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| In [[mathematics]], a '''duality''', generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an [[Involution (mathematics)|involution]] operation: if the dual of ''A'' is ''B'', then the dual of ''B'' is ''A''. Such involutions sometimes have [[fixed point (mathematics)|fixed points]], so that the dual of ''A'' is ''A'' itself. For example, [[Desargues' theorem]] in [[projective geometry]] is self-dual in this sense.
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| In mathematical contexts, ''duality'' has numerous meanings<ref>See Atiyah (2007)</ref> although it is “a very pervasive and important concept in (modern) mathematics”<ref>{{harvnb|Kostrikin|2001}}</ref> and “an important general theme that has manifestations in almost every area of mathematics.”<ref name="PCM187L">{{harvnb|Gowers|2008|loc=p. 187, col. 1}}</ref>
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| 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. 189, col. 2}}</ref>
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| Duality can also be seen as a [[functor]], at least in the realm of vector spaces. There it is allowed to assign to each space its dual space and the pullback construction allows to assign for each arrow {{nowrap|''f'': ''V'' → ''W''}}, its dual {{nowrap|''f''<sup>∗</sup>: ''W''<sup>∗</sup> → ''V''<sup>∗</sup>}}.
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| ==Order-reversing dualities== | |
| A particularly simple form of duality comes from [[order theory]]. The [[duality (order theory)|dual]] of a [[poset]] ''P'' = (''X'', ≤) is the poset ''P<sup>d</sup>'' = (''X'', ≥) comprising the same ground set but the [[converse relation]]. Familiar examples of dual partial orders include
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| * the subset and superset relations ⊂ and ⊃ on any collection of sets,
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| * the ''divides'' and ''multiple-of'' relations on the [[integers]], and
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| * the ''descendant-of'' and ''ancestor-of'' relations on the set of humans.
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| A concept defined for a partial order ''P'' will correspond to a ''dual concept'' on the dual poset ''P<sup>d</sup>''. For instance, a [[minimal element]] of ''P'' will be a [[maximal element]] of ''P<sup>d</sup>'': 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]].
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| A particular order reversal of this type occurs in the [[power set|family of all subsets]] of some set ''S'': if <math>\bar A=S\setminus A</math> denotes the [[complement set]], then ''A'' ⊂ ''B'' if and only if <math>\bar B\subset \bar A</math>. 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]]. In [[logic]], one may represent a [[truth assignment]] to the variables of an unquantified formula as a set, the variables that are true for the assignment. A truth assignment satisfies the formula if and only if the complementary truth assignment satisfies the [[De Morgan dual]] of its formula. The existential and universal quantifiers in logic are similarly dual.
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| A partial order may be interpreted as a [[category (mathematics)|category]] in which there is an arrow from ''x'' to ''y'' in the category if and only if ''x'' ≤ ''y'' in the partial order. The order-reversing duality of partial orders can be extended to the concept of a [[dual category]], the category formed by reversing all the arrows in a given category. Many of the specific dualities described later are dualities of categories in this sense.
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| According to Artstein-Avidan and Milman,<ref>{{harvnb|Artstein-Avidan|Milman|2007}}</ref><ref>{{harvnb|Artstein-Avidan|Milman|2008}}</ref> a ''duality transform'' is just an [[involutive antiautomorphism]] <math> \mathcal T </math> of a [[partially ordered set]] ''S'', that is, an [[Order theory#Functions between orders|order-reversing]] involution <math> \mathcal T : S \to S. </math> Surprisingly, in several important cases these simple properties determine the transform uniquely up to some simple symmetries. If <math> \mathcal T_1, \mathcal T_2 </math> are two duality transforms then their [[Function composition|composition]] is an [[Order isomorphism|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<sup>''R''</sup> are induced by permutations of ''R''. The papers cited above treat only sets ''S'' of functions on ''R''<sup>''n''</sup> satisfying some condition of convexity and prove that all order automorphisms are induced by linear or affine transformations of ''R''<sup>''n''</sup>.
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| ==Dimension-reversing dualities==
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| [[File:Dual Cube-Octahedron.svg|thumb|200px|The features of the cube and its dual octahedron correspond one-for-one with dimensions reversed.]]
