} : {{var|S}} → {{var|S}}}}.<ref>{{harvnb|Artstein-Avidan|Milman|2007}}</ref><ref>{{harvnb|Artstein-Avidan|Milman|2008}}</ref> In several important cases these simple properties determine the transform uniquely up to some simple symmetries. For example, if {{math|{{var|f}}{{sub|1}}}}, {{math|{{var|f}}{{sub|2}}}} are two duality transforms then their [[Function composition|composition]] is an [[Order isomorphism|order automorphism]] of {{mvar|S}}; thus, any two duality transforms differ only by an order automorphism. For example, all order automorphisms of a [[power set]] {{math|1={{var|S}} = 2{{sup|{{var|R}}}}}} are induced by permutations of {{mvar|R}}.

A concept defined for a partial order {{mvar|P}} will correspond to a ''dual concept'' on the dual poset {{math|{{var|P}}{{sup|{{var|d}}}}}}. For instance, a [[minimal element]] of {{mvar|P}} will be a [[maximal element]] of {{math|{{var|P}}{{sup|{{var|d}}}}}}: minimality and maximality are dual concepts in order theory. Other pairs of dual concepts are [[upper and lower bounds]], [[lower set]]s and [[upper set]]s, and [[ideal (order theory)|ideals]] and [[filter (mathematics)|filters]].

In topology, [[open set]]s and [[closed set]]s 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 ==
[[File:Dual Cube-Octahedron.svg|thumb|200px|The features of the cube and its dual octahedron correspond one-for-one with dimensions reversed.]]
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 of this is the duality of the [[Platonic solid]]s, 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 [[vertex (geometry)|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 [[pole and polar|polar reciprocation]], any [[convex polyhedron]], or more generally any [[convex polytope]], corresponds to a [[dual polyhedron]] or dual polytope, with an {{mvar|i}}-dimensional feature of an {{mvar|n}}-dimensional polytope corresponding to an {{math|({{var|n}} − {{var|i}} − 1)}}-dimensional feature of the dual polytope. The incidence-preserving nature of the duality is reflected in the fact that the [[face lattice]]s of the primal and dual polyhedra or polytopes are themselves [[#order-theoretic duality|order-theoretic duals]]. Duality of polytopes and order-theoretic duality are both [[Involution (mathematics)|involution]]s: 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.

[[File:Duals graphs.svg|right|thumb|240px|A [[planar graph]] in blue, and its [[dual graph]] in red.]]
From any 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 embedding]]s 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 {{mvar|S}} of points in the plane between the [[Delaunay triangulation]] of {{mvar|S}} and the [[Voronoi diagram]] of {{mvar|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 (mathematics)|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 {{nowrap|<math>\mathbb R^n</math>),}} a system of linear constraints (specifying that the point lie in a [[Half-space (geometry)|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.
{{Clear}}

