# Talk:Duality (mathematics)

WikiProject Mathematics (Rated C-class, High-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 C Class
 High Importance
Field: General

## This entire article is completely vacuous and artificial.

The definition (if it can even be called one) given states that:

Generally speaking, dualities translate concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion.

This could describe any of a number of different mathematical operations. An attempt is made to be more specific by characterizing dualities as involutions, but some of the items listed below are not involutions (the dual of a linear space for example). It seems like the only thing that the examples have in common is that they contain the word "dual".

I have never seen the word "dual" or the concept of duality itself defined or discussed in mathematical literature. This article certainly doesn't cite any instances of such discussions. In fact, the word is actually used in many different ways in mathematics. Thus to try to define duality as a single mathematical concept is misleading and incorrect.

I would like to suggest deleting this article or changing the title to "Mathematical terms containing the word dual".

68.89.168.74 (talk) 05:26, 7 March 2008 (UTC)

## Explanation of edit

So I put in this explanation -- "Usually duality is associated with some sort of general operation, where finding the "dual" of an object twice retrieves the original object (hence the "duality"). This does not preclude the possibilty that an object is its own dual, but there should be at least some objects which are distinct from their duals."

I'm not sure this is the best wording, but I think it conveys the "feeling" behind duality which transcends the categories in place.

Demorgan's dual: objects - binary operators, operation - swapping and and or and negating. Dual polyhedron: objects - polyhedra, operation: faces to vertices Dual spaces: objects - vector spaces, operation: taking linear functionals

## older stuff

Further dualities include;

Charles Matthews 09:43, 21 Apr 2004 (UTC)

Actually I take the dual in dual numbers to imply double, rather than any duality.

Charles Matthews 07:03, 28 Apr 2004 (UTC)

What does the phrase "is dual to" mean? helohe 21:05, 7 Jun 2005 (UTC)

## category theory

Shouldn't there be a reference to the notion of 'dual' from category theory? If someone with more expertise does not add it in the next days, I will give it a go. Ringan 22:50, 16 October 2005 (UTC)Erik Douglas

I'm not an expert but I definitely think that someone should add something about autonomous categories. I'll start by adding a link to it.82.243.57.92 (talk) 20:31, 20 November 2014 (UTC)

## "Geometric" duality?

The planar graph example in the "geometric" section isn't particularly geometric: it's topological. I nuked the word 'geometric' from the link for that reason, but now it doesn't fit in its section. Bhudson 22:45, 8 December 2006 (UTC)

## duality as involution

I have noticed that in several of the most important examples of mathematical 'duality', an involution operation is involved. In ${\displaystyle L_{p}}$ spaces, for example, the Riesz representation theorem may be thought of as an involution: "take some function ${\displaystyle f\in L_{p}}$ and form a functional by integrating it against ${\displaystyle g\in L_{q}}$ where ${\displaystyle 1/p+1/q=1}$." When this operation is done twice it returns us to f. Planar graph duality has a similar property: exchanging vertices and faces twice returns us to the original graph, so duality is an involution on the class of planar graphs. De Morgan duality may be thought of in this way as well, etc. etc.

It seems to me that duality follows this pattern sufficiently often that it should be framed as such in the introduction. My recent edits reflect that. Hopefully you all will find it a valuable direction to take the article. 69.215.17.209 17:44, 22 April 2007 (UTC)

## Duality not always an involution

It may be worth noting that the dual operator is not always an involution. For example, for finite dimensional vector spaces it holds true that ${\displaystyle (V^{*})^{*}=V.}$ However, this is not true for infinite dimensional vector spaces(seeDual_space#The_infinite_dimensional_case). Alexander.fairley 08:30, 19 May 2007 (UTC)

