I have been thinking for sometime about asking this question, but because I did not want to have two "big-list" questions open at the same time, I did not ask this one. Now its time has come.

Wikipedia has a good page on several forms of "duality" in mathematics, which outlines several notions of duality (geometric, in convex analysis, topology, set theory, etc.) I am very interested in getting help with the following goal:

Collect an annotated list of various notions of duality that occur in mathematics, with the ultimate aim of describing the notions in a way that makes it easier to recognize and intuitively build connections between the various notions of duality. Also welcome are comments / answers that highlight how a particular notion of duality can be extremely useful (in proving theorems, in applications, for computational reasons, etc.)

Some additional context

I got thinking about this question after reading the following amazing paper: The concept of duality in convex analysis, and the characterization of the Legendre transform, by Shiri Artstein-Avidan and Vitali Milman, where the authors talk about duality in more abstract terms (though, largely in the setting of convex analysis). Motivated by their abstract treatment got me thinking whether such abstract treatments of duality have been investigated for other types of duality, which eventually led to this question.

Thus, in line with with the Avidan-Milman results, one may also ask similar questions about other types of duality (i.e., one tries to characterize why and how a chosen notion of duality is the only "natural" choice under a set of axiomatic requirements).

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    This looks like a very good big list question. Also it is a case where closing a question (Survit's earlier memorable big list question) was beneficial. – Gil Kalai Aug 26 '11 at 11:21
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    Several subsequent works to the Artshtein-Milman paper can be found in papers 21-30 here: math.tau.ac.il/~shiri/publications.html – Gil Kalai Aug 26 '11 at 11:39
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    Great question. I've also wondered about this for some time. – Cole Leahy Aug 28 '11 at 16:53
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    The paper ["A history of duality in algebraic topology" by Becker and Gottlieb] (math.purdue.edu/~gottlieb/Bibliography/53.pdf) is a very nice read. Several concepts of duality are discussed, along with their interactions. – Bruno Stonek Feb 20 '17 at 15:46
  • @BrunoStonek I should have read your comment instead of writing my answer below. Thanks for the link! – Sebastian Goette Feb 20 '17 at 19:10

27 Answers 27

The (1) Fourier transform, (2) mirror symmetry, (3) electric-magnetic duality, and the (4) Pontrjagin and (5) Langlands dualities of Lie groups are all seen to be interrelated by the proposal of Strominger-Yau-Zaslow for mirror symmetry and the work of Kapustin-Witten (foreshadowed by Montonen-Olive) framing the geometric Langlands program in physical terms.

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    Also, the work of Batyrev-Borisov relates mirror symmetry to (6) polar duality of polytopes. – Michael Thaddeus Aug 26 '11 at 0:56
  • In the same vein as electromagnetic duality (physics), what about wave particle duality? – Spice the Bird Aug 27 '11 at 3:43
  • Related to mirror symmetry: quantamagazine.org/… – Tom Copeland Apr 9 at 21:54

Projective geometry. Is that the first use of the term "dual" in mathematics?

  • I suspect some notions were present in the works of Aristotle and Plato, although more in a vein of natural science. While there are possibilities for answering this question mathematically, I think the poster would benefit from talking with or reading history of science type researchers first. Gerhard "Ask Me About Design Systems" Paseman, 2011.08.25 – Gerhard Paseman Aug 26 '11 at 0:39
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    Is this a comment or an answer? Could you expand it a bit---would be very helpful. Thanks – Suvrit Aug 26 '11 at 16:13
  • Suvrit, if that was directed at me, I will expand my remarks using an answer format later today. Gerhard "Ask Me About System Design" Paseman, 2011.08.26 – Gerhard Paseman Aug 26 '11 at 16:53
  • A nice example is then Sylvester–Gallai theorem. It was posed as a problem by J. J. Sylvester in 1893 and first solved by Melchior by proving the dual problem. – jjcale Aug 26 '11 at 20:07
  • I think so. Projective duality, i.e. on a projective plane nothing changes interchanging "point" with "line" is the primitive example of mathematical duality. – Leo Alonso Oct 7 '11 at 16:54

There are various dualities arising in elementary logic:

  • the duality between $\forall$ and $\exists$, as expressed by the validity $$\neg\forall x\ \neg\varphi(x)\iff \exists x\ \varphi(x).$$

  • the duality between $\wedge$ (and) and $\vee$ (or), as expressed via the de Morgan laws $$\neg(p\wedge q)\iff (\neg p)\vee(\neg q).$$

  • the duality in modal logic between possibility and necessity, as expressed via $$\neg\Diamond\varphi\iff\square\neg\varphi,$$ (that is: $\varphi$ is not possible if and only if $\neg\varphi$ is necessary), a principle which has manifestations for any of the diverse interpretations of these modal operators satisfying this equivalence.