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| 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 ''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 lattice]]s of the primal and dual polyhedra or polytopes are themselves [[duality (mathematics)#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.
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| [[File:Duals graphs.svg|right|thumb|240px|A [[planar graph]] in blue, and its [[dual graph]] in red.]]
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| 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 ''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.
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| In topology, [[Poincaré duality]] also reverses dimensions; it corresponds to the fact that, if a topological [[manifold]] is respresented 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'' − ''k'')th [[cohomology]] group.
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| [[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).]]
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| Another example of a dimension-reversing duality arises in [[projective geometry]].<ref>{{harvnb|Veblen|Young|1965}}.</ref> In the [[projective plane]], 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: in terms of the [[incidence matrix]] of the points and lines in the plane, this operation is just that of forming the [[transpose]]. Transformations of this type exist also in any higher dimension; one way to construct them is to use the same [[pole and polar|polar transformations]] that generate polyhedron and polytope duality. Due to this ability to replace any configuration of points and lines with a corresponding configuration of lines and points, there arises a general principle of [[duality (projective geometry)|duality in projective geometry]]: given any theorem in 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]].
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| The points, lines, and higher dimensional subspaces ''n''-dimensional projective space may be interpreted as describing the linear subspaces of an (''n'' + 1)-dimensional [[vector space]]; if this vector space is supplied with an [[inner product]] the transformation from any linear subspace to its perpendicular subspace is an example of a projective duality. The [[Hodge dual]] extends this duality within an inner product space by providing a canonical correspondence between the elements of the [[exterior algebra]].
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| 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 '''R'''<sup>''n''</sup>), 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.
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| {{-}}
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| ==Duality in logic and set theory==
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| 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 ∀ [[quantifier]]s. These are dual because ∃''x''.¬''P''(''x'') and ¬∀''x''.''P''(''x'') are equivalent for all predicates ''P'': if there exists an ''x'' for which ''P'' fails to hold, then it is false that ''P'' holds for all ''x''. From this fundamental logical duality follow several others:
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| * 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.
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| * 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, <math>\bigwedge (\neg x_i) = \neg\bigvee x_i</math>. The left side is true if and only if ∀''i''.¬''x''<sub>''i''</sub>, and the right side if and only if ¬∃''i''.''x''<sub>''i''</sub>.
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| * In [[modal logic]], □''p'' means that the proposition ''p'' is "necessarily" true, and <math>\Diamond p</math> 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'' in which ''p'' is true", while "''p'' is necessarily true" means "for all worlds ''W'', ''p'' is true". The duality of □ and <math>\Diamond</math> 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.
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| Other analogous dualities follow from these:
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| * Set-theoretic union and intersection are dual under the [[set complement]] operator ⋅<sup>''C''</sup>. That is, ''A<sup>C</sup>'' ∩ ''B<sup>C</sup>'' = (''A'' ∪ ''B'')<sup>''C''</sup>, and more generally, <math>\bigcap A_\alpha^C = \left(\bigcup A_\alpha\right)^C</math>. This follows from the duality of ∀ and ∃: an element ''x'' is a member of <math>\bigcap A_\alpha^C</math> if and only if ∀α.¬''x''∈''A''<sub>α</sub>, and is a member of <math>\left(\bigcup A_\alpha\right)^C</math> if and only if ¬∃α.''x''∈''A''<sub>α</sub>.
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| [[Topology]] inherits a duality between [[open set|open]] and [[closed set|closed subset]]s 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 (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 ''U'' is equal to the closure of the complement of ''U''.
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| 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]].
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| * [[Birkhoff's representation theorem]] relating [[distributive lattice]]s and [[partial order]]s
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| ==Dual objects==
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| 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 the one 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''<sup>∗</sup>=Hom(''X'', ''D'') is referred to as the ''dual'' of ''X''. It may or may not be true that the ''bidual'', that is to say, the dual of the dual, ''X''<sup>∗∗</sup> = (''X''<sup>∗</sup>)<sup>∗</sup> is isomorphic to ''X'', as the following example, which is underlying many other dualities, shows: the [[dual vector space]] ''V''<sup>∗</sup> of a ''K''-[[vector space]] ''V'' is defined as
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| :''V''<sup>∗</sup> = Hom (''V'', ''K'').