I agree. In addition: Fourier transform (mentioned in the "Analytic dualities" section) is not quite an involution; rather, its fourth power is the identity. Well, this is not far from being involution. However, the Laplace transform (also mentioned there) is not at all involutive, is it? Boris Tsirelson (talk) 18:31, 10 March 2009 (UTC)
In my work, "duality" is usually a pair of transforms, not a single transform. The short definition of duality I would use in many areas is "A duality between two categories is an equivalence of categories between the first and the opposite category of the second." For instance, the Fourier transform and the inverse Fourier transform seem like a duality to me. Alexander.fairley's comment on the double dual even applies to the example given above for L^p spaces; the dual of the dual of L^1 is not L^1. JackSchmidt (talk) 18:41, 10 March 2009 (UTC)
I think a duality should exhibit a surprising isomorphism between two structures that can be useful to understand both. Fourier transform handles amplitude/frequency. Projective geometry handles points/lines. Vector space duality handles vectors/functionals (in finite dimensions). In most of these cases, it doesn't make any sense to talk about the same transform being applied twice. JackSchmidt (talk) 19:03, 10 March 2009 (UTC)
Whether something is an involution may also depend on how it is defined. For instance, in Birkhoff's representation theorem (a duality of categories between finite partial orders and finite distributive lattices), the correspondence from one type of object to the other seems on the face of it very different: to go from a partial order to a lattice, take the collection of downward-closed subsets of the order, but to go from a lattice to a partial order, take the subset of join-irreducible elements of the lattice. From another point of view, though, both directions are the same operation: to go from X to its dual, take Hom(X,2). —David Eppstein (talk) 20:07, 10 March 2009 (UTC)

## Some more algebra dualities

In module theory, an RS bimodule M sets up a (possibly cruddy) duality between the module categories of R and S. When the duality is an equivalence of nice (sub)categories, then it is called Morita duality (see Morita equivalence). When R=S is a commutative k-algebra and the duality is nice enough, M is called a canonical module. If R=S=M is a commutative Gorenstein ring, then the duality is nice enough. The zero dimensional case specializes to something that is an alternative definition of zero dimensional Gorenstein in the non-commutative case: a ring has a perfect duality (as in MR0097427) iff the ring is a quasi-Frobenius ring (see Frobenius algebra). Dualizing complexes are used to handle more general cases, as in MR1799866.

When R is a k=S algebra and M=k, then one often gets important dualities, for instance Northcott's inverse polynomials to describe injective modules over Noetherian rings, and the dual module in the representation theory of finite groups. JackSchmidt (talk) 19:28, 8 March 2009 (UTC)

Cool. Put it to the article! Actually, I have just pinged Arcfrk, PaulTanenbaum and Stca74 (since I know they were interested in duality) to gather around with the hope to make this a GA (or more). Let's work on this pretty cool topic. I guess we will eventually have a hard time as there are so many dualities... Jakob.scholbach (talk) 21:36, 8 March 2009 (UTC)
In case someone wants to start adding before I get a chance. In §19C of:
A number of "classical dualities" are given:
• Galois duality between field extensions and Galois groups, probably also Galois connections in general (not involutive)
• Vector space duality on finite dimensional vector spaces over a division ring (and an explicit statement that duality does not work at the level of general vector spaces of possibly infinite dimension) (involutive if the division ring is a commutative field)
• Pontryagin duality, both for locally compact Hausdorff topological abelian groups and for finite abelian groups. (both are involutive)
• Gelfand-Naimark duality between C*-algebras and compact Hausdorff spaces (not involutive), not mentioned in current revision of this article, and related to the target of the redirect Gelfand duality.
• k-duality of k-algebras (not involutive unless algebra is commutative)
• Quasi-frobenius ring duality (not involutive unless ring is commutative)
It goes on to talk about Morita duality (a nice sort of duality for some semiperfect rings) in §19D. JackSchmidt (talk) 18:57, 10 March 2009 (UTC)

## Duality as Hom(-,X)

It seems like a lot of the dualities have an expression as Hom(-,X), where actually there are two transforms Hom_C(-,X) and Hom_D(-,X) where X is an object that somehow lies in both C and D. Most of the module dualities are like that and the Birkhoff's representation theorem duality is like that.

Maybe there is some theorem on representable functors that might be relevant?

I think many dualities do not have this form too, so don't take this as a definition. JackSchmidt (talk) 20:16, 10 March 2009 (UTC)

I guess the prototypical examples of this are two that you didn't mention, duality of vector spaces over a field F as Hom(-,F) and Pontryagin duality as Hom(-,C*). —David Eppstein (talk) 20:58, 10 March 2009 (UTC)
Hm... The only idea I can propose for now: let's gather a couple of dualities (and write a little bit about them), then we can perhaps find some ways to organize the zoo. As far as I see now, we won't be able to give a "definition" of dualities, but only a list of possible meanings. Some dualities may be instances of several principles at a time etc. Jakob.scholbach (talk) 22:14, 10 March 2009 (UTC)
It might also be interesting to look at dualities in related sciences, such as physics. For example Duality_(electricity_and_magnetism) looks pretty interesting. It's not clear to me what the mathematical content behind this is, but I'd not be surprised if it is maths. Jakob.scholbach (talk) 22:14, 10 March 2009 (UTC)