== Duality in logic and set theory ==
In logic, functions or relations {{mvar|A}} and {{mvar|B}} are considered dual if {{math|1={{var|A}}(¬{{var|x}}) = ¬{{var|B}}({{var|x}})}}, where ¬ is [[logical negation]]. The basic duality of this type is the duality of the ∃ and ∀ [[Quantifier (logic)|quantifiers]] in classical logic. These are dual because {{math|∃{{var|x}}.¬{{var|P}}({{var|x}})}} and {{math|¬∀{{var|x}}.{{var|P}}({{var|x}})}} are equivalent for all predicates {{var|P}} in classical logic: if there exists an {{var|x}} for which {{var|P}} fails to hold, then it is false that {{var|P}} holds for all {{var|x}} (but the converse does not hold constructively). From this fundamental logical duality follow several others:
* A formula is said to be ''[[satisfiability|satisfiable]]'' in a certain model if there are assignments to its [[free variable]]s 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 {{math|∧}} and {{math|∨}} operators are dual in this sense, because {{math|(¬{{var|x}} ∧ ¬{{var|y}})}} and {{math|¬({{var|x}} ∨ {{var|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, {{math|1={{big|∧}} (¬ {{var|x}}{{sub|{{var|i}}}}) = ¬{{big|∨}} {{var|x}}{{sub|{{var|i}}}}}}.  The left side is true if and only if {{math|∀{{var|i}}.¬{{var|x}}{{sub|{{var|i}}}}}}, and the right side if and only if ¬∃''i''.''x''<sub>''i''</sub>.
* In [[modal logic]], {{math|□{{var|p}}}} means that the proposition {{mvar|p}} is "necessarily" true, and {{math|◊{{var|p}}}} that {{mvar|p}} is "possibly" true.  Most interpretations of modal logic assign dual meanings to these two operators.  For example in [[Kripke semantics]], "{{mvar|p}} is possibly true" means "there exists some world {{mvar|W}} such that {{mvar|p}} is true in {{mvar|W}}", while "{{mvar|p}} is necessarily true" means "for all worlds {{mvar|W}}, {{mvar|p}} is true in {{mvar|W}}".  The duality of {{math|□}} and {{math|◊}} then follows from the analogous duality of {{math|∀}} and {{math|∃}}. 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 {{math|{{sup|∁}}}}. That is, {{math|1={{var|A}}{{sup|∁}} ∩ {{var|B}}{{sup|∁}} = ({{var|A}} ∪ {{var|B}}){{sup|∁}}}}, and more generally, {{math|1=⋂ {{var|A}}{{sub|{{var|α}}}}{{sup|∁}} = (⋃ {{var|A}}{{sub|{{var|α}}}}){{sup|∁}}}}. This follows from the duality of {{math|∀}} and {{math|∃}}: an element {{mvar|x}} is a member of {{math|⋂ {{var|A}}{{sub|{{var|α}}}}{{sup|∁}}}} if and only if {{math|∀{{var|α}}.¬{{var|x}} ∈ {{var|A}}{{sub|{{var|α}}}}}}, and is a member of {{math|(⋂ {{var|A}}{{sub|{{var|α}}}}){{sup|∁}}}} if and only if {{math|¬∃{{var|α}}. {{var|x}} ∈ {{var|A}}{{sub|{{var|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 the 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, {{math|{{var|V}} → {{var|V}}{{sup|**}}}}, 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 {{slink||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 {{slink|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 isomophism, the module is called reflexive.

For [[topological vector space]]s (including [[normed vector space]]s), there is a separate notion of a [[Dual_space#Continuous_dual_space|topological dual]], denoted {{tmath|V'}} to distinguish from the algebraic dual {{var|V}}{{sup|*}}, with different possible topologies on the dual, each of which defines a different bidual space {{tmath|V' '}}. In these cases the canonical evaluation map {{tmath|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 {{mvar|X}}, the set of morphisms {{math|Hom ({{var|X}}, {{var|D}})}} into some fixed object {{mvar|D}}, with a structure similar to that of {{mvar|X}}. This is sometimes called [[internal Hom]]. In general, this yields a true duality only for specific choices of {{mvar|D}}, in which case {{math|1={{var|X}}{{sup|*}} = Hom ({{var|X}}, {{var|D}})}} is referred to as the ''dual'' of {{mvar|X}}. There is always a map from {{mvar|X}} to the ''bidual'', that is to say, the dual of the dual,
<math display="block">X \to X^{**} := (X^*)^* = \operatorname{Hom}(\operatorname{Hom}(X, D), D).</math>
It assigns to some {{math|{{var|x}} ∈ {{var|X}}}} the map that associates to any map {{math|{{var|f}} : {{var|X}} → {{var|D}}}} (i.e., an element in {{math|Hom({{var|X}}, {{var|D}})}}) the value {{math|{{var|f}}({{var|x}})}}.
Depending on the concrete duality considered and also depending on the object {{mvar|X}}, this map may or may not be an isomorphism.

=== Dual vector spaces revisited ===
The construction of the dual vector space
<math display="block">V^* = \operatorname{Hom}(V, K)</math>
mentioned in the introduction is an example of such a duality. Indeed, the set of morphisms, i.e., [[linear map]]s, forms a vector space in its own right. The map {{math|{{var|V}} → {{var|V}}{{sup|**}}}} mentioned above is always injective. It is surjective, and therefore an isomorphism, if and only if the [[Hamel dimension|dimension]] of {{mvar|V}} is finite. This fact characterizes finite-dimensional vector spaces without referring to a basis.

