# The concept of duality

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.)

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).

• 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
• 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
• Great question. I've also wondered about this for some time. – Cole Leahy Aug 28 '11 at 16:53
• 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

I'm surprised no one has mentioned this one yet. Duality between simply connected Riemannian symmetric spaces of compact and noncompact type. I personally think this is a perfect example because it demonstrates how useful duality can be in mathematics:

1. First of all, this is a genuine duality, meaning, it gives a bijective correspondence between isometry classes of simply connected symmetric spaces of compact type on one side and symmetric spaces of noncompact type on the other (these ones are automatically simply connected). And, surely, $$M^{**} \cong M$$.
2. $$M$$ and $$M^*$$ share several basic properties in common: dimension, rank, (the identity component of the) isotropy subgroup of the isometry group, which in this situation is the same as the holonomy group. There is also a bijective correspondence between totally geodesic submanifolds of $$M$$ and $$M^*$$. Duality respects de Rham decomposition and thus irreducibility.
3. In particular, if you want to classify all (simply connected) symmetric spaces, it suffices to classify irreducible ones of just one of the two types, since you get all irreducible spaces of the other type for free by duality, and any (simply connected) symmetric space will then be the product of irreducible ones of these two types and a Euclidean space. If I'm not mistaken, this is exactly what Élie Cartan did in his classification of symmetric spaces in 1926: he obtained a list of compact irreducible ones, and the rest followed.
4. Finally, since symmetric spaces of (non)compact type are quotients of (non)compact semisimple Lie groups, the methods one uses to study spaces of these two types are quite different. You may have a geometrical problem (e.g., classify polar or cohomogeneity-one isometric actions) that is easier to solve on one of the two types, and then duality enables you to transfer some of the results to spaces of the other type to obtain at least a partial solution there.

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.

• Is there really a historical connection between these two uses of the horseshoe? If so, was there a reason for the reversal of the direction, i.e., for the $\supset$ of propositional logic corresponding to he $\subseteq$ of set theory? – Andreas Blass Dec 12 '19 at 1:12
• @Andreas Blass. Yes there was a historical connection. This explains the history: philosophy.stackexchange.com/questions/31029/…⊃-selected-for-material-implication/31031 – abo Dec 12 '19 at 7:05
• The truncated link above: philosophy.stackexchange.com/questions/31029/… – LSpice Jan 19 at 19:37

The contravariant powerset functor $$P : C^{\text{op}} \rightarrow C$$ is the canonical example of duality.[1]

There are several intereresting duality principles which 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

• ...between sets and complete atomic Boolean algebras; has been generalized by Paré and Mikkelsen to a duality between any elementary topos and its internal complete atomic Heyting algebras. – მამუკა ჯიბლაძე Jan 19 at 6:28

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

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.