I'd like to see these gathered into a section, showing how they're really all examples of the same phenomenon. I think the situation is something like this, but I'm putting it here rather than directly editing it into the article because I don't know the category theory literature very well and don't want to be committing original research. In its most basic form, duality of categories is just the same as duality of orders, already discussed briefly in the geometric duality section: just reverse all the arrows to get the dual category. But in many situations, this duality has more structure. In particular, for any X in any category C with small hom-sets, Hom(-,X) is an arrow-reversing functor (to a category in which the objects are hom-sets: for any arrow Y->Z in C, composition with that arrow gives a function Hom(Z,X)->Hom(Y,X)). But suppose X has the following two properties: (1) for all pairs of arrows f≠g between the same Y and Z, there exists an h from Z to X such that fh≠gh, and (2) for all pairs of arrows f≠g between the same Y and Z, there exists an h from X to Y such that hf≠hg. Then in this case, by property (1), Hom(-,X) is an arrow-reversing isomorphism of categories. Further, because it's an isomorphism of categories, X* = Hom(X,X) has the same two properties (they're dual to each other), so Hom(-,X*) returns back to the original category. —David Eppstein (talk) 21:46, 12 March 2009 (UTC)

Doesn't (1), (2) just mean that X is a generator and a cogenerator? Ringspectrum (talk) 09:11, 13 March 2009 (UTC)
But I don't see why (1) implies that Hom(-,X) is an isomorphism of categories. I think it is just equivalent to Hom(-,X) being faithful. Ringspectrum (talk) 09:15, 13 March 2009 (UTC)
I was assuming that the image of Hom(-,X) is the category of Hom-sets, so what it doesn't explain to me is why it's sometimes an interesting category in its own right. Anyway, probably what we should be doing is looking for a reliable source that makes a story like this about how these dualities are all related, and using that story, rather than trying to make up our own. —David Eppstein (talk) 14:23, 13 March 2009 (UTC)
The following example helped me understand the problem. Roughly speaking, David's description leads to a (possibly transfinite) monotonically decreasing sequence of full categories Cn,Dn such that Hom(-,X) and Hom(-,X*) describe faithful functors between Cn and Dn. On the off chance that Cn and Dn have limits (they become eventually constant), then Hom(-,X) and Hom(-,X*) define a duality between the limits. Unfortunately the limit of Cn need not be C0 and need not be a nice category at all (with explicit examples in Lam). I'll describe it in terms of modules:
Assuming the categories are actually subcategories of the category of modules over a ring, (1) and (2) do not quite define a duality, but they point out exactly what needs to be done. Condition (1) makes the map 1-1, and you implicity assume that the map is onto (by choosing its codomain to be its range, no biggie). But then you need to do the same thing for Hom(-,X*), and by then you might have had to shrink the domain of Hom(-,X) again. So you repeat and shrink the codomain of Hom(-,X), then its domain, etc. etc. An equilibrium is reached when you take the source category to be the category of X-reflexive modules and and the codomain category to be the category of X*-reflexive modules. However, in this new setting you still have to be careful that X and X* are both cogenerators (where I name things the module theory way). This sets up something a little weaker than Morita duality, but in the case of modules where you skip the traveling to the equilibirium and just start with a module U and its categories of U-reflexive and U*-reflexive modules, then you get a categorical duality, as in Lam's Corollary 19.40.
A big problem is that all this shrinking of the domain and codomain might have left you with some really nasty categories, not the one you started with, not one that is closed under quotient modules, submodules, and extensions. On the off chance, that the resulting categories *are* closed, then you call the duality a Morita duality. JackSchmidt (talk) 15:54, 13 March 2009 (UTC)
What's the name of the book of Lam you're referring to? Ringspectrum (talk) 19:44, 13 March 2009 (UTC)
Oh, I mentioned it above. Lectures on modules and rings, (Lam 1999). JackSchmidt (talk) 20:48, 13 March 2009 (UTC)

## Duality for Z^

I've got questions to

Hn(G, M) × H1−n (G, Hom (M, Q/Z)) → Q/Z

Is M supposed to be a discrete G-module? Is it group cohomology (for profinite/arbitrary groups?) or Tate cohomology? How is it proved? Direct proof for n = 1 and then dimension shift? Ringspectrum (talk) 20:32, 11 March 2009 (UTC)