==== Isomorphisms of {{math|''V''}} and {{math|''V''{{sup|∗}}}} and inner product spaces ====
A vector space {{math|''V''}} is isomorphic to {{math|''V''{{sup|∗}}}} precisely if {{math|''V''}} is finite-dimensional. In this case, such an isomorphism is equivalent to a non-degenerate [[bilinear form]]
<math display="block">\varphi: V \times V \to K</math>
In this case {{math|''V''}} is called an [[inner product space]].
For example, if {{math|''K''}} is the field of [[real number|real]] or [[complex number]]s, any [[positive definite]] bilinear form gives rise to such an isomorphism. In [[Riemannian geometry]], {{math|''V''}} is taken to be the [[tangent space]] of a [[manifold]] and such positive bilinear forms are called [[Riemannian metric]]s. 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 {{math|''n''}}-dimensional vector space, the Hodge star operator maps [[differential form|{{math|''k''}}-forms]] to {{math|(''n'' − ''k'')}}-forms. This can be used to formulate [[Maxwell's equation]]s. In this guise, the duality inherent in the inner product space exchanges the role of [[magnetic field|magnetic]] and [[electric field]]s.

==== Duality in projective geometry ====
[[File:Complete-quads.svg|thumb|240px|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 plane]]s, it is possible to find [[Transformation (geometry)|geometric transformation]]s 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.<ref>{{harvnb|Veblen|Young|1965}}.</ref> For such planes there arises a general principle of [[duality (projective geometry)|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.<ref>{{Harvard citations|last1=Veblen|last2=Young|year=1965|loc = Ch. I, Theorem 11}}</ref> 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 [[Line-line intersection|intersection point]] of these two lines". For further examples, see [[Duality (projective geometry)#Dual theorems|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 <math>\mathbb{RP}^2</math> correspond to one-dimensional subvector spaces <math>V \subset \mathbb R^3</math><ref>More generally, one can consider the projective planes over any field, such as the complex numbers or [[finite field]]s or even [[division ring]]s.</ref> while the lines in the projective plane correspond to subvector spaces <math>W</math> of dimension 2. The duality in such projective geometries stems from assigning to a one-dimensional <math>V</math> the subspace of <math>(\mathbb R^3)^*</math> consisting of those linear maps <math>f: \mathbb R^3 \to \mathbb R</math> which satisfy <math>f (V) = 0</math>. 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 <math>(\mathbb R^3)^*</math>.

The (positive definite) bilinear form
<math display="block">\langle \cdot , \cdot \rangle : \R^3 \times \R^3 \to \R, \langle x , y \rangle = \sum_{i=1}^3 x_i y_i</math>
yields an identification of this projective plane with the <math>\mathbb{RP}^2</math>. Concretely, the duality assigns to <math>V \subset \mathbb R^3</math> its [[orthogonal subspace|orthogonal]] <math>\left\{w \in \R^3, \langle v, w \rangle = 0 \text{ for all } v \in V\right\}</math>. The explicit formulas in [[duality (projective geometry)|duality in projective geometry]] arise by means of this identification.

=== Topological vector spaces and Hilbert spaces ===
In the realm of [[topological vector space]]s, a similar construction exists, replacing the dual by the [[Dual vector space#Continuous dual space|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 <math>X</math> that is canonically isomorphic to its bidual <math>X''</math> is called a [[reflexive space]]:
<math display="block">X\cong X''.</math>

Examples:
* As in the finite-dimensional case, on each [[Hilbert space]] {{mvar|H}} its [[inner product]] {{math|⟨⋅, ⋅⟩}} defines a map <math display="block">H \to H^*, v \mapsto (w \mapsto \langle w,v \rangle),</math> which is a [[bijection]] due to the [[Riesz representation theorem]]. As a corollary, every Hilbert space is a [[Reflexive space#Reflexive Banach spaces|reflexive Banach space]].
* The [[Reflexive space#Reflexive Banach spaces|dual normed space]] of an [[Lp spaces|{{math|{{var|L}}{{sup|{{var|p}}}}}}-space]] is {{math|{{var|L}}{{sup|{{var|q}}}}}} where {{math|1=1/{{var|p}} + 1/{{var|q}} = 1}} provided that {{math|1 ≤ {{var|p}} < ∞}}, but the dual of {{math|{{var|L}}{{sup|∞}}}} is bigger than {{math|{{var|L}}{{sup|1}}}}. Hence {{math|{{var|L}}{{sup|1}}}} is not reflexive.
* [[Distribution (mathematics)|Distributions]] are linear functionals on appropriate spaces of functions. They are an important technical means in the theory of [[partial differential equation]]s (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.<ref>See [[elliptic regularity]].</ref> All the standard spaces of distributions — <math>{\mathcal D}'(U)</math>, <math>{\mathcal S}'(\R^n)</math>, <math>{\mathcal C}^\infty(U)'</math> — are reflexive locally convex spaces.<ref>{{harvtxt|Edwards|1965|loc=8.4.7}}.</ref>

