The concept of duality

Question

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 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).

Answer 1

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

Answer 2

A simple kind of duality in logic is between implication and set-theoretic inclusion, which explains why the horseshoe $\supset$ is found in both contexts. The most natural way to think about it, is the reverse of the actual usage:

A $\supset$ (implies) B if A $\subseteq$ (is a subset of) B

So for instance,

x is a person implies x is a mammal, since the set of people is a subset of the set of mammals.

Answer 3

Polytope duality

This relates both, the duality of convex sets (polar dual) and duality of lattices (via the face lattice of a polytope).

In the case of 3-dimensional polyhedra, it is related to the duality of planar graphs.

Answer 4

The transposition of Young tableaux yields a kind of duality, internal to the set of partitions of an integer $n$.

Answer 5

From "Hilbert series and Gelfand duality" by Vadim Schechtman (CiteSeerX, arXiv:0908.4533):

In their great work on spherical functions ... Berezin and Gelfand wrote:

”... there exists a deep duality between the function ... giving the law of multiplication in the center of the [infinitesimal] group ring [of a semisimple Lee group]and the function ... giving multiplication of representations.

... an analogous duality exists between matrix elements of ... an irreducible representation of the group SU(2) ... and the so-called ”Clebsch-Gordan coefficients”...

Another example of such a duality are the formulas of Gelfand-Tsetlin for matrix elements of irreducible representations of the algebra of complex matrices with trace 0 and the formulas for coordinates in the group of unitary matrices... In all of these cases the duality consists in the fact that functions of discrete arguments satisfy finite difference equations analogous to differential equations satisfied by functions of real variables that correspond to them.”

The second of the above examples may be expressed by saying that we have a duality between the classical orthogonal polynomials (Jacobi etc.) and their discrete analogues (Hahn etc.). (In fact, all of the above examples admit a similar reformulation.)

The main purpose of the present note is to propose an example illustrating that exactly this type of dual polynomials appears in certain Hilbert series.

Answer 6

The contravariant powerset functor, i.e. covariant functor $P : C^{\text{op}} \rightarrow C$ defined as $P(A)=2^A$ and $P(f)(g) = g \circ f$ is the canonical example of duality.[1] The category $C$ is usually $Set$, but generalizations to toposes exist.

There are several intereresting duality principles that amount to application of the contravariant powerset functor. The duality between products and coproducts in category theory can be seen as application of the contravariant powerset functor on the definition of the coproduct. $P(A+B) \cong PA \times PB$ and \begin{gather*} P([f,g] : A+B \rightarrow D) = \\ \langle Pf,Pg\rangle : PD \rightarrow P(A + B). \end{gather*} Similarly for injections to the coproduct: \begin{gather*} P(i_1 : A \rightarrow A+B) = \\ \pi_1 : P(A+B) \rightarrow PA. \end{gather*} By expanding $P(A+B) \cong PA \times PB$ and using the coproduct laws, these are seen to be the product laws, which justifies use of that notation for products above, where you often suppress explicit notation for the isomorphism. The other projection is similar. This kind of application of the contravariant powerset functor can be done for all finite colimits.

For exponentials, the situation is much different. Trying to find operation $B \setminus A$ (not the set subtraction, but close) such that the contravariant powerset functor produces an exponential, something along the lines of $P(B \setminus A) \cong PA \Rightarrow PB$ is a cause of much confusion about duality, since coexponentials are not very natural concept [2] and can cause havoc when combined with, say, a topos. Inverting the arrows in $C^{op}$, in essence $B \setminus A \cong B \Rightarrow A$, seems obvious solution, but there aren't necessarily exponentials in $C^{op}$, and (speculation:) the notion of exponential as internalization of hom-sets somehow inverts as well. There are several important notions of topology (open sets, bounded sets) that seem suitable for such definition in terms of exponential $PA \Rightarrow PB$. This also interacts with problems defining a left adjoint to a partially applied coproduct functor $F \dashv A + -$ (which might also be called subtraction if it exists). I can see two possible outcomes for such construct. Either the power set hierarchy expands without limit and iterating the exponential (which is similar to iterating the power set) produces ever larger categories. Or the possibilities for such expansion are limited, and some principle, e.g. Rice's theorem, limits the expressive power of repeated towers of exponentials.

[1] Lawvere, Rosebrugh: "Sets for mathematics"

[2] Crolard: Subtractive logic