Each of these dualities arises in the conjugation of one logical quantifier or operation with $\neg$.

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    And in fact, each of these three notions is related, the first being a kind of infinitary version of the second, and the third a consequence of the fact that semantics for $\Box$ is obtained by universal quantification, and semantics for $\Diamond$ by existential quantification. – Adam Bjorndahl Aug 26 '11 at 1:44
  • Yes, I agree with that. – Joel David Hamkins Aug 26 '11 at 1:54
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    Repeatedly applying the duality between $\forall$ and $\exists$, you get determinacy of games of bounded finite length. Dualizing infinite strings of quantifiers --- which is not generally valid --- is a way of expressing determinacy of infinite games. – Andreas Blass Aug 26 '11 at 4:56
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    Another ubiquitous manifestation of the duality of $\exists$ and $\forall$ are the corresponding dual sides of the arithmetic or projective hierarchies, contrasting $\Sigma^1_n$ and $\Pi^1_n$. – Joel David Hamkins Aug 26 '11 at 10:46
  • These have their analogues in the (equivalent) dual $F_\sigma$ and $G_\delta$ hiearchies. – Joel David Hamkins Aug 26 '11 at 22:46

Galois connections (nLab,wikipedia). This is really just an adjuction between one category and the opposite of another, where these categories are preorders. A Galois correspondence is when this adjunction is an equivalence of categories.

Stone duality (nLab,wikipedia). This is best explained by the linked page, but one I will point out is that one has as a small part of this duality, $FinSet \simeq FinBool^{op}$ (the category of finite sets is equivalent to the opposite of the category of finite boolean algebras), which has as a corollary, the category of Stone spaces is equivalent to that of profinite sets.

There is the nLab page duality, but one can see by searching the nLab, there are a number of other pages that people might find useful.

What would be useful here is a list of mechanisms lying behind these appearances of duality. So we have (at least)

  1. Duality pairing
  2. Dualizing object
  3. Maximal fixed subcategories of an adjunction
  4. Arrow reversal

Then we could look at any relations between these mechanisms, such as between 2 and 3, maps into a dualizing object form the functors for an adjunction.

Atiyah in his talk Duality in Mathematics and Physics says

"Fundamentally, duality gives two different points of view of looking at the same object. There are many things that have two different points of view and in principle they are all dualities."

So perhaps we need

5 . Something is seen in two different ways

The Dynkin diagram for $SL_n$ is a string of $n-1$ dots, we can view it from either end as point, line, plane, etc. Put another way, the symmetry of the diagram corresponds to an outer automorphism which account for the duality of projective geometry.

I wonder if 'deeper' dualities come from more intricate processes of seeing something from two points of view. Frenkel gives a very accessible talk What Do Fermat's Last Theorem and Electro-magnetic Duality Have in Common? where he explains that the duality of Geometric Langlands arises from compactifying a 6d quantum field theory in two different ways onto 2d surfaces.

  • This answer is very helpful, and indeed it strikes at the heart of some of the things that I've been trying to understand, or get a big-picture of. At a more philosophical level though, one could ask the rhetorical question: why care about duality? (not a very good question though). – Suvrit Sep 1 '11 at 15:42
  • Some thoughts on your question are here: golem.ph.utexas.edu/category/2011/05/… – David Corfield Sep 1 '11 at 15:57
  • Does such a characterization of duality imply that there exists a general, rather abstract, notion of automorphism group such that any two dual objects have isomorphic automorphism groups? – Sylvain JULIEN Aug 9 '15 at 12:55

In the study of convexity and convex polyhedra there are three (related) important notions of duality

1) Polar duality

This is a map assigning to every convex set $K$ containing the origin its polar dual: $K^*$ which is the set of all points whose inner product with every point in $K$ is at most 1.