Some true statements are in Cartan-Eilenberg, Chapter XII, Section 6 (Duality). They concern finite groups and Tate cohomology. Based on the indexing in this article and in C-E, I suspect this article is using Tate cohomology. I think checking it for G=1 and n=0,1 establishes that it must be Tate. JackSchmidt (talk) 21:13, 11 March 2009 (UTC)
For G finite, it can be found in {Template:Neukirch et al. CNF}, Chapter III (Tate cohomology). But here we have ${\displaystyle G={\hat {\mathbf {Z} }}}$. Ringspectrum (talk) 09:44, 12 March 2009 (UTC)
It is in Milne, Arithmetic Duality theorems (see the article for the ref), I.1.10. Jakob.scholbach (talk) 10:04, 12 March 2009 (UTC)

## Is there a notable duality thing going on with ...

It is certainly notable that transposing a Young tableau gives another one, a fact that leads to easy bijective proofs of facts like that the number of partitions in which the number of pieces has property X is the same as the number of partitions in which the largest piece has property X. I don't know the connection to representation theory so well but I imagine that the fact that every irrep corresponds to some other irrep is similarly important. But I don't know whether any of this is usually called a duality. Maybe the representation theory part has something to do with Schur–Weyl duality? —David Eppstein (talk) 22:46, 11 March 2009 (UTC)
The transpose of a Young tableau is also called the dual partition. Every (ordinary) irrep of a finite symmetric group is self-dual (that is, has character values in the real numbers), so it is not the standard representation theoretic dual Hom_k(-,k). There is another abelian group acting on the irreps of a finite group, Irr(G/[G,G]), which in this case has exactly one non-identity element, the "sign" representation. The irrep corresponding to a dual partition is the tensor product of the sign rep with the irrep corresponding to the original partition. Schur–Weyl duality is different, as it relates the usage of partitions to describe conjugacy classes and irreps in the symmetric group to a similar usage in the general linear group. It is probably worth mentioning though. JackSchmidt (talk) 14:12, 12 March 2009 (UTC)

## Article is missing logical and set-theoretic dualities

Sets are dual with their complements, and the set intersection and union operators are dual; this is analogous to the logical duality of ∧ and ∨. Also not mentioned are logical dualities between ∀ and ∃, and between certain pairs of modal logical operators such as ⋄ and □; temporal logic also contains pairs of dual temporal modes. Validity and satisfiability of formulas are dual notions and are frequently referred to as such.

In topology, open and closed sets are dual, and interiors are dual to closures.

Dominus (talk) 08:15, 18 March 2009 (UTC)

Go ahead and add to the article! As you see, we are in the midst of revamping it all... Perhaps this would match nicely with the Birkhoff distributive lattice duality? Jakob.scholbach (talk) 08:37, 18 March 2009 (UTC)
Well, I'm not sure I made it better, but at least I made it longer. —Dominus (talk) 15:57, 18 March 2009 (UTC)

It seems to me that what all these examples have in common is that we have some sort of complementation or negation operator, say "~", which satisfies ~~x = x, and two operators, say A and B, related by Ax = ~B~x. For the set examples "~" is the set complement operator; for the logic examples it's logical negation.

It was tempting to try to explain this in category-theoretic terms, with A and B being functors, and "~" being a natural isomorphism of A and B, but this explanation isn't general enough. Topological closure and interior fit the pattern of the previous paragraph, but neither one is a functor. —Dominus (talk) 14:55, 18 March 2009 (UTC)

## Nice work

I looked over the article as a whole today, and I was very surprised at how good the article is now. There are still lots of places that need touching up, but I think the current article makes it clear that duality is a rich and interesting subject no matter how it is approached. This is one of very few mathematics articles on enwiki I have read that really seems to address a wide range of views in a coherent and engaging manner. Nice work. JackSchmidt (talk) 19:29, 18 March 2009 (UTC)

## More to do: category theory

Category theory books invariably discuss the fundamental duality properties of categories, that if C is a category, then so is Cop, often called the "dual category" of C. Products are then dual to disjoint sums, initial objects to terminal objects, pullbacks to pushouts, monomorphisms to epimorphisms, limits to colimits, isomorphisms to isomorphisms, and, in general, X-constructions to co-X-constructions.

An example of this duality is that having proved that if 1 is a terminal object in some category, the product X × 1 exists and is isomorphic to X, one need not prove separately that if 0 is an initial object, the coproduct X + 0 is isomorphic to X; it follows immediately by duality. —Dominus (talk) 19:34, 18 March 2009 (UTC)

This is already done in the dual categories section. JackSchmidt (talk) 20:39, 18 March 2009 (UTC)
I think it needs work. —Dominus (talk) 20:50, 18 March 2009 (UTC)

## More to do: lattices, upper/lower bounds?

The lattice axioms are symmetric between upper and lower bounds, and so one has dualities between meet and join, beetween minima and maxima, and so forth.