=== Further dual objects ===
The [[dual lattice]] of a [[Lattice (group)|lattice]] {{mvar|L}} is given by
<math display="block">\operatorname{Hom} (L, \mathbb{Z}),</math>
the set of linear functions on the [[real vector space]] containing the lattice that map the points of the lattice to the integers <math>\mathbb{Z}</math>. This is used in the construction of [[toric variety|toric varieties]].<ref>{{Harvard citations| last1=Fulton | year=1993|nb=yes}}</ref> The [[Pontryagin dual]] of [[locally compact]] [[topological group]]s ''G'' is given by
<math display="block">\operatorname{Hom} (G, S^1),</math>
continuous [[group homomorphisms]] with values in the circle (with multiplication of complex numbers as group operation).

== Dual categories ==
{{main article|Dual (category theory)}}

=== 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 [[category (mathematics)|categories]] {{mvar|C}} and {{mvar|D}}:
{{block indent|{{math|{{var|F}}: {{var|C}} → {{var|D}}}}}}
which for any two objects ''X'' and ''Y'' of ''C'' gives a map
{{block indent|{{math|Hom{{sub|{{var|C}}}}({{var|X}}, {{var|Y}}) → Hom{{sub|{{var|D}}}}({{var|F}}({{var|Y}}), {{var|F}}({{var|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]] {{math|{{var|C}}{{sup|op}}}} of {{mvar|C}}, and {{mvar|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.<ref>{{Harvnb|Mac Lane|1998|loc=Ch. II.1}}.</ref> Therefore, any duality between categories {{mvar|C}} and {{mvar|D}} is formally the same as an equivalence between {{mvar|C}} and {{math|{{var|D}}{{sup|op}}}} ({{math|{{var|C}}{{sup|op}}}} and {{mvar|D}}). However, in many circumstances the opposite categories have no inherent meaning, which makes duality an additional, separate concept.<ref>{{Harvard citations | last1=Lam|year=1999|loc=§19C}}</ref>

A category that is equivalent to its dual is called ''self-dual''. An example of self-dual category is the category of [[Hilbert space]]s.<ref name="AdamekRosicky1994">{{cite book|author1=Jiří Adámek|author2=J. Rosicky|title=Locally Presentable and Accessible Categories|url=https://books.google.com/books?id=iXh6rOd7of0C&pg=PA62|year=1994|publisher=Cambridge University Press|isbn=978-0-521-42261-1|page=62}}</ref>

Many [[category theory|category-theoretic]] notions come in pairs in the sense that they correspond to each other while considering the opposite category. For example, [[Cartesian product]]s {{math|{{var|Y}}{{sub|1}} × {{var|Y}}{{sub|2}}}} and [[disjoint union]]s {{math|{{var|Y}}{{sub|1}} ⊔ {{var|Y}}{{sub|2}}}} of sets are dual to each other in the sense that
{{block indent|{{math|1=Hom ({{var|X}}, {{var|Y}}{{sub|1}} × {{var|Y}}{{sub|2}}) = Hom ({{var|X}}, {{var|Y}}{{sub|1}}) × Hom ({{var|X}}, {{var|Y}}{{sub|2}})}}}}
and
{{block indent|{{math|1=Hom ({{var|Y}}{{sub|1}} ⊔ {{var|Y}}{{sub|2}}, {{var|X}}) = Hom ({{var|Y}}{{sub|1}}, {{var|X}}) × Hom ({{var|Y}}{{sub|2}}, {{var|X}})}}}}
for any set {{mvar|X}}. This is a particular case of a more general duality phenomenon, under which [[limit (category theory)|limits]] in a category {{mvar|C}} correspond to [[colimit]]s in the opposite category {{math|{{var|C}}{{sup|op}}}}; further concrete examples of this are [[epimorphism]]s vs. [[monomorphism]], in particular [[factor module]]s (or groups etc.) vs. [[submodule]]s, [[direct product]]s vs. [[Direct sum of groups|direct sums]] (also called [[coproduct]]s 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 module|projective]] and [[injective module]]s in [[homological algebra]],<ref>{{Harvard citations | last1=Weibel |year=1994|txt}}</ref> [[fibration]]s and [[cofibration]]s in topology and more generally [[model category|model categories]].<ref>{{Harvard citations | last1=Dwyer | last2=Spaliński | year=1995|txt}}</ref>

