In classical geometry, duality
is a basic feature of the axioms. For 2 dimensional marked surfaces, swap points for face elements and lines for lines. Since the simplest concept of dimension is totally ordered, in an $n$ dimensional space one can always swap points for $n$-faces, lines for $(n-1)$-faces, and so on.
What would a ternary
analogue of projective duality look like? Even without fixing upon axioms, it is clear that the ordered nature of dimension cannot so easily accommodate a ternary transformation between points and lines and surface elements. That is, we would like to consider a triality which takes points to lines, lines to faces, and faces to points. So we choose to generalise dimension to values in the higher ordinals, as necessary. First, however, let us consider the case of points, lines and faces with assigned dimensions of 0,1 and 2 respectively.
Are there any obvious collections of 0, 1 and 2 cells that allow for such ternary transformations?
Consider the sphere with 2 hemispheres, 2 marked points and 2 half equators. This object is self-ternary
in the sense that a triality takes the space to itself. By the way, when orienting the geometric elements, this kind of arrow in an $n$-category is known as a globule
. It is a simple choice of arrow with well defined sources
in each dimension. It is also an operad polytope.
Now consider the humble cube. Ordinary duality takes the cube to the octahedron
, which is equally symmetric. Triality
should specify three spaces with (8,12,6) then (6,8,12) then (12,6,8) points, lines and faces. It is possible to maintain a marking of the genus zero sphere by adjusting the number of points that lie on a line. Thus the three Euler
characteristics are given by
$8 - 12 + 6 = 2$
$6 - 2 \times 8 + 12 = 2$
$12 - 3 \times 6 + 8 = 2$
which helps solve the cube triality. For instance, note that since the average number of edges to a face in the third case is 4.5, we know that there are probably 4 pentagons and 4 squares.