In certain ordered sets there is a duality between < and >. For example R possesses the least-upper-bound property. The dual "greatest lower bound" property is equivalent, perhaps even equivalent in general; I forget. —Dominus (talk) 19:34, 18 March 2009 (UTC)

I think this is different from the Birkhoff duality (which currently is given short schrift). It might be nice to mention that lots of lattice-y categories are self-dual (full category is, I think) and this allows switching ≤ and ≥ in theorems. JackSchmidt (talk) 21:01, 18 March 2009 (UTC)
It is different. The reversing-all-order-relations duality is mentioned briefly in the geometric section (it comes up there from the reversal of the face lattice of a polyhedron and its dual) and is closely related to the reverse-all-arrows dualities of categories. Birkhoff instead treats a partial order as dual to a different kind of object, a distributive lattice; it is one of a number of Stone dualities relating topological spaces to order-theoretic algebras. In this case, although the Birkhoff article doesn't really mention it, a finite distributive lattice can be interpreted as the family of open sets of a finite topology, and the corresponding partial order can be interpreted as the set of points of the topology. —David Eppstein (talk) 22:19, 18 March 2009 (UTC)
It is indeed different. I went ahead and added a section on order-theoretic duality because it is widely used in order theory. I put it ahead of geometric duality, not because I think this sense is any more important, but because the latter mentions it. And the order-theoretic flavor should certainly be easily understood by a novice reader.
By the way, I'm not sure that the free-standing article even ought to remain. Probably should just be folded in here.—PaulTanenbaum (talk) 03:20, 20 March 2009 (UTC)

## More to do: tesselations

Every tesselation has a dual tesselation. In particular, the dual of the Voronoi tesselation is the Delaunay triangulation. —Dominus (talk) 19:48, 18 March 2009 (UTC)

The dual relationship (dual graphs) is already discussed in the geometric duality, but this sounds like a very nice example and very likely a prettier picture to include in this article near that paragraph. JackSchmidt (talk) 20:58, 18 March 2009 (UTC)
I think File:Delaunay Voronoi.png (right) would be a good example to use to illustrate this. It's probably worth pointing out that the primal and dual edges don't necessarily cross each other in this form. Voronoi-Delaunay duality can actually be seen as an instance of convex polyhedron duality, not just planar graph duality: both diagrams are projections onto the plane of polar polyhedra in 3d. —David Eppstein (talk) 22:24, 18 March 2009 (UTC)

## Closed and open sets?

Recently, closed and open sets are mentioned as an example of duality. While it is possible to call this a duality I wonder whether people actually are doing so. I have never heard the d-word in this context. For the sake of conciseness we ought, IMO, not include things that are somehow reminding us of dualities, but are not really considered dualities. Comments? Jakob.scholbach (talk) 08:33, 23 March 2009 (UTC)

Another point: currently set theory (complement) is mentioned as a consequence of logic, which I find odd. Instead, I think, the duality of quantifiers in logic etc. derives from the set theoretical gadget, right? Jakob.scholbach (talk) 08:33, 23 March 2009 (UTC)

I think of the set theoretical complementation as closely related to order-theoretic duality, since it is reversing the order of the subset relation on the sets. —David Eppstein (talk) 13:57, 23 March 2009 (UTC)
I mentioned closed and open sets specifically because I had seen them mentioned elsewhere as an example of duality, not because they "somehow reminded me of duality but are not really considered dual". Are you trying to say that you think that section needs references? If so, why don't you just say that? —Dominus (talk) 16:57, 23 March 2009 (UTC)
"A set W and an operation □...is usually called a topological space and □ an interior operation on this space. The dual operation, ⋄, defined by ⋄X = -□-X... is called the closure operation..."Modal Logic.
"Any condition placed on the members of [the topology] F can be transformed, according to the De Morgan laws, into a condition on the members of the dual family F*. In case F is a topology, the members of the dual family are called closed sets..." Topological and Uniform Spaces
"We observe here there is a certain duality between theorems concerning open and closed sets. In particular, open and closed, interior and closure, union and intersection, are dual categories under the operation of complementation." Introduction to Riemann Surfaces
"Theorem 1.1: The union of an arbitrary collection of open subsets of C is an open set. ... There is a proposition dual to Theorem 1.1 dealing with closed sets." An Introduction to Complex Function Theory
I didn't mean anything in particular, I was just wondering. The references are reassuring, anyways. Can you put that to the article (where possible with precise references as the ones given above)? Thx. Jakob.scholbach (talk) 17:56, 23 March 2009 (UTC)