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| The set of morphisms, i.e., [[linear map]]s, is a vector space in its own right. There is always a natural, injective map ''V'' → ''V''<sup>∗∗</sup> given by ''v'' ↦ (''f'' ↦ ''f''(''v'')), where ''f'' is an element of the dual space. That map is an isomorphism if and only if the [[Hamel dimension|dimension]] of ''V'' is finite.
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| 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. A topological vector space that is canonically isomorphic to its bidual is called [[reflexive space]].
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| The [[dual lattice]] of a [[Lattice (group)|lattice]] ''L'' is given by
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| :Hom(''L'', '''Z'''),
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| which 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
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| :Hom(''G'', ''S''<sup>1</sup>),
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| continuous [[group homomorphisms]] with values in the circle (with multiplication of complex numbers as group operation).
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| ==Dual categories==
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| ===Opposite category and adjoint functors===
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| 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]] ''C'' and ''D'':
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| : ''F'': ''C'' → ''D''
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| which for any two objects ''X'' and ''Y'' of ''C'' gives a map
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| : Hom<sub>''C''</sub>(''X'', ''Y'') → Hom<sub>''D''</sub>(''F''(''Y''), ''F''(''X''))
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| 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''<sup>op</sup> 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.<ref>{{Harvnb|Mac Lane|1998|loc=Ch. II.1}}.</ref> Therefore, any duality between categories ''C'' and ''D'' is formally the same as an equivalence between ''C'' and ''D''<sup>op</sup> (''C''<sup>op</sup> and ''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>
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| 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 ''Y''<sub>1</sub> × ''Y''<sub>2</sub> and [[disjoint union]]s ''Y''<sub>1</sub> ⊔ ''Y''<sub>2</sub> of sets are dual to each other in the sense that
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| :Hom(''X'', ''Y''<sub>1</sub> × ''Y''<sub>2</sub>) = Hom(''X'', ''Y''<sub>1</sub>) × Hom(''X'', ''Y''<sub>2</sub>)
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| and
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| :Hom(''Y''<sub>1</sub> ⊔ ''Y''<sub>2</sub>, ''X'') = Hom(''Y''<sub>1</sub>, ''X'') × Hom(''Y''<sub>2</sub>, ''X'')
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| for any set ''X''. This is a particular case of a more general duality phenomenon, under which [[limit (category theory)|limits]] in a category ''C'' correspond to [[colimit]]s in the opposite category ''C''<sup>op</sup>; 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>
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| Two [[functors]] ''F'': ''C'' → ''D'' and ''G'': ''D'' → ''C'' are [[adjoint functor|adjoint]] if for all objects ''c'' in ''C'' and ''d'' in ''D''
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| :Hom<sub>''D''</sub>(F(''c''), ''d'') ≅ Hom<sub>''C''</sub>(''c'', ''G''(''d'')),
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| in a natural way. Actually, the correspondence of limits and colimits is an example of adjoints, since there is an adjunction
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| :<math>\operatorname{colim}: C^I \leftrightarrow C: \Delta \, </math>
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| 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 diagramm which has ''c'' at all places. Dually,
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| :<math>\Delta: C^I \leftrightarrow C: \lim. \, </math>
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| ===Examples===
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| For example, there is a duality 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
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| :(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>
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| Compare with [[noncommutative geometry]] and [[Gelfand duality]].
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| In a number of situations, the objects of two categories linked by a duality are [[partially ordered]], i.e., there is some notion of an object "being smaller" than another one. In such a situation, a duality that respects the orderings in question is known as a [[Galois connection]]. An example is the standard duality in [[Galois theory]] ([[fundamental theorem of Galois theory]]) between [[field extension]]s and [[subgroup]]s of the [[Galois group]]: 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>
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| [[Pontryagin duality]] gives a duality on the category of [[locally compact]] [[abelian group]]s: given any such group ''G'', the [[character group]]
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| :χ(''G'') = Hom(''G'', ''S''<sup>1</sup>)
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| 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
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| :''G'' ≅ χ(χ(''G'')).<ref>{{Harvard citations | last1=Loomis|year=1953 | loc=p. 151, section 37D}}</ref>
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| Moreover, [[discrete group]]s correspond to [[compact group|compact abelian group]]s; finite groups correspond to finite groups. Pontryagin is the background to [[Fourier analysis]], see below.