Two [[functor]]s {{math|{{var|F}}: {{var|C}} → {{var|D}}}} and {{math|{{var|G}}: {{var|D}} → {{var|C}}}} are [[adjoint functor|adjoint]] if for all objects ''c'' in ''C'' and ''d'' in ''D''
{{block indent|{{math|Hom{{sub|{{var|D}}}}(F({{var|c}}), {{var|d}}) ≅ Hom{{sub|{{var|C}}}}({{var|c}}, {{var|G}}({{var|d}}))}},}}
in a natural way.  Actually, the correspondence of limits and colimits is an example of adjoints, since there is an adjunction
{{block indent|{{math|colim: {{var|C}}{{sup|{{var|I}}}} ↔ {{var|C}}: Δ}}}}
between the colimit functor that assigns to any diagram in {{mvar|C}} indexed by some category {{mvar|I}} its colimit and the diagonal functor that maps any object {{mvar|c}} of {{mvar|C}} to the constant diagram which has {{mvar|c}} at all places. Dually,
{{block indent|{{math|Δ: {{var|C}} ↔ {{var|C}}{{sup|{{var|I}}}}: lim}}.}}

=== Spaces and functions ===

[[Gelfand duality]] is a duality between commutative [[C*-algebra]]s ''A'' and [[compact space|compact]] [[Hausdorff space]]s ''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.<ref>{{Harvnb|Negrepontis|1971}}.</ref>

In a similar vein there is a duality in [[algebraic geometry]] between [[commutative ring]]s and [[affine scheme]]s: to every commutative ring ''A'' there is an affine spectrum, [[spectrum of a ring|Spec ''A'']]. Conversely, given an affine scheme ''S'', one gets back a ring by taking global sections of the [[structure sheaf]] O<sub>''S''</sub>. In addition, [[ring homomorphism]]s are in one-to-one correspondence with morphisms of affine schemes, thereby there is an equivalence
: (Commutative rings)<sup>op</sup> ≅ (affine schemes)<ref>{{Harvard citations | last1=Hartshorne | year=1966 | loc=Ch. II.2, esp. Prop. II.2.3|nb=yes}}</ref>
Affine schemes are the local building blocks of [[scheme (mathematics)|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.<ref>{{Harvard citations | last1=Joyal | last2=Street|year=1991 |txt}}</ref>

=== 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.<ref>See {{Harvard citations | last1=Lang|year=2002|loc=Theorem VI.1.1}} for finite Galois extensions.</ref>

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 space]]s and spatial [[locale (mathematics)|locales]].
* [[Birkhoff's representation theorem]] relating [[distributive lattice]]s and [[partial order]]s

=== Pontryagin duality ===
[[Pontryagin duality]] gives a duality on the category of [[locally compact]] [[abelian group]]s: given any such group ''G'', the [[character group]]
: ''χ''(''G'') = Hom (''G'', ''S''<sup>1</sup>)
given by continuous group homomorphisms from ''G'' to the [[circle group]] ''S''<sup>1</sup> 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'')).<ref>{{Harvard citations | last1=Loomis|year=1953 | loc=p. 151, section 37D}}</ref>
Moreover, [[discrete group]]s correspond to [[compact group|compact abelian group]]s; 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 [[mathematical analysis|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:
<math display="block">\widehat{f}(\xi) := \int_{-\infty}^\infty f(x)\ e^{- 2\pi i x \xi} \, dx, </math>
and conversely
<math display="block">f(x) = \int_{-\infty}^\infty \widehat{f}(\xi)\ e^{2 \pi i x \xi} \, d\xi.</math>
If ''f'' is an [[Lebesgue integration|''L''<sup>2</sup>-function]] on '''R''' or '''R'''<sup>''N''</sup>, say, then so is <math>\widehat{f}</math> and <math>f(-x) = \widehat{\widehat{f}}(x)</math>. Moreover, the transform interchanges operations of multiplication and [[convolution]] on the corresponding [[function space]]s. A conceptual explanation of the Fourier transform is obtained by the aforementioned Pontryagin duality, applied to the locally compact groups '''R''' (or '''R'''<sup>''N''</sup> etc.): any character of '''R''' is given by ''ξ'' ↦ ''e''<sup>−2''πixξ''</sup>. The dualizing character of Fourier transform has many other manifestations, for example, in alternative descriptions of [[quantum mechanics|quantum mechanical]] systems in terms of coordinate and momentum representations.
* [[Laplace transform]] is similar to Fourier transform and interchanges [[linear operator|operators]] of multiplication by polynomials with constant coefficient [[linear differential operator]]s.
* [[Legendre transformation]] is an important analytic duality which switches between [[velocity|velocities]] in [[Lagrangian mechanics]] and [[momentum|momenta]] in [[Hamiltonian mechanics]].