## Fourier transform is not an involution (still)

I must repeat: Fourier transform (mentioned in the "Analytic dualities" section) is not quite an involution; rather, its fourth power is the identity. Indeed, the inversion formula is different (very similar, but different) from the original formula. Boris Tsirelson (talk) 20:37, 23 March 2009 (UTC)

## Dual lattice?

The section on dual objects has a sentence that begins: "The dual lattice of a lattice L is given by Hom(L, Z)..." with the phrase dual lattice being a link to an article on crystalography, but it's not clear to me that that article is what the person who wrote this sentence wished to point to. If I could figure out which sense of lattice is being discussed here, I'd disambiguate the passage myself. Can anybody pls clean this up, since all other references to lattices in this article deal with the poset-kind of lattice.—PaulTanenbaum (talk) 02:03, 27 March 2009 (UTC)

Yes, that was me. Perhaps we should simply remove that. (I personally think, the crystallography article should have some other name, but the dual lattice in the mathematical sense is not so important, on the other hand, that we have to include it here). Jakob.scholbach (talk) 10:17, 27 March 2009 (UTC)
Hmmm, dual lattice is not important enough to include? The crystallographic reciprocal lattice is basically the same as the Poisson summation formula, which has inspired the Selberg trace formula. I would say it's very important... I don't yet have a good idea though where in the article it should appear. I suspect that even the planar duality of graphs was inspired by reciprocal lattices. --GaborPete (talk) 07:32, 3 April 2009 (UTC)

## Returning to the first question: what is duality?

I'm not trained in category theory, so I have troubles following the discussion on Duality as Hom(-,X) exactly. But from what I understand, I think the point of duality really should be the pair of transforms "Hom_C(-,X) and Hom_D(-,X), where X is an object that somehow lies in both C and D". This was also the conclusion of Is duality always an involution? discussion, it seems. But the definition presently given in the leading paragraph would basically fit ANY correspondence or translation between two theories, say, expressing classical line-circle geometric questions algebraically with coordinate geometry. True, mentioning that it is typically an involution sort of means that we do the same translation mechanism to get from theory or object A to theory or object B as vice versa, and this is not true or meaningful for most correspondences. But still, the main underlying idea should be stressed more, I think (and maybe has to be first found?). --GaborPete (talk) 08:29, 3 April 2009 (UTC)

A couple of weeks ago in the Math WikiProject page I asked if it is true that every duality basically comes from the idea of a linear pairing. In the answers I was given basically two counterexamples: 1. planar and polyhedral duality; 2. the Boolean or set theoretical duality given by negation or complement. That thread has gotten archived, so let me continue here:

First, to clarify: by "linear duality" I meant two related things: an n-dim vector space has an n-dimensional dual; a k-dimensional subspace has an (n-k)-dimensional corresponding subspace in the dual space. Are these somehow dual notions of duality?
Anyway, the planar and polyhedron duality seems to me basically the same as Poincaré duality, and that does come from a linear pairing, through the de Rahm cohomology, say. Is this an oversimplifying point of view?
Regarding the Boolean duality, I have to agree, I don't see a direct connection. (I have not thought about the linear logic stuff...) On the other hand, I would say that Boolean duality is such a basic construction of the mind that it might have been the inspiration to construct and look for all the other dualities. Maybe the article should have this duality, well-known to every human, more at the beginning, to explain the main idea? --GaborPete (talk) 08:29, 3 April 2009 (UTC)

Finally, a paper I have just noticed: Shiri Artstein - Vitali Milman: A CHARACTERIZATION OF THE CONCEPT OF DUALITY [1]. See also Topic 8 among the research interests of Shiri Artstein, [2]. --GaborPete (talk) 08:29, 3 April 2009 (UTC)

From a quick glance it does not seem that the paper you cite covers all the aspects of duality. But it does seem to give a nice overview of the analytic parts. Would you volunteer to write about that in the article? I will try to come up with references for the facts already mentioned. I propose we do some brainstorming what other relevant dualities might still be missing and then think about a good global architecture of the article. Jakob.scholbach (talk) 13:44, 6 April 2009 (UTC)
It seems GaborPete is busy, thus I did it, see the new section "A more general approach: concept of duality". Boris Tsirelson (talk) 19:43, 30 May 2009 (UTC)
I moved this new paragraph to be part of the "order-reversing dualities" section. I don't see why it should be considered any more general than some of the other dualities discussed here. (E.g. the category-theoretic ones.) —David Eppstein (talk) 21:16, 30 May 2009 (UTC)

## Dualising object

Dualising object redirects to here, but there is no explanation of the concept here. Cheers, 128.232.228.113 (talk) 17:06, 3 May 2011 (UTC).