On polytopes it induces an order reversing map on the face lattices. This operation has subtle relations to mirror-symmetry and Koszul duality.

Web sources (1; 2; 3; 4)

2) Gale transform

This is an operation to move from n points in $R^d$ to n points in $R^k$ where k=n-d-1. It is especially useful if the original n points are in convex position to study the convex polytope they define. (Web-sources: 1, 2; 3; 4)

3) Linear programming duality

This is an operation to move from a linear programming problem to a dual problem which have the same solution.

(We sources: 1; 2; 3;)

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    And don't forget Legendre transform! It is used in e.g. in Hamiltonian mechanics. Recently it also got a lot of attention from people studying the complex Monge-Ampere equation on manifolds and related concepts of energy. – Margaret Friedland Aug 26 '11 at 15:47
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    And there is also the still rather mysterious duality between the Brunn-Minkowski theory and the dual Brunn-Minkowski theory introduced by Erwin Lutwak. – Deane Yang Sep 1 '11 at 14:54
  • Do you have a link explaining the connections among polar duality, mirror symmetry, and Koszul duality? Link 4 above is broken. – Tom Copeland Apr 9 at 21:58

I think that the obvious one between spaces (topological, differentiable, algebraic, etc.) and the rings of structure preserving functions on them should be mentioned.

Differential geometry: Eigenvalues of Laplace operators $\Leftrightarrow$ length of closed geodesics

representation theory: irreducible representations $\Leftrightarrow$ conjugacy classes in a group

Number theory: primes $\Leftrightarrow$ zeros of $L$ functions

Quantum mechanics: particles $\Leftrightarrow$ waves

Argument principle in complex analysis: contour integrals $\Leftrightarrow$ residues

Index theory: topological index $\Leftrightarrow$ analytic index

Algebraic geometry: algebraic cycles $\Leftrightarrow$ motives

Most of them can be found in: www.claymath.org/cw/arthur/pdf/52.pdf

Trace fomulas like Poisson summation formula, Arthur's trace formula, Selberg's trace formula, Gutzwiller trace formula, Lefschetz trace formula, Weil's explicit formula quantify these relations.

There is always a sort of Fourier uncertainty involved, so a one-to-one correspondance between "geometric objects" and "spectral objects" is not available, except perhaps for the symmetric group $S_n$ via Young diagrams.

Koszul duality is a useful duality. For example, one can cite

  1. Koszul duality of quadratic algebras (due to Priddy) which is related to inversion of formal power series.

  2. Koszul duality of quadratic operads (due to Ginzburg and Kapranov) which is related to reversion of formal power series or plethystic reversion.

  3. Koszul duality of cyclic quadratic operads (due to Getzler and Kapranov) which is related to Legendre duality and Legendre transform.

One can see (1 and 2) that Koszul duality is often related to the notion of inversion $g \mapsto g^{-1}$ in a group.

  • Of course, the Legendre transformation can also be related to compositional inversion. – Tom Copeland Jan 29 '17 at 17:12

This answer has a heavy bias towards logical structures. The simplest notion I know is order-theoretic duality.

  1. The dual of an order is the inverse relation of the order (less-than vs. greater-than, subset vs. superset)
  2. Greatest lower bounds and least upper bounds (minimum vs. maximum, intersection vs. union, conjunction vs. disjunction)
  3. Bottom and top
  4. Least and greatest fixed points
  5. Additive and multiplicative maps

In structures containing negation, we have De Morgan duality, such as the examples from logic given by Joel David Hamkins.

I do not know if 'duality' is the right term, but I think of adjunctions as duals too. To add to the answer of David Roberts:

  1. Conjunction and implication (both with one argument fixed) are adjoints
  2. Existential and universal quantification are adjoints to a certain form of substitution
  3. Strongest postconditions and weakest liberal preconditions in programming language semantics
  4. Sets of models and sets of formulae
  5. A lattice and its image under a closure operator

In settings with a notion of time, there are temporal dualities from the interaction of the past and the future. There are several examples in temporal and modal logics.