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| * [[Tannaka-Krein duality]], a non-commutative analogue of Pontryagin duality<ref>{{Harvard citations | last1=Joyal | last2=Street|year=1991 |txt}}</ref>
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| * [[Gelfand duality]] relating commutative [[C*-algebra]]s and [[compact space|compact]] [[Hausdorff space]]s
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| Both Gelfand and Pontryagin duality can be deduced in a largely formal, category-theoretic way.<ref>{{Harvnb|Negrepontis|1971}}.</ref>
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| == Analytic dualities ==
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| In [[mathematical analysis|analysis]], problems are frequently solved by passing to the dual description of functions and operators.
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| [[Fourier transform]] switches between functions on a vector space and its dual:
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| :<math>\hat{f}(\xi) := \int_{-\infty}^{\infty} f(x)\ e^{- 2\pi i x \xi}\,dx, </math>
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| and conversely
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| :<math>f(x) = \int_{-\infty}^{\infty} \hat{f}(\xi)\ e^{2 \pi i x \xi}\,d\xi.</math>
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| If ''f'' is an [[Lebesgue integration|''L''<sup>2</sup>-function]] on '''R''' or '''R'''<sup>''N''</sup>, say, then so is <math>\hat{f}</math> and <math>f(-x) = \hat{\hat{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>−2πi''x''ξ</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.
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| * [[Laplace transform]] is similar to Fourier transform and interchanges [[linear operator|operators]] of multiplication by polynomials with constant coefficient [[linear differential operator]]s.
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| * [[Legendre transformation]] is an important analytic duality which switches between [[velocity|velocities]] in [[Lagrangian mechanics]] and [[momentum|momenta]] in [[Hamiltonian mechanics]].
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| ==Poincaré-style dualities==
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| 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
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| :''A'' ⊗ ''B'' → ''K''.
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| For [[perfect pairing]]s, there is, therefore, an isomorphism of ''A'' to the [[dual vector space|dual]] of ''B''.
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| For example, [[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''')
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| :H<sup>''i''</sup>(X) ⊗ H<sup>2''n''−''i''</sup>(X) → '''C''',
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| 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
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| :<math>(\gamma, \omega) \mapsto \int_\gamma \omega</math>
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| (integrating a differential ''k''-form over an 2''n''−''k''-(real)-dimensional cycle) is a perfect pairing.
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| 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> 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 both these dualities can be done using [[derived category|derived categories]] and certain [[image functors for sheaves|direct and inverse image functors of sheaves]], applied to locally constant sheaves (with respect to the classical analytical topology in the first case, and with respect to the [[étale topology]] in the second case).
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| 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]] 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>\hat {\mathbf Z}</math>, the [[profinite completion]] of '''Z''', the integers. Therefore, the perfect pairing (for any [[G-module|''G''-module]] ''M'')
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| :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>
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| 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]]).<ref>{{Harvard citations|last1=Mazur|year=1973|txt}}; {{Harvard citations|last=Milne|year=2006|txt}}</ref>
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| [[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>
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| * [[Alexander duality]]
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| ==See also==
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| * [[List of dualities]]
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| * [[Duality principle (disambiguation)]]
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| * [[Dual (category theory)]]
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| * [[Dual number]]s, a certain [[associative algebra]]; the term "dual" here is synonymous with ''double'', and is unrelated to the notions given above.
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| * [[Duality (electrical engineering)]]
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| * [[Lagrange duality]]
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| * [[Dual code]]
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| * [[Dual lattice]]
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| * [[Dual basis]]
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| * [[Dual abelian variety]]
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| * [[Adjoint functor]]
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| * [[Dualizing module]]
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| ===Duality in general===
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| * Atiyah, Michael (2007), [http://www.fme.upc.edu/arxius/butlleti-digital/riemann/071218_conferencia_atiyah-d_article.pdf Duality in Mathematics and Physics], lecture notes from the Institut de Matematica de la Universitat de Barcelona (IMUB).