## How would you call this duality?

16×16 matrix of 1×4 matrices

Below the dual 1×4 matrix
of 16×16 matrices

When you have an m×n matrix of p×q matrices, it is possible to define a dual p×q matrix of m×n matrices. Compare: v:User:Mate2code/hat#Dual_matrix I often use this to take a closer look at the binary digits, when the elements of a matrix are binary numbers. That can be seen in the nimber multiplication table on the right, with the permuted binary Walsh matrices below. Does anyone know, how this kind of duality is called? It's not limited to matrices. Mate2code (talk) 22:10, 17 October 2011 (UTC)

## Dual graphs

Dual graphs are discussed in the article, but only in the case where the embedding surface is a plane (or equivalently a sphere). There is no mention of dual graphs in other surfaces, although the term is used in WP articles, e.g. Dyck graph. In view of this omission, it is not surprising that there is also no mention of Petrie duals. Maproom (talk) 22:21, 2 January 2015 (UTC)

## Dual cone

I think the recent addition of the "Dual cone" section, and particularly its position near the top of the "Introductory examples", is unfortunate:

• It is not an example of duality: the relation is not symmetric.
• It is obscure. Dual polytopes, and dual graphs, will be more familiar to most readers.
• It is poorly explained. The text does not make it sufficiently clear that the position of the cone depends on the origin of the space. The diagram does not make it clear which region constitutes the dual cone.

Maproom (talk) 10:40, 24 July 2015 (UTC)

