Dualities
Table of Contents
Projective duality
We consider a two-dimensional projective plane. Such a plane can be introduced by a set of axioms, similar to Hilbert's axioms for Euclidean geometry.
Projective plane axioms
Often we speak about the two-dimensional projective plane and think about the extended Euclidean plane (real or complex), but there are others. Famously, there are two-dimensional projective planes where Desargues' theorem does not hold. Here however, we will look at planes where Desargues' theorem holds. For this case Coxeter gives the following six axioms:
Axiom 1: Any two distinct points are incident with just one line.
Axiom 2: Any two lines are incident with at least one point.
Axiom 3: There exist four points of which no three are collinear.
Axiom 4: The three diagonal points of a complete quadrangle are never collinear.
Axiom 5: If two triangles are in perspective from a point, they are in perspective from a line.
Axiom 6: If a projectivity leaves invariant each of three distinct points on a line, it leaves invariant every point on the line.
The first axiom simply says that there is a unique line through two distinct points, like in Euclidean geometry. The second axiom excludes the possibility to have lines that never meet (parallel lines). This distinguishes the geometry from the Euclidean plane.
The quadrangle that appears in the fourth axiom consists of four points (called vertices) joined by six distinct lines. The third axiom ensures that such a configuration of points exists. The six lines of the quadrangle intersect in three additional points which are called the diagonal points of the quadrangle. In the following figure the points \(P\), \(Q\), \(R\) and \(S\) are the vertices of the quadrangle and the points \(A\), \(B\) and \(C\) are the diagonal points:
The fifth axiom is one direction of Desargues' theorem.
In the sixth axiom a projectivity is a transformation of lines and points in the projective plane by means of elementary correspondences. These elementary correspondences work as follows. Consider a point \(P\) and a line \(\ell\) not incident with \(P\). To each point \(A\) on \(\ell\) we can associate a line \(a\) obtained by joining \(A\) with \(P\). We can continue by adding another line \(m\) (again not incident with \(P\)). The line \(a\) intersects \(m\) in the point \(A'\), so the points \(A\) and \(A'\) also correspond to each other. If we continue by adding another point \(Q\) to the picture we obtain the line \(a'\) by joining \(Q\) with \(A'\) and so on. A finite sequence of such correspondences is called a projectivity.
Principle of duality
The duality principle (in two dimensions) states that every theorem still holds true if we exchange the words "points" and "lines" as well as some other pairs of words such as "join" and "meet". An incomplete list of words that have to be exchanged is shown in the following table:
| point \(A\) | \(\leftrightarrow\) | line \(a\) |
| join | \(\leftrightarrow\) | meet |
| collinear | \(\leftrightarrow\) | concurrent |
| vertex | \(\leftrightarrow\) | side |
| triangle | \(\leftrightarrow\) | triangle |
| quadrangle | \(\leftrightarrow\) | quadrilateral |
Let us see how this works for the axioms above.
The dual of Axiom 1 becomes:
Any two distinct lines are incident with just one point.
This holds true because of the second axiom (it is a specialization of the second axiom to the case where the lines are distinct). Similarly, the dual of Axiom 2 becomes:
Any two points are incident with at least one line.
If the points are distinct this is just the first axiom. If they are the same then any line through that point will work (such a line exists because the third axioms ensures that there are more points through which we could draw a line).
Dualizing the third axiom we get the following statement:
There exist four lines of which no three are concurrent.
We can take these four lines to be the sides \(\langle P, Q\rangle\), \(\langle Q, R\rangle\), \(\langle R, S\rangle\) and \(\langle S, P\rangle\) of the quadrangle with vertices \(P\), \(Q\), \(R\) and \(S\) shown in the figure above. If any three of them were concurrent, three of the vertices would have to be on the same line, but then the figure is not a complete quadrangle.
The dual of Axiom 4 is the following:
The three diagonal lines of a complete quadrilateral are never concurrent.
A complete quadrilateral is the dual of a complete quadrangle: It consists of four lines in the lines meeting in six distinct points. These six points can be joined by three additional lines which are called the diagonal lines of the quadrilateral. In the figure below the four lines \(p\), \(q\), \(r\) and \(s\) in red are the sides of the quadrilateral and the green lines \(a\), \(b\) and \(c\) are the diagonal lines. The diagonal lines are concurrent only if the six intersection points are not distinct.
The dual of the fifth axiom is nothing but the other direction of Desargues' theorem.
Finally, the dual of the sixth axiom becomes:
If a projectivity leaves invariant each of three distinct lines through a point, it leaves invariant every line through the point.
Note that here we also had to replace "point on a line" with "line through a point". This could have been avoided by using the more symmetric formulation "incident with" everywhere.
All the axioms from above imply their dual statements (even though we have not shown this here for all of them). Consequently, every theorem derived from them has a dual theorem which is also true. An example of theorems that are dual to each other are the theorems by Menelaus and Ceva.
The duality principle also extends to higher dimensions. In an \(n\)-dimensional projective space the dual of a \(p\)-dimensional subspace is a \((n - p - 1)\)-dimensional subspace. For example, in three dimensions, points and planes are dual to each other, while lines are dual to lines.