==Homology and cohomology==
Theorems showing that certain objects of interest are the [[dual space]]s (in the sense of linear algebra) of other objects of interest are often called ''dualities''. Many of these dualities are given by a [[bilinear function|bilinear pairing]] of two ''K''-vector spaces
: ''A'' ⊗ ''B'' → ''K''.
For [[perfect pairing]]s, there is, therefore, an isomorphism of ''A'' to the [[dual vector space|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''')
: H<sup>''i''</sup>(X) ⊗ H<sup>2''n''−''i''</sup>(X) → '''C''',
where ''n'' is the (complex) dimension of ''X''.<ref>{{Harvard citations | last1=Griffiths | last2=Harris | year=1994|nb=yes|loc=p. 56}}</ref> Poincaré duality can also be expressed as a relation of [[singular homology]] and [[de Rham cohomology]], by asserting that the map
: <math>(\gamma, \omega) \mapsto \int_\gamma \omega</math>
(integrating a differential ''k''-form over a (2''n''&nbsp;−&nbsp;''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 ''k''th [[homology (mathematics)|homology]] group and the (''n''&nbsp;−&nbsp;''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'''<sub>ℓ</sub>-coefficients instead.<ref>{{Harvard citations | last1=Milne |year=1980|loc=Ch. VI.11|nb=yes}}</ref> This is further generalized to possibly [[singular variety|singular varieties]], using [[intersection cohomology]] instead, a duality called [[Verdier duality]].<ref>{{Harvard citations | last1=Iversen | year=1986|nb=yes|loc=Ch. VII.3, VII.5}}</ref> [[Serre duality]] or [[coherent duality]] are similar to the statements above, but applies to cohomology of [[coherent sheaves]] instead.<ref>{{Harvard citations | last1=Hartshorne | year=1966 | loc=Ch. III.7|nb=yes}}</ref>

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 category|derived categories]] and certain [[image functors for sheaves|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 field|finite]], [[local field|local]]  and [[global field]]s (also known as [[Galois cohomology]], since étale cohomology over a field is equivalent to [[group cohomology]] of the ([[absolute Galois group|absolute]]) [[Galois group]] of the field) admit similar pairings. The absolute Galois group ''G''('''F'''<sub>''q''</sub>) of a finite field, for example, is isomorphic to <math>\widehat {\mathbf Z}</math>, the [[profinite completion]] of '''Z''', the integers. Therefore, the perfect pairing (for any [[G-module|''G''-module]] ''M'')
: H<sup>''n''</sup>(''G'', ''M'') × H<sup>1−''n''</sup> (''G'', Hom (''M'', '''Q'''/'''Z''')) → '''Q'''/'''Z'''<ref>{{Harvard citations|last=Milne|year=2006|loc=Example I.1.10|txt}}</ref>
is a direct consequence of [[Pontryagin duality]] of finite groups. For local and global fields, similar statements exist ([[Local Tate duality|local duality]] and global or [[Poitou&ndash;Tate duality]]).<ref>{{Harvard citations|last1=Mazur|year=1973|txt}}; {{Harvard citations|last=Milne|year=2006|txt}}</ref>

== See also ==
{{div col|colwidth=20em}}
* [[Adjoint functor]]
* [[Autonomous category]]
* [[Convex body]] and polar body.
* [[Dual abelian variety]]
* [[Dual basis]]
* [[Dual (category theory)]]
* [[Dual code]]
* [[Duality (electrical engineering)]]
* [[Duality (optimization)]]
* [[Dualizing module]]
* [[Dualizing sheaf]]
* [[Dual lattice]]
* [[Dual norm]]
* [[Dual number]]s, a certain [[associative algebra]]; the term "dual" here is synonymous with ''double'', and is unrelated to the notions given above.
* [[Dual system]]
* [[Koszul duality]]
* [[Langlands dual]]
* [[Linear programming#Duality]]
* [[List of dualities]]
* [[Matlis duality]]
* [[Petrie polygon#Petrie dual|Petrie duality]]
* [[Pontryagin duality]]
* [[S-duality]]
* [[T-duality]], [[Homological mirror symmetry|Mirror symmetry]]
{{div col end}}