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| *{{Springer|title=Duality|id=D/d034120|first=A. I.|last=Kostrikin|authorlink=Alexei Kostrikin}}.
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| *{{citation |contribution=III.19 Duality |title=[[The Princeton Companion to Mathematics]] |pages=187–190 |first=Timothy |last=Gowers |publisher=Princeton University Press |year=2008}}.
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| * {{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=http://www.ams.org/bull/2001-38-04/S0273-0979-01-00913-2/ | mr=1848254 | year=2001 | journal=American Mathematical Society. Bulletin. New Series | issn=0002-9904 | volume=38 | issue=4 | pages=389–408}} (a non-technical overview about several aspects of geometry, including dualities)
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| ===Duality in algebraic topology ===
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| *James C. Becker and Daniel Henry Gottlieb, [http://www.math.purdue.edu/~gottlieb/Bibliography/53.pdf A History of Duality in Algebraic Topology]
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| ===Specific dualities===
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| * {{Citation | last1=Artstein-Avidan | first1=Shiri | 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=2648–2666 | doi=10.1016/j.jfa.2007.11.008}}. Also [http://www.math.tau.ac.il/~shiri/publications.html author's site].
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| * {{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}}. Also [http://www.math.tau.ac.il/~shiri/publications.html author's site].
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| * {{Citation | last1=Dwyer | first1=William G. | last2=Spaliński | first2=J. | title=Handbook of algebraic topology | 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}}
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| * {{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}}
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| * {{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}}
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| * {{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Residues and Duality | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics '''20''' | year=1966 | pages=20–48}}
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| * {{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=[[Algebraic Geometry (book)|Algebraic Geometry]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 | oclc=13348052 | mr=0463157 | year=1977}}
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| * {{Citation | last1=Iversen | first1=Birger | title=Cohomology of sheaves | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Universitext | isbn=978-3-540-16389-3 | mr=842190 | year=1986}}
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| * {{Citation | last1=Joyal | first1=André | author1-link=André Joyal | last2=Street | first2=Ross | author2-link=Ross Street | title=Category theory (Como, 1990) | url=http://www.maths.mq.edu.au/~street/CT90Como.pdf | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture notes in mathematics | mr=1173027 | year=1991 | volume=1488 | chapter=An introduction to Tannaka duality and quantum groups | pages=413–492}}
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| * {{Citation | last1=Lam | first1=Tsit-Yuen | title=Lectures on modules and rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | mr=1653294 | year=1999}}
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| * {{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Algebra | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-95385-4 | mr=1878556 | year=2002 | volume=211}}
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| * {{Citation | last1=Loomis | first1=Lynn H. | title=An introduction to abstract harmonic analysis | publisher=D. Van Nostrand Company, Inc. | location=Toronto-New York-London | year=1953 | pages=x+190}}
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| * {{Citation | last1=Mac Lane | first1=Saunders | author1-link=Saunders Mac Lane | title=[[Categories for the Working Mathematician]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-0-387-98403-2 | year=1998}}
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| * {{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. Quatrième Série | issn=0012-9593 | volume=6 | pages=521–552}}
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| * {{Citation | last1=Milne | first1=James S. | title=Étale cohomology | publisher=[[Princeton University Press]] | isbn=978-0-691-08238-7 | year=1980}}
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| * {{Citation | last1=Milne | first1=James S. | title=Arithmetic duality theorems | url=http://www.jmilne.org/math/Books/adt.html | publisher=BookSurge, LLC | location=Charleston, SC | edition=2nd | isbn=978-1-4196-4274-6 | mr=2261462 | year=2006}}
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| * {{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}}
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| * {{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. New York-Toronto-London | mr=0179666 | year=1965}}
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| * {{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}}
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| [[Category:Duality theories|*]]
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| [[ja:双対]]
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| [[ru:Двойственность]]
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