Some representation theorems for lattices are ancestors of dualities. For example, Stone's representation theorem for Boolean algebras is now usually referred to as a duality. There are various dualities relating families of lattices with families of discrete structures.

  1. Complete, atomic, Boolean algebras and powersets [Lindenbaum and Tarski]
  2. Finite distributive lattices and finite posets [Birkhoff]
  3. Completely distributive, algebraic lattices and posets [Raney, others I cannot recall]
  4. Boolean algebras with operators and sets with relations [Jónsson and Tarski]
  5. Distributive algebras with operators and ordered sets with relations [Gehrke and Jónsson (though there may be earlier work)]

The list goes on. Such results are sometimes called discrete dualities. There is much recent work on discrete duality in terms of what are called canonical extensions. These duality results often include a topological component.

  1. Boolean algebras and Stone spaces [Stone]
  2. Distributive lattices and Priestley spaces [Priestley]
  3. Heyting algebras and Esakia spaces [Esakia]
  4. Topological representations of arbitrary lattices [Urquhart]
  5. Extensions of Stone and Priestley duality to lattices with operators
  6. Dualities arising in Modal logic [Goldblatt]

One 'analogy between analogies' is that of a dualising object. The term schizophrenic object has also been used in this context.

Porst and Tholen's article Concrete Dualities discusses some of these and other dualities and the connection to adjunctions. Other references are Peter Johnstone's book Stone Spaces and Clarke and Davey's book Natural Dualities for the Working Algebraist.

Finite-dimensional linear spaces. A particular feature in this case is that the (algebraic) dual of a finite-dimensional vector space, namely the space of linear maps from the vector space into the base field, is isomorphic to the original space (since it is of the same dimensionality) but not canonically so. In contrast, the bi-dual (the dual of the dual) is canonically isomorphic to the original space, and so may be identified with it.

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    Hi John, please see if you can possibly expand your answer to cover some of the broader aims of my question. – Suvrit Aug 26 '11 at 17:34
  • @Suvrit: I don't if it's what you're looking for, but I've expanded it now. – John Bentin Aug 27 '11 at 7:08

I am personally fond of matroid duality. Let $M$ be a matroid with ground set $E$. The dual of $M$ is the matroid with ground set $E$ and whose bases are the complements of bases of $M$. It is easy to verify that the dual of $M$ is indeed a matroid and we immediately have that $M^{dd}=M$.

Matroid duality illustrates that deletion and contraction are actually dual operations. That is, deletion corresponds to contraction in the dual and vice versa.

It also nicely generalizes duality for planar graphs. That is, if $G$ is a planar graph, and $M(G)$ is the cycle matroid of $G$, then $M^d(G)=M(G^d)$ (here, $G^d$ is the planar dual of $G$).

Finally, here is a proof for free of Euler's Formula via matroid duality. Let $G$ be a connected planar graph with edge set $E$. Suppose that $G$ has $v$ vertices, $e$ edges and $f$ faces. Let $r$ be the rank function of the cycle matroid of $G$ and let $r_d$ be the rank function of the dual of the cycle matroid of $G$. Then

\[ e=r(E)+r_d(E)=(v-1)+(f-1). \]

I enjoyed a series of talks by Bernd Sturmfels on some such interrelationships, which it looks like are written up in a paper by Rostalski and Sturmfels called "Dualities in Convex Algebraic Geometry."

Abstract: Convex algebraic geometry concerns the interplay between optimization theory and real algebraic geometry. Its objects of study include convex semialgebraic sets that arise in semidefinite programming and from sums of squares. This article compares three notions of duality that are relevant in these contexts: duality of convex bodies, duality of projective varieties, and the Karush-Kuhn-Tucker conditions derived from Lagrange duality. We show that the optimal value of a polynomial program is an algebraic function whose minimal polynomial is expressed by the hypersurface projectively dual to the constraint set. We give an exposition of recent results on the boundary structure of the convex hull of a compact variety, we contrast this to Lasserre’s representation as a spectrahedral shadow, and we explore the geometric underpinnings of semidefinite programming duality.