Duality in category theory
A category \(\mathcal{A}\) consists of a collection of objects \(\operatorname{ob}(\mathcal{A})\) with arrows (or morphisms) between them. The collection of arrows between two objects \(A, B \in \operatorname{ob}(\mathcal{A})\) is denoted \(\mathcal{A}(A, B)\). Each object has an identity arrow \(1_A \in \mathcal{A}(A, A)\). An arrow \(f\) from \(A\) to \(B\) can be composed with an arrow \(g\) from \(B\) to \(C\). The composition is an arrow from \(A\) to \(C\) and written \(g \circ f\).
\begin{CD} A @>f>> B @>g>> C \end{CD} \begin{CD} A @>{g \circ f}>> C \end{CD}For every category \(\mathcal{A}\) there is an opposite (or dual) category \(\mathcal{A}^{\mathrm{op}}\). The objects in the opposite category are the same as in the original one, \(\operatorname{ob}(\mathcal{A}^{\mathrm{op}}) = \operatorname{ob}(\mathcal{A})\), but all the arrows go in the opposite direction. If \(f\) is an arrow \(A \to B\) in \(\mathcal{A}\), then there is a corresponding arrow \(f^{\mathrm{op}}: B \to A\) in \(\mathcal{A}^{\operatorname{op}}\). The identity arrows stay the same.
The composition in the opposite category theory is "reversed" in the following sense: Consider two composable arrrow in \(\mathcal{A}^{\mathrm{op}}\):
\begin{align} \begin{CD} A @>{f^{\mathrm{op}}}>> B @>{g^{\mathrm{op}}}>> C \end{CD} \label{eq:composition-aop} \end{align}Their composition is a map from \(A\) to \(C\), but since we are composing in the opposite category we use the new symbol \(\odot\) for it:
\begin{align} \begin{CD} A @>{g^{\mathrm{op}} \odot f^{\mathrm{op}}}>> C \end{CD} \end{align}In the category \(\mathcal{A}\) the corresponding arrows go in the other direction,
\begin{align} \begin{CD} A @<{f}<< B @<{g}<< C \end{CD} \label{eq:composition-a} \end{align}and composing them with \(\circ\) gives a map from \(C\) to \(A\):
\begin{align} \begin{CD} A @<{f \circ g}<< C \end{CD} \end{align}The composition \(\odot\) in \(\mathcal{A}^{\mathrm{op}}\) is defined such that the maps \(g^{\mathrm{op}} \odot f^{\mathrm{op}}\) and \(f \circ g\) correspond to each other:
\begin{align} g^{\mathrm{op}} \odot f^{\mathrm{op}} = \left(f \circ g\right)^{\mathrm{op}} \end{align}Sometimes the superscript \({}^{\mathrm{op}}\) is not written and the same symbol \(f\) is used for \(f^{\mathrm{op}}\). The arrow in \(\mathcal{A}^{\operatorname{op}}\) from \(A\) to \(C\) is then \(f \circ g\) which is "reversed" if one only looks at the diagram \eqref{eq:composition-aop} in \(\mathcal{A}^{\mathrm{op}}\).
Projective duality as a duality of categories
Can we understand projective duality as a duality of some categories? More precisely, does the operation of exchanging the words "points" and "lines" in the projective plane1 correspond to reversing the arrows in some category?
Projective duality in the plane can be formulated in terms of an incidence structure which is a triple \((P, L, I)\), where \(P\) is the set of points, \(L\) is the set of lines and \(I \subseteq P \times L\) is a relation. By definition \((p, \ell) \in I\) if and only if the point \(p\) and the line \(\ell\) are incident (i.e. if \(p\) lies on \(\ell\)). The dual of the incidence relation \(I\) is defined as
\begin{align} \check{I} = \left\{(\ell, p) : (p, \ell) \in I\right\} \subseteq L \times P. \end{align}The statement that the projective plane obeys the duality principle then becomes that the incidence structures \((P, L, I)\) and \((L, P, \check{I})\) are isomorphic.
We can build a category \(\mathcal{A}\) from the incidence structure \((P, L, I)\) as follows: We take the objects to be all the points and all the lines, i.e. \(\operatorname{ob}(\mathcal{A}) = P \cup L\), and declare that there is an arrow \(p \to \ell\) from a point to a line if \((p, \ell) \in I\). Each point and each line also gets an identity arrow.2
The opposite category \(\mathcal{A}^{\mathrm{op}}\) also has as objects all the points and all the lines of the plane but the arrows are reversed. For each arrow \(p \to \ell\) there is now an arrow \(\ell \to p\) and going back to the incidence relations we therefore have \((\ell, p) \in \check{I}\). The opposite category \(\mathcal{A}^{\mathrm{op}}\) therefore corresponds precisely to the dual incidence structure \((L, P, \check{I})\).
Of course the only thing we have achieved is to convince ourselves that the duality operation in projective geometry can be understood as switching to the opposite category for some category. We have not shown anything new and in particular we have not even formulated the axioms in a categorical language. I suspect that this has been done somewhere but I have not been able to find a reference.
References and acknowledgements
- H.S.M. Coxeter: Projective Geometry
- Tom Leinster: Basic Category Theory
I am very grateful for comments by and discussions with an amazing referee.