I disagree with you regarding all points. It is the very purpose of mentioning this duality that not every duality is involutive, such as the one on vector spaces, for example. To properly define dual polytopes you need quite some space (as opposed to hand-waving where the vertices of the dual polytope lie). Likewise for dual graphs (look up what precise assumptions you need to even define the dual graph). The origin of the space is implicit in talking about points in R^2. Other than that it is also entirely irrelevant, any choice of origin will do. Finally, the picture does display the dual cone in red.
The purpose of this introductory list is to give examples of dualities whose structural properties appear again and again. Moreover, the rationale behind this list is to increase their generality and mathematical complexity. The dual cone is just fitting right there, I think. Jakob.scholbach (talk) 12:24, 24 July 2015 (UTC)
If the dual cone example started by saying that the example is not involutive, I might agree with your first point. t doesn't. It is therefore likely to confuse the readers of this "introductory example". Yes, for dual graph, you need a space, but not one with a metric and an origin. The picture displays the dual cone in red, but the reader may fail to guess this, the label C* is in the lowest of the three triangles into which the red area is divided. You say "any choice of origin will do"; it seems to me that this is false, the position of the cone is determined by the origin, its apex is at the origin.
For an expert like yourself who already knows about dual cones and the more familiar properties of duality, the example may be a good one. But in a list of "introductory examples", I believe it is confusing. It would be better to have an example which most readers are likely to understand, for example the way that the octahedron is the dual of the cube. Maproom (talk) 16:32, 24 July 2015 (UTC)
What exactly do you consider confusing about the dual cone?
Actually I just learned about the dual cone while working on this article. I chose it as a prototypical example, because it is completely elementary, and the closely related to the duality in optimization. I plan to work on the further sections to make this clearer.
My comment about the origin was intended to say that the precise choice of the origin does not matter. Choosing one origin of course does matter. Jakob.scholbach (talk) 21:39, 26 July 2015 (UTC)
I am sorry, Jacob, but Maproom is right: the "dual cone" as presented is not an example of duality in the sense of the article, and the picture does more harm than good. I suggest that you delete this subsection. There is a standard dual cone construction for convex cones (related to polarity for convex sets containing the origin) that qualifies far better under this rubric; whether or not it is an introductory example is arguable. Arcfrk (talk) 05:11, 27 July 2015 (UTC)
Well, we can certainly tweak the section and emphasize it is a duality in the narrow sense on cones. If you dislike the picture, maybe you can create a better one? Jakob.scholbach (talk) 15:45, 27 July 2015 (UTC)
The key word is "convex". Applying this construction to an arbitrary set (even a cone) suffers from the same defect as taking orthogonal complement to an arbitrary subset C of a linear space: namely, the result depends only on the span of C; only in the case of polar duality, convex span replaces the span. In effect, you compose a genuine duality between convex sets (or linear subspaces) with a "forgetful map" from general sets/cones (respectively, general subsets) to convex (respectively, linear) ones — this is what makes the construction asymmetric. And yes, the convex duality is more sophisticated than the linear one, so it would be logical to treat it later in the article. Arcfrk (talk) 07:14, 28 July 2015 (UTC)
There is no formal definition of "duality" in the article; so it should cover all mathematical concepts which are called "dual". Apparently, "dual cone" is such "an example of duality in the sense of the article". I don't understand the problems that Maproom and Arcfrk have with it, except that it wasn't familar to them (it was neither to me), and its explanation could be improved (I'd like to contribute when this dispute is settled) . - Jochen Burghardt (talk) 16:25, 27 July 2015 (UTC)
As I said before, the purpose of the "Introductory examples" section should be to introduce the subject of the article to readers, using as far as possible examples that may already be familiar to them. Complement of a subset is good for this; so would dual polyhedron be. That section should not be used to teach unfamiliar material to them. I don't object to "dual cone" being explained later in the article, particularly now that that subsection is rather clearer. Maproom (talk) 17:09, 27 July 2015 (UTC)
Ok, I see. Do you suggest another non-involutionary introductory example, or do you think we don't need one (in the introduction)? - Jochen Burghardt (talk) 17:24, 27 July 2015 (UTC)
I am still not convinced that the dual polyhedron is actually easier than the dual cone. As I said before, precisely even defining the dual polyhedron requires quite some space. It is not a very prototypical example. For example, it is harder (it seems) to understand the fact that in practically all dualities you have a reversal of morphisms.
@Arcfrk: of course every cone is convex here (and closed). I guess the every day meaning of a cone presupposes this, but I had already added a footnote to make this precise. This is a technical point which deserves to be briefly mentioned, but it is not a fundamental difficulty.
I suggest we all spend more time on the article itself than on the talk space. Jakob.scholbach (talk) 08:04, 28 July 2015 (UTC)
The question is not whether "dual polyhedron is easier than the dual cone", it's whether it is more familiar to the typical reader. The introductory section is not there to present unfamiliar and interesting examples of duality, it's there to help the reader grasp what duality is. Maproom (talk) 08:21, 28 July 2015 (UTC)
Exactly. I think the reader will not benefit from an early mention of the dual polyhedron in the sense that he / she will not grasp better what duality is about in genereal. The duality of polyhedrons is simply not a prototypical duality, therefore it does not help that much to understand what the general phenomenon of duality is. You can talk about it using elementary language, yes, but the same is true for the dual cone. The dual cone is easier to precisely define. The dual cone highlights the fact that dualities are not in all cases involutions. In short, the dual cone is a better example than the dual polyhedron.
If you are still not convinced, I heartily invite you to write an introductory section (possibly starting from what we have in the article) about the dual polyhedron; maybe here in the talk space. You will see that making it precise is not that easy; the space you will spend is better invested in other dualities. Jakob.scholbach (talk) 12:29, 29 July 2015 (UTC)
I believe that the article would be improved by removing the "dual cone" example from the introductory section. The essence of duality is that it's reflexive. Ok, the concept is extended to non-reflexive subjects such as dual vector spaces. But the introductory section should be addressed to readers who are not yet familiar with the concept. Using an introductory example which is obscure, illustrated by a diagram that's hard to understand, and above all is not reflexive and does not even mention that it's not reflexive, is likely to induce readers to give up and look for some other explanation of duality. Maproom (talk) 19:00, 17 November 2016 (UTC)

I'm not sure that the dual cone example should be removed, but I am certain it should be moved down. It is not an introductory example, and it should not be the second example on the page. An introductory example should be something familiar to the reader, which gives the correct general idea, to be refined later. The dual cone construction may be a good example to show that not every duality is an involution, but if so it does not make sense to explain that before the reader has a clear idea what a duality is. The dual cone construction is unfamiliar even to many mathematicians. I will shortly move it downwards. —Mark Dominus (talk) 21:41, 18 November 2016 (UTC)

OK, move it down then. Jakob.scholbach (talk) 18:57, 19 November 2016 (UTC)
I've changed my mind. I think the dual cone and Galois group sections should both be moved down, but the way the article is structured at present there isn't a good place to move them to. And I think some of the more familiar examples should be moved up. But I don't want to undertake a wholesale reorganization of the article's examples, so I'm going to leave it alone, at least for now. —Mark Dominus (talk) 20:45, 29 November 2016 (UTC)