Duality is the corner stone of the theory of Distributions

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    I think this point of view on distributions can be a bit misleading despite being true. Distribution theory can be regarded as an extension of linear operations on functions by continuity to more general objects (distributions). For example, one often needs to compute "integrals" such as $\int u(x) v(x) dx$ where both $u$ and $v$ are distributions whose wavefront sets don't collide. They key point is not that $u$ and $v$ are dual; rather that the number is the limit when you take classical approximations. Hormander's book expresses this view in many ways. Anyhow, it's a good example. – Phil Isett Aug 27 '11 at 16:02

A very simple and important notion of duality is the following.

Start with a collection $F$ of subsets of a ground set $X$.

Now, define the blocker $F^*$ of $F$ as follows:

$F^*=${$ X \backslash A: A \notin F $}.

In words, we take the complements of all sets not in $F$.

This notion is very important in combinatorial optimization and polyhedral combinatorics. It is also a simple manifestation of Alexander duality from algebraic topology.


Addendum (Adam Bjorndahl):

This construction can be viewed as a generalization of the quantifier duality $$\forall \equiv \lnot \exists \lnot.$$

As above, fix a set $X$. For $F \subseteq 2^{X}$, define the formula $(\text{F}x) \ \phi(x)$ to mean that $$\{x \in X : \phi(x)\} \in F.$$ So $(\text{F}x) \ \phi(x)$ might be read "for $F$-many $x$, property $\phi$ holds". Three special cases deserve some attention.

  • When $F = \{X\}$, we recover the usual "for all" quantifier. Succinctly, $\forall = \{X\}$.

  • Dualizing, we obtain $$\lnot (\text{F}x) \lnot \phi(x) \iff \{x \in X : \lnot \phi(x)\} \notin F;$$ thus if $A = \{x \in X : \phi(x)\}$, we have $$\lnot (\text{F}x) \lnot \phi(x) \iff A \in F^{*} \iff (\text{F}^{\*}x) \phi(x),$$ where $F^{\*}$ is the blocker of $F$.

  • Finally, if $U \subset 2^{X}$ is an ultrafilter on $X$, then $$\lnot (\text{U}x) \lnot \phi(x) \iff (\text{U}x)\phi(x),$$ which exhibits ultrafilters as self-dual quantifiers, a perspective I find appealing.

The Jónsson–Tarski duality between Boolean algebras with operators (in particular, modal algebras) and general frames. (A variant of this, called Esakia duality, has topological frames instead of general frames. There is also an analogous duality of Heyting algebras and intuitionistic frames, which I never remember whose name it bears.) This duality is the basis of the Kripke semantics for modal, intuitionistic, and other nonclassical logics.

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    and then there are all those variants of Priestley duality... – Peter Arndt Aug 26 '11 at 17:16

One "duality principle" that occurs in category theory is that of Isbel Duality. My feeling is (feel free to correct me if I am wrong) is that this encapsulates stone duality, Gelfand duality, and the duality of affine schemes and commutative rings in the same disscusion. Let $\mathcal{C}$ be a (small) category. Then the presheaves on $\mathcal{C}$ and the co-presheaves on $\mathcal{C}$ are somehow dual to one another. Conceptually, one thinks of the presheves as spaces and co-presheaves as quantities. This is something that I am trying to understand myself. Some nice articals at n-lab are:

http://ncatlab.org/nlab/show/space+and+quantity

http://ncatlab.org/nlab/show/Isbell%20duality

The Curry–Howard isomorphism between typed $\lambda$-calculus and intuitionistic proofs.

How about the duality between proofs and models?

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    How about it? It's a great duality, e.g. its incarnation as Gabriel-Ulmer duality, see ncatlab.org/nlab/show/Gabriel-Ulmer+duality – Peter Arndt Aug 26 '11 at 9:45
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    @Alex: This sounds like a nice entry -- could you elaborate? – Cam McLeman Aug 26 '11 at 13:32
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    Yes, this sounds quite interesting. Please expand if you have a few moments. Thanks. – Suvrit Aug 26 '11 at 16:20
  • The duality is simple: each new axiom or proof technique specifies a subset of true statements, and each model or (counter)example specifies a superset over true statements, i.e. all statements that are true for this model/example. When a bunch of proof techniques and a bunch of models are such that the subset = the superset, you have complete duality. BTW, to those big in categories: do they have a notion of probability? For example imagine a category of extendable computer programs whose meaning is never finally instantiated, and a notion of complexity for them. Just curious... – Alex Aug 26 '11 at 21:38
  • What you appear to describe in the first sentence is just the Galois connection between classes of models and sets of formulas. What do proofs have to do with it? – Emil Jeřábek Aug 29 '11 at 11:15

The duality between measure and category in the set theory of the reals.