== Notes ==
{{reflist}}

== References ==
{{sfn whitelist|CITEREFKostrikin2001}}

=== Duality in general ===
* {{cite web |last=Atiyah |first=Michael |date=2007 |title=Duality in Mathematics and Physics lecture notes from the Institut de Matematica de la Universitat de Barcelona (IMUB) |url=http://www.fme.upc.edu/ca/arxius/butlleti-digital/riemann/071218_conferencia_atiyah-d_article.pdf }}
* {{Springer|title=Duality|id=D/d034120|first=A. I.|last=Kostrikin|authorlink=Alexei Kostrikin}}.
* {{citation |contribution=III.19 Duality |title=The Princeton Companion to Mathematics |pages=187–190 |first=Timothy |last=Gowers |publisher=Princeton University Press |year=2008|title-link=The Princeton Companion to Mathematics }}.
* {{Citation | doi=10.1090/S0273-0979-01-00913-2 | last1=Cartier | first1=Pierre | title=A mad day's work: from Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry | url=https://www.ams.org/bull/2001-38-04/S0273-0979-01-00913-2/ | mr=1848254 | year=2001 | journal=Bulletin of the American Mathematical Society |series=New Series | issn=0002-9904 | volume=38 | issue=4 | pages=389–408| doi-access=free }} (a non-technical overview about several aspects of geometry, including dualities)

=== Duality in algebraic topology ===
* James C. Becker and Daniel Henry Gottlieb, [http://www.math.purdue.edu/~gottlieb/Bibliography/53.pdf A History of Duality in Algebraic Topology]