  • see Oxtoby's book, Measure and Category – Gerald Edgar Aug 27 '11 at 13:04

In control theory there exists the duality controllability and observability. It is very well understood in the context of linear control theory, not so much for nonlinear systems. It is related to the linear space duality between vectors and functionals, but more work in understanding it from a more general perspective would be welcome.

The Dual Graph of a plane graph $G$ is a graph that has a vertex corresponding to each face of G, and an edge joining two neighboring faces for each edge in $G$.

From Quantum Field Theory III: Gauge Theory by Eberhard Zeidler (Springer, 4/2011):

A) Preface (pg. XII):

"It turns out that cohomology and homology have their roots in the rules for electrical circuits formulated by Kirchhoff in 1847."

B) Ch. 22. Electrical Circuits as a Paradigm in Homology and Cohomology :

(pg. 1009)

"The study of electrical networks rests upon preliminary theory of graphs. ... My purpose here is to show that actually this theory is nothing else than the first chapter of classical algebraic topology theory and may be very advantageously treated as such by the well known methods of that science." -- Solomon Lefschetz (as quoted by Zeidler) from Applications of Algebraic Topology: Graphs and Networks, the Picard-Lefschetz Theory, and Feynman Algorithms (Springer, 1975) by Solomon Lefschetz.

(i) Electric currents $J$ are 1-cycles: $\partial J=0$.

(ii) Voltages $V$ are 1-coboundaries: $V=-dU$ (U is the electrostatic potential).

(iii) There exists a duality relation between electric currents and voltages: $<V|J>=0$.

(iv) If the electrical current is connected, then we get $\beta^0=1$ for the zeroth Betti number. In the general case, $\beta_0$ is equal to the number of connectivity components of the electrical circuit.

(v) If the electrical circuit has $s_0$ nodes and $s_1$ connections, then the Euler characteristic is given by $\chi=s_0-s_1$.

(vi) This yields the first Betti number $\beta_1=\beta_0-\chi$.

(vii) The space of electric currents is a linear space of dimension $\beta_1$.

(pg. 1020)

  • Homology describes the geometry of the electric circuit; in particular, the first Betti number is equal to the number of essential loops (also called 1-cycles).
  • Cohomology describes the physics of the circuit (i.e., cohomology describes voltage and hence the electric currents, by Ohm's law).
  • There exists a crucial duality relation between homology and cohomology which reflects the influence of the geometry of an electrical circuit on its physics (based on the duality relation (22.17) below).

$dU(J)=U(\partial J)$ for all 1-chains $J$. (22.17)

C) Ch. 23. The Electromagnetic Field and the de Rham Cohomology:

(pg. 1027)

De Rham cohomology reformulates and generalizes the fundamental theorem of calculus due to Newton and Leibniz to differential forms on manifolds. In terms of physics, this describes the existence of potentials. The key role is played by Poincare's cohomology rule and the generalized Stokes integral theorem.

Disclaimer: the following is probably a special case of the more abstract notions in other answers above like this one. But I think it still worthwhile to expand on it a bit.

There is a notion of duality in symmetric monoidal categories. The concept is explained for example in "Duality, Trace and Transfer" by Dold and Puppe or here. All endomorphisms of dualisable objects have traces, represented as endomorphisms of the unit object. Typical examples:

In the category of vector spaces $(\mathcal{Vect},\otimes,\Bbbk)$, a vector space is strongly dualisable if and only if it is finite-dimensional ($V$ strongly dualisable implies $(V^*)^*\cong V$, see also this answer). For modules over a ring, strongly dualisable means finitely generated projective. The trace of an endomorphism is the usual trace. Note that endomorphisms of infinite-dimensional vector spaces can have a trace, but for example, the identity typically hasn't. Similar dualities exist in other tensor categories, for example, categories of representations.

In the stable homotopy category of spaces, all finite CW complexes (finite CW spectra) have a strong dual, the Spanier-Whitehead dual. The trace of an endomorphism is its Lefschetz number, represented as an endomorphism of the sphere spectrum. Given a cohomology theory (like singular cohomology, $K$-theory or bordism), the homology of a dualisable space is isomorphic to the cohomology of its dual.

In the case of compact manifolds, the Spanier-Whitehead dual is the Thom space of the stable normal bundle (or rather, the Thom spectrum of normal bundles), and Spanier Whitehead duality translates into Poincaré duality (of course, there are many other proofs/explanations of Poincaré duality). Comparing the trace in the homotopy category with traces in cohomology, one gets (versions of the) Poincaré-Hopf theorem and the Lefschetz fixpoint theorem.

Serre duality

Grothendieck duality

Verdier duality

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    Thanks for your answer. However, could you expand for the non cognoscenti? – Suvrit Aug 26 '11 at 16:21

Umbral compositional inversion is a type of duality related to multiplicative and compositional inversion of functions and to matrix inversion, which is very useful in deriving algebraic relations and other identities among polynomial sequences important in number theory, special functions, and enumerative combinatorics as well as operator calculi.

If a pair $p_n(x)$ and $q_n(x)$ of polynomial Sheffer sequences is an umbral inverse pair (UIP), then

$$p_n(q.(x))= x^n=q_n(p.(x)),$$

where, e.g., $p_n(q.(x))=\sum^n_{k=0} \; p_{n,k} \cdot q_k(x).$

This implies that the pair of lower triangular matrices comprised of the coefficients of these polynomial sequences are an inverse pair.

An important UIP of binomial Sheffer sequences are the Bell / Touchard polynomials, comprised of the Stirling numbers of the second kind, and the falling factorials, comprised of the Stirling numbers of the first kind. An important UIP of Appell Sheffer sequences are the Bernoulli polynomials and the reciprocal integer polynomials.

For a binomial Sheffer sequence of polynomials, defined by the exponential generating function

$$e^{h(t)x}=e^{p.(x)t},$$

with $h(0)=0$ and $(p.(x)^n)=p_n(x)$, the umbral compositional inverse Sheffer sequence is given by

$$e^{h^{(-1)}(t)x}=e^{q.(x)t},$$

where $h$ and $h^{(-1)}$ are a compositional inverse pair. E.g., $(h(t),h^{-1}(t))=(e^t-1,\ln(1+t))$ generate the Bell and falling factorial pair.

For an Appell Sheffer sequence, defined by the e.g.f.

$$f(t)e^{x \cdot t}= e^{u.(x)t},$$

with $f(0)=1$, its umbral compositional inverse Appell sequence is given by

$$\frac{1}{f(t)}e^{x \cdot t} = e^{v.(x)t}.$$

For example, $f(t)=\frac{t}{e^t-1}$ for the Bernoulli polynomials $B_n(x)$, so

$$e^{v.(x)t}=\frac{e^t-1}{t}e^{xt}$$ for the UI dual, the reciprocal integer polynomials,

$$v_n(x)=\frac{(1+x)^{n+1}-x^{n+1}}{n+1}.$$

Consequently, using the simple convolution properties of Appell sequences and the UI relation,

$$ \frac{(1+B.(x))^{n+1}-B.(x)^{n+1}}{n+1} = \frac{(B.(1+x))^{n+1}-B.(x)^{n+1}}{n+1} = x^n = \frac{d}{dx} \frac{x^{n+1}}{n+1},$$

so we can formally identify for any power series

$$S(B.(1+x))-S(B.(x)) = S^{'}(x).$$

There are many interesting and useful dual operators associated with these UIPs (e.g., the first derivative and the forward finite difference operator and its inverse are related to the Bernoulli UIP), and the umbral inverse property alone is often very useful in giving simple, concise derivations of properties of the polynomials and their coefficients. Interweaving the two types of inversions, multiplicative and compositional, a simple formula for the Bernoulli polynomials and their associated base numbers, the Bernoulli numbers, in terms of the Stirling numbers is easily derived in my blog post Compositional Inverse Operators and Sheffer Sequences.

The duality between projective modules and injective modules, also the duality between divisible abelian groups and free abelian groups.

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