===Specific dualities===
* {{Citation | last1=Artstein-Avidan | first1=Shiri | author1-link = Shiri Artstein | last2=Milman | first2=Vitali | author2-link=Vitali Milman | year=2008 | title=The concept of duality for measure projections of convex bodies | journal=Journal of Functional Analysis | volume=254 | issue=10 | pages=2648–66 | doi=10.1016/j.jfa.2007.11.008| doi-access=free | citeseerx=10.1.1.417.3470 }}. Also [http://www.math.tau.ac.il/~shiri/publications.html author's site].
* {{Citation | last1=Artstein-Avidan | first1=Shiri | last2=Milman | first2=Vitali | author2-link=Vitali Milman | year=2007 | title=A characterization of the concept of duality | journal=Electronic Research Announcements in Mathematical Sciences | volume=14 | pages=42–59 | url=http://www.aimsciences.org/journals/pdfs.jsp?paperID=2887&mode=full | access-date=2009-05-30 | archive-url=https://web.archive.org/web/20110724230910/http://www.aimsciences.org/journals/pdfs.jsp?paperID=2887&mode=full | archive-date=2011-07-24 | url-status=dead }}. Also [http://www.math.tau.ac.il/~shiri/publications.html author's site].
* {{Citation | last1=Dwyer | first1=William G. | author1-link=William Gerard Dwyer | last2=Spaliński | first2=Jan | title=Handbook of algebraic topology | chapter-url=http://hopf.math.purdue.edu/cgi-bin/generate?/Dwyer-Spalinski/theories | publisher=North-Holland | location=Amsterdam | mr=1361887 | year=1995 | chapter=Homotopy theories and model categories | pages=73–126 | access-date=2009-03-11 | archive-date=2021-02-09 | archive-url=https://web.archive.org/web/20210209105759/https://hopf.math.purdue.edu/cgi-bin/generate?%2FDwyer-Spalinski%2Ftheories | url-status=dead }}
* {{Citation | last1=Fulton | first1=William | author1-link=William Fulton (mathematician) | title=Introduction to toric varieties | publisher=[[Princeton University Press]] | isbn=978-0-691-00049-7 | year=1993}}
* {{Citation | last1=Griffiths | first1=Phillip | author1-link=Phillip Griffiths | last2=Harris | first2=Joseph | author2-link=Joe Harris (mathematician) | title=Principles of algebraic geometry | publisher=[[John Wiley & Sons]] | location=New York | series=Wiley Classics Library | isbn=978-0-471-05059-9 | mr=1288523 | year=1994}}
* {{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Residues and Duality | publisher=[[Springer-Verlag]]  | series=Lecture Notes in Mathematics  |volume=20 | year=1966 | pages=20–48 |isbn=978-3-540-34794-1}}
* {{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Algebraic Geometry | publisher=[[Springer-Verlag]] | isbn=978-0-387-90244-9 | oclc=13348052 | mr=0463157 | year=1977| title-link=Algebraic Geometry (book) }}
* {{Citation | last1=Iversen | first1=Birger | title=Cohomology of sheaves | publisher=[[Springer-Verlag]] | series=Universitext | isbn=978-3-540-16389-3 | mr=842190 | year=1986}}
* {{Citation | last1=Joyal | first1=André | author1-link=André Joyal | last2=Street | first2=Ross | author2-link=Ross Street  | title=Category theory |doi=10.1007/BFb0084235 | chapter-url=http://www.maths.mq.edu.au/~street/CT90Como.pdf | publisher=[[Springer-Verlag]] |isbn=978-3-540-46435-8 | series=Lecture Notes in Mathematics | mr=1173027 | year=1991 | volume=1488 | chapter=An introduction to Tannaka duality and quantum groups | pages=413–492}}
* {{Citation | last1=Lam | first1=Tsit-Yuen | title=Lectures on modules and rings | publisher=[[Springer-Verlag]] | series=Graduate Texts in Mathematics |volume=189 | isbn=978-0-387-98428-5 | mr=1653294 | year=1999}}
* {{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Algebra | publisher=[[Springer-Verlag]] | series=Graduate Texts in Mathematics | isbn=978-0-387-95385-4 | mr=1878556 | year=2002 | volume=211}}
* {{Citation | last1=Loomis | first1=Lynn H. | title=An introduction to abstract harmonic analysis | publisher=D. Van Nostrand  | year=1953 | pages=x+190| hdl=2027/uc1.b4250788 | hdl-access=free }}
* {{Citation | last1=Mac Lane | first1=Saunders | author1-link=Saunders Mac Lane | title=Categories for the Working Mathematician | publisher=[[Springer-Verlag]] | edition=2nd | isbn=978-0-387-98403-2 | year=1998| title-link=Categories for the Working Mathematician }}
* {{Citation | last1=Mazur | first1=Barry | author1-link=Barry Mazur | title=Notes on étale cohomology of number fields | mr=0344254 | year=1973 | journal=Annales Scientifiques de l'École Normale Supérieure |series=Série 4 | issn=0012-9593 | volume=6 | issue=4 | pages=521–552| doi=10.24033/asens.1257 | doi-access=free }}
* {{Citation | last1=Milne | first1=James S. | title=Étale cohomology | publisher=[[Princeton University Press]] | isbn=978-0-691-08238-7 | year=1980 | url-access=registration | url=https://archive.org/details/etalecohomology00miln }}
* {{Citation | last1=Milne | first1=James S. | title=Arithmetic duality theorems | url=http://www.jmilne.org/math/Books/adt.html | publisher=BookSurge, LLC | location=Charleston, South Carolina | edition=2nd | isbn=978-1-4196-4274-6 | mr=2261462 | year=2006}}
* {{Citation | doi=10.1016/0021-8693(71)90105-0 | last1=Negrepontis | first1=Joan W. | title=Duality in analysis from the point of view of triples | mr=0280571 | year=1971 | journal=Journal of Algebra | issn=0021-8693 | volume=19 | issue=2 | pages=228–253| doi-access= }}
* {{ citation | first1 = Walter | last1 = Rudin | year = 1976 | isbn = 0-07-054235-X | title = Principles Of Mathematical Analysis | edition = 3rd | publisher = [[McGraw-Hill]] | location = New York }}
* {{Citation | last1=Veblen | first1=Oswald | author1-link=Oswald Veblen | last2=Young | first2=John Wesley | title=Projective geometry. Vols. 1, 2 | publisher=Blaisdell Publishing Co. Ginn and Co. | mr=0179666 | year=1965}}
* {{Citation | last1=Weibel | first1=Charles A. | title=An introduction to homological algebra | publisher=[[Cambridge University Press]] | isbn=978-0-521-55987-4 | mr=1269324 | year=1994}}
* {{cite book
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 | year = 1965
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}}

[[Category:Duality theories|*]]

[[